al detalla theory
DESCRIPTION
Dr. Detalla Math TheoryTRANSCRIPT
![Page 1: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/1.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MISSING TERMS IN CLASSICAL
INEQUALITIES
ALNAR L. DETALLA
Department of MathematicsCollege of Arts and SciencesCentral Mindanao University
8710 Musuan, Bukidnon
May 21, 2010
A. L. Detalla Missing Terms in Classical Inequalities
![Page 2: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/2.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
INRODUCTION
In 1920, G. H. Hardy proved the following:
A. L. Detalla Missing Terms in Classical Inequalities
![Page 3: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/3.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
INRODUCTION
In 1920, G. H. Hardy proved the following:
A. L. Detalla Missing Terms in Classical Inequalities
![Page 4: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/4.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
INRODUCTION
In 1920, G. H. Hardy proved the following:Let 1 < p < ∞ and denote
A. L. Detalla Missing Terms in Classical Inequalities
![Page 5: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/5.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
INRODUCTION
In 1920, G. H. Hardy proved the following:Let 1 < p < ∞ and denote
F (t) :=
∫ t
0f(x)dx, for ǫ < p− 1
∫∞t
f(x)dx, for ǫ > p− 1
A. L. Detalla Missing Terms in Classical Inequalities
![Page 6: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/6.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
INRODUCTION
In 1920, G. H. Hardy proved the following:Let 1 < p < ∞ and denote
F (t) :=
∫ t
0f(x)dx, for ǫ < p− 1
∫∞t
f(x)dx, for ǫ > p− 1
f : is a non-negative measurable function on (0,∞). Then
A. L. Detalla Missing Terms in Classical Inequalities
![Page 7: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/7.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
INRODUCTION
In 1920, G. H. Hardy proved the following:Let 1 < p < ∞ and denote
F (t) :=
∫ t
0f(x)dx, for ǫ < p− 1
∫∞t
f(x)dx, for ǫ > p− 1
f : is a non-negative measurable function on (0,∞). Then
∫ ∞
0
F p(t)tǫ−pdt ≤ C
∫ ∞
0
f p(t)tǫdt (1)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 8: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/8.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
INRODUCTION
In 1920, G. H. Hardy proved the following:Let 1 < p < ∞ and denote
F (t) :=
∫ t
0f(x)dx, for ǫ < p− 1
∫∞t
f(x)dx, for ǫ > p− 1
f : is a non-negative measurable function on (0,∞). Then
∫ ∞
0
F p(t)tǫ−pdt ≤ C
∫ ∞
0
f p(t)tǫdt (1)
where C > 0 independent of f .
A. L. Detalla Missing Terms in Classical Inequalities
![Page 9: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/9.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫ ∞
0
F p(t)tǫ−pdt ≤ C
∫ ∞
0
f p(t)tǫdt (1)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 10: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/10.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫ ∞
0
F p(t)tǫ−pdt ≤ C
∫ ∞
0
f p(t)tǫdt (1)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 11: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/11.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫ ∞
0
F p(t)tǫ−pdt ≤ C
∫ ∞
0
f p(t)tǫdt (1)
C =(
p
|ǫ−p+1|
)p
by: E. Landau (1930)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 12: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/12.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫ ∞
0
F p(t)tǫ−pdt ≤ C
∫ ∞
0
f p(t)tǫdt (1)
C =(
p
|ǫ−p+1|
)p
by: E. Landau (1930)
NOTE: (1) can be written as
A. L. Detalla Missing Terms in Classical Inequalities
![Page 13: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/13.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫ ∞
0
F p(t)tǫ−pdt ≤ C
∫ ∞
0
f p(t)tǫdt (1)
C =(
p
|ǫ−p+1|
)p
by: E. Landau (1930)
NOTE: (1) can be written as
∫ ∞
0
|u(t)|ptǫ−pdt ≤(
p
|ǫ− p+ 1|
)p ∫ ∞
0
|u′(t)|ptǫdt (2)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 14: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/14.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫ ∞
0
F p(t)tǫ−pdt ≤ C
∫ ∞
0
f p(t)tǫdt (1)
C =(
p
|ǫ−p+1|
)p
by: E. Landau (1930)
NOTE: (1) can be written as
∫ ∞
0
|u(t)|ptǫ−pdt ≤(
p
|ǫ− p+ 1|
)p ∫ ∞
0
|u′(t)|ptǫdt (2)
where u′(t) := dudt, u ∈ Cc((0,∞)), p > 1, ǫ 6= p− 1.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 15: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/15.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫ ∞
0
F p(t)tǫ−pdt ≤ C
∫ ∞
0
f p(t)tǫdt (1)
C =(
p
|ǫ−p+1|
)p
by: E. Landau (1930)
NOTE: (1) can be written as
∫ ∞
0
|u(t)|ptǫ−pdt ≤(
p
|ǫ− p+ 1|
)p ∫ ∞
0
|u′(t)|ptǫdt (2)
where u′(t) := dudt, u ∈ Cc((0,∞)), p > 1, ǫ 6= p− 1.
We call (2) the classical Hardy inequality
A. L. Detalla Missing Terms in Classical Inequalities
![Page 16: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/16.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
for ǫ = 0, (2) becomes
A. L. Detalla Missing Terms in Classical Inequalities
![Page 17: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/17.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
for ǫ = 0, (2) becomes
A. L. Detalla Missing Terms in Classical Inequalities
![Page 18: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/18.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
for ǫ = 0, (2) becomes
∫ ∞
0
|u′(t)|pdt ≥(
p− 1
p
)p ∫ ∞
0
|u(t)|p|t|p dt (3)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 19: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/19.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
for ǫ = 0, (2) becomes
∫ ∞
0
|u′(t)|pdt ≥(
p− 1
p
)p ∫ ∞
0
|u(t)|p|t|p dt (3)
from (3) a Hardy inequality of higher dimension can bederived which is a classical Sobolev embedding inequality.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 20: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/20.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
for ǫ = 0, (2) becomes
∫ ∞
0
|u′(t)|pdt ≥(
p− 1
p
)p ∫ ∞
0
|u(t)|p|t|p dt (3)
from (3) a Hardy inequality of higher dimension can bederived which is a classical Sobolev embedding inequality.For n > 2,
A. L. Detalla Missing Terms in Classical Inequalities
![Page 21: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/21.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
for ǫ = 0, (2) becomes
∫ ∞
0
|u′(t)|pdt ≥(
p− 1
p
)p ∫ ∞
0
|u(t)|p|t|p dt (3)
from (3) a Hardy inequality of higher dimension can bederived which is a classical Sobolev embedding inequality.For n > 2,
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx, ∀u ∈ W 1,20 (Ω). (4)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 22: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/22.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
for ǫ = 0, (2) becomes
∫ ∞
0
|u′(t)|pdt ≥(
p− 1
p
)p ∫ ∞
0
|u(t)|p|t|p dt (3)
from (3) a Hardy inequality of higher dimension can bederived which is a classical Sobolev embedding inequality.For n > 2,
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx, ∀u ∈ W 1,20 (Ω). (4)
Lp-version of (4) where 1 ≤ p < n is
A. L. Detalla Missing Terms in Classical Inequalities
![Page 23: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/23.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
for ǫ = 0, (2) becomes
∫ ∞
0
|u′(t)|pdt ≥(
p− 1
p
)p ∫ ∞
0
|u(t)|p|t|p dt (3)
from (3) a Hardy inequality of higher dimension can bederived which is a classical Sobolev embedding inequality.For n > 2,
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx, ∀u ∈ W 1,20 (Ω). (4)
Lp-version of (4) where 1 ≤ p < n is
∫
Ω
|∇u(x)|pdx ≥(
n− p
p
)p ∫
Ω
|u(x)|p|x|p dx, ∀u ∈ W 1,p
0 (Ω).
(5)A. L. Detalla Missing Terms in Classical Inequalities
![Page 24: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/24.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
A higher order generalization of (4) was proven by Rellich
A. L. Detalla Missing Terms in Classical Inequalities
![Page 25: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/25.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
A higher order generalization of (4) was proven by Rellich
A. L. Detalla Missing Terms in Classical Inequalities
![Page 26: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/26.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
A higher order generalization of (4) was proven by RellichFor n > 4,
A. L. Detalla Missing Terms in Classical Inequalities
![Page 27: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/27.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
A higher order generalization of (4) was proven by RellichFor n > 4,
∫
Ω
|∆u(x)|2dx ≥ n2(n− 4)2
16
∫
Ω
|u(x)|2|x|4 dx,
∀u ∈ W 2,20 (Ω).
(6)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 28: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/28.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
A higher order generalization of (4) was proven by RellichFor n > 4,
∫
Ω
|∆u(x)|2dx ≥ n2(n− 4)2
16
∫
Ω
|u(x)|2|x|4 dx,
∀u ∈ W 2,20 (Ω).
(6)
Lp-version of (6) is
A. L. Detalla Missing Terms in Classical Inequalities
![Page 29: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/29.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
A higher order generalization of (4) was proven by RellichFor n > 4,
∫
Ω
|∆u(x)|2dx ≥ n2(n− 4)2
16
∫
Ω
|u(x)|2|x|4 dx,
∀u ∈ W 2,20 (Ω).
(6)
Lp-version of (6) is
∫
Ω
|∆u(x)|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx,
∀u ∈ W 2,p0 (Ω).
(7)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 30: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/30.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
W l,p(Ω):function space on Ω whose generalized derivatives∂γu of order ≤ l satisfies
A. L. Detalla Missing Terms in Classical Inequalities
![Page 31: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/31.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
W l,p(Ω):function space on Ω whose generalized derivatives∂γu of order ≤ l satisfies
A. L. Detalla Missing Terms in Classical Inequalities
![Page 32: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/32.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
W l,p(Ω):function space on Ω whose generalized derivatives∂γu of order ≤ l satisfies
||u||W l,p(Ω) =∑
|γ|≤l
(∫
Ω
|∂γu(x)|pdx)
1p
< ∞
A. L. Detalla Missing Terms in Classical Inequalities
![Page 33: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/33.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
W l,p(Ω):function space on Ω whose generalized derivatives∂γu of order ≤ l satisfies
||u||W l,p(Ω) =∑
|γ|≤l
(∫
Ω
|∂γu(x)|pdx)
1p
< ∞
W l,p0 (Ω) denotes the completion of C∞
0 (Ω) in W l,p(Ω).
A. L. Detalla Missing Terms in Classical Inequalities
![Page 34: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/34.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Hardy-Sobolev Inequalities
A. L. Detalla Missing Terms in Classical Inequalities
![Page 35: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/35.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Hardy-Sobolev Inequalities
A. L. Detalla Missing Terms in Classical Inequalities
![Page 36: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/36.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Hardy-Sobolev Inequalities
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx, ∀u ∈ W 1,20 (Ω) (4)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 37: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/37.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Hardy-Sobolev Inequalities
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx, ∀u ∈ W 1,20 (Ω) (4)
∫
Ω
|∇u(x)|pdx ≥(
n− p
p
)p ∫
Ω
|u(x)|p|x|p dx, ∀u ∈ W 1,p
0 (Ω) (5)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 38: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/38.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Hardy-Sobolev Inequalities
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx, ∀u ∈ W 1,20 (Ω) (4)
∫
Ω
|∇u(x)|pdx ≥(
n− p
p
)p ∫
Ω
|u(x)|p|x|p dx, ∀u ∈ W 1,p
0 (Ω) (5)
Hardy-Sobolev-Rellich Inequalities
A. L. Detalla Missing Terms in Classical Inequalities
![Page 39: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/39.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Hardy-Sobolev Inequalities
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx, ∀u ∈ W 1,20 (Ω) (4)
∫
Ω
|∇u(x)|pdx ≥(
n− p
p
)p ∫
Ω
|u(x)|p|x|p dx, ∀u ∈ W 1,p
0 (Ω) (5)
Hardy-Sobolev-Rellich Inequalities
A. L. Detalla Missing Terms in Classical Inequalities
![Page 40: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/40.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Hardy-Sobolev Inequalities
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx, ∀u ∈ W 1,20 (Ω) (4)
∫
Ω
|∇u(x)|pdx ≥(
n− p
p
)p ∫
Ω
|u(x)|p|x|p dx, ∀u ∈ W 1,p
0 (Ω) (5)
Hardy-Sobolev-Rellich Inequalities
∫
Ω
|∆u(x)|2dx ≥ n2(n− 4)2
16
∫
Ω
|u(x)|2|x|4 dx, ∀u ∈ W 2,2
0 (Ω) (6)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 41: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/41.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Hardy-Sobolev Inequalities
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx, ∀u ∈ W 1,20 (Ω) (4)
∫
Ω
|∇u(x)|pdx ≥(
n− p
p
)p ∫
Ω
|u(x)|p|x|p dx, ∀u ∈ W 1,p
0 (Ω) (5)
Hardy-Sobolev-Rellich Inequalities
∫
Ω
|∆u(x)|2dx ≥ n2(n− 4)2
16
∫
Ω
|u(x)|2|x|4 dx, ∀u ∈ W 2,2
0 (Ω) (6)
∫
Ω
|∆u(x)|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx (7)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 42: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/42.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (4)
NOTATIONS
A. L. Detalla Missing Terms in Classical Inequalities
![Page 43: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/43.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (4)
NOTATIONS
A. L. Detalla Missing Terms in Classical Inequalities
![Page 44: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/44.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (4)
NOTATIONS
For t > 0 and ρ ≥ 2,
A. L. Detalla Missing Terms in Classical Inequalities
![Page 45: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/45.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (4)
NOTATIONS
For t > 0 and ρ ≥ 2,
A1(t) = log Rt, A2(t) = logA1(t), . . . Aρ(t) = logAρ−1(t)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 46: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/46.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (4)
NOTATIONS
For t > 0 and ρ ≥ 2,
A1(t) = log Rt, A2(t) = logA1(t), . . . Aρ(t) = logAρ−1(t)
and
A. L. Detalla Missing Terms in Classical Inequalities
![Page 47: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/47.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (4)
NOTATIONS
For t > 0 and ρ ≥ 2,
A1(t) = log Rt, A2(t) = logA1(t), . . . Aρ(t) = logAρ−1(t)
and
e1 = e, e2 = ee1 , . . . eρ = eeρ−1
A. L. Detalla Missing Terms in Classical Inequalities
![Page 48: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/48.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
THEOREM 1 ( A. Detalla, T. Horiuchi and H. Ando(2005))
A. L. Detalla Missing Terms in Classical Inequalities
![Page 49: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/49.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
THEOREM 1 ( A. Detalla, T. Horiuchi and H. Ando(2005))
A. L. Detalla Missing Terms in Classical Inequalities
![Page 50: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/50.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
THEOREM 1 ( A. Detalla, T. Horiuchi and H. Ando(2005))
Assume γ ≥ 2 and n ≥ 2. If R ≥ supΩ |x|ek then thereexists sharp remainder terms such that
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx+1
4
∫
Ω
u(x)2
|x|2[
A1(|x|)−γ +
(
A1(|x|)A2(|x|))−γ
+ · · ·+(
A1(|x|)A2(|x|) . . . Ak(|x|))−γ]
dx.
(8)
for any u ∈ W 1,20 (Ω).
A. L. Detalla Missing Terms in Classical Inequalities
![Page 51: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/51.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx+1
4
∫
Ω
u(x)2
|x|2[
A1(|x|)−γ +
(
A1(|x|)A2(|x|))−γ
+ · · ·+(
A1(|x|)A2(|x|) . . . Ak(|x|))−γ]
dx (8)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 52: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/52.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx+1
4
∫
Ω
u(x)2
|x|2[
A1(|x|)−γ +
(
A1(|x|)A2(|x|))−γ
+ · · ·+(
A1(|x|)A2(|x|) . . . Ak(|x|))−γ]
dx (8)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 53: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/53.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx+1
4
∫
Ω
u(x)2
|x|2[
A1(|x|)−γ +
(
A1(|x|)A2(|x|))−γ
+ · · ·+(
A1(|x|)A2(|x|) . . . Ak(|x|))−γ]
dx (8)
REMARK
In inequality (8), 14is best constant for all k-missing terms
and γ ≥ 2 is sharp
A. L. Detalla Missing Terms in Classical Inequalities
![Page 54: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/54.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 55: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/55.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
THEOREM 2 ( Adimurthi, N. Chaudhuri and M.Ramaswamy (2001))
Let Ω be a bounded domain in Rn with 0 ∈ Ω and n ≥ 2.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 56: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/56.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
THEOREM 2 ( Adimurthi, N. Chaudhuri and M.Ramaswamy (2001))
Let Ω be a bounded domain in Rn with 0 ∈ Ω and n ≥ 2.
Noncritical Case(1 < p < n)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 57: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/57.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
THEOREM 2 ( Adimurthi, N. Chaudhuri and M.Ramaswamy (2001))
Let Ω be a bounded domain in Rn with 0 ∈ Ω and n ≥ 2.
Noncritical Case(1 < p < n)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 58: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/58.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
THEOREM 2 ( Adimurthi, N. Chaudhuri and M.Ramaswamy (2001))
Let Ω be a bounded domain in Rn with 0 ∈ Ω and n ≥ 2.
Noncritical Case(1 < p < n)
Let R ≥ supΩ
(
|x|e 2p
)
. Then there exist K > 0 depending on
n, p, and R such that for any u ∈ W 1,p0 (Ω) and γ ≥ 2
A. L. Detalla Missing Terms in Classical Inequalities
![Page 59: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/59.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
THEOREM 2 ( Adimurthi, N. Chaudhuri and M.Ramaswamy (2001))
Let Ω be a bounded domain in Rn with 0 ∈ Ω and n ≥ 2.
Noncritical Case(1 < p < n)
Let R ≥ supΩ
(
|x|e 2p
)
. Then there exist K > 0 depending on
n, p, and R such that for any u ∈ W 1,p0 (Ω) and γ ≥ 2
∫
Ω
|∇u(x)|pdx ≥(
n− p
p
)p ∫
Ω
|u(x)|p|x|p dx+
K
∫
Ω
|u(x)|p|x|p
(
logR
|x|
)−γ
dx(9)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 60: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/60.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 61: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/61.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
THEOREM 2 ( Adimurthi, N. Chaudhuri and M.Ramaswamy (2001))
Let Ω be a bounded domain in Rn with 0 ∈ Ω and n ≥ 2.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 62: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/62.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
THEOREM 2 ( Adimurthi, N. Chaudhuri and M.Ramaswamy (2001))
Let Ω be a bounded domain in Rn with 0 ∈ Ω and n ≥ 2.
Critical Case (p = n)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 63: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/63.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
THEOREM 2 ( Adimurthi, N. Chaudhuri and M.Ramaswamy (2001))
Let Ω be a bounded domain in Rn with 0 ∈ Ω and n ≥ 2.
Critical Case (p = n)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 64: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/64.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
THEOREM 2 ( Adimurthi, N. Chaudhuri and M.Ramaswamy (2001))
Let Ω be a bounded domain in Rn with 0 ∈ Ω and n ≥ 2.
Critical Case (p = n)
Let R ≥ supΩ
(
|x|e 2n
)
. Then for any u ∈ W 1,n0 (Ω)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 65: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/65.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
THEOREM 2 ( Adimurthi, N. Chaudhuri and M.Ramaswamy (2001))
Let Ω be a bounded domain in Rn with 0 ∈ Ω and n ≥ 2.
Critical Case (p = n)
Let R ≥ supΩ
(
|x|e 2n
)
. Then for any u ∈ W 1,n0 (Ω)
∫
Ω
|∇u(x)|ndx ≥(
n− 1
n
)n ∫
Ω
|u(x)|n|x|n
(
logR
|x|
)−n
dx
(10)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 66: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/66.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Corollary
Let 1 < p < n, and let
Fp =
f : Ω → R+|f ∈ L∞
loc(Ω \ 0) with
lim sup|x|→0
|x|pf(x)(
log1
|x|
)2
< ∞
A. L. Detalla Missing Terms in Classical Inequalities
![Page 67: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/67.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Corollary
Let 1 < p < n, and let
Fp =
f : Ω → R+|f ∈ L∞
loc(Ω \ 0) with
lim sup|x|→0
|x|pf(x)(
log1
|x|
)2
< ∞
A. L. Detalla Missing Terms in Classical Inequalities
![Page 68: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/68.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Corollary
Let 1 < p < n, and let
Fp =
f : Ω → R+|f ∈ L∞
loc(Ω \ 0) with
lim sup|x|→0
|x|pf(x)(
log1
|x|
)2
< ∞
If f ∈ Fp, ∃ λ(f) > 0 such that for u ∈ W 1,p0 (Ω)
∫
Ω
|∇u|pdx ≥(
n− p
p
)p ∫
Ω
|u(x)|p|x|2p dx+λ(f)
∫
Ω
|u(x)|pf(x)dx
A. L. Detalla Missing Terms in Classical Inequalities
![Page 69: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/69.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Corollary
Let 1 < p < n, and let
Fp =
f : Ω → R+|f ∈ L∞
loc(Ω \ 0) with
lim sup|x|→0
|x|pf(x)(
log1
|x|
)2
< ∞
If f ∈ Fp, ∃ λ(f) > 0 such that for u ∈ W 1,p0 (Ω)
∫
Ω
|∇u|pdx ≥(
n− p
p
)p ∫
Ω
|u(x)|p|x|2p dx+λ(f)
∫
Ω
|u(x)|pf(x)dx
If f /∈ Fp and if |x|pf(x)(
log 1|x|
)2
tends to ∞ as |x| → 0,
then no inequality of the above type can hold.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 70: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/70.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
A. L. Detalla Missing Terms in Classical Inequalities
![Page 71: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/71.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
APPLICATION
The results will be used to analyze the behaviour of thefirst eigenvalue of the weighted eigenvalue problem for theoperator
A. L. Detalla Missing Terms in Classical Inequalities
![Page 72: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/72.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
APPLICATION
The results will be used to analyze the behaviour of thefirst eigenvalue of the weighted eigenvalue problem for theoperator
A. L. Detalla Missing Terms in Classical Inequalities
![Page 73: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/73.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
APPLICATION
The results will be used to analyze the behaviour of thefirst eigenvalue of the weighted eigenvalue problem for theoperator
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 74: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/74.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
APPLICATION
The results will be used to analyze the behaviour of thefirst eigenvalue of the weighted eigenvalue problem for theoperator
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
as µ →(
n−p
p
)p
.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 75: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/75.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 76: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/76.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
L∇µis related to the variational problem
A. L. Detalla Missing Terms in Classical Inequalities
![Page 77: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/77.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
L∇µis related to the variational problem
A. L. Detalla Missing Terms in Classical Inequalities
![Page 78: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/78.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
L∇µis related to the variational problem
infu∈K
∫
Ω
(
|∇u|p − µ|u|p|x|p
)
dx
A. L. Detalla Missing Terms in Classical Inequalities
![Page 79: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/79.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
L∇µis related to the variational problem
infu∈K
∫
Ω
(
|∇u|p − µ|u|p|x|p
)
dx
where K is given by
A. L. Detalla Missing Terms in Classical Inequalities
![Page 80: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/80.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
L∇µis related to the variational problem
infu∈K
∫
Ω
(
|∇u|p − µ|u|p|x|p
)
dx
where K is given by
K =
u ∈ W 1,p0 (Ω) :
∫
Ω
|u(x)|pf(x)dx = 1
,
f : is a weight function.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 81: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/81.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 82: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/82.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
Consider the weighted eigenvalue problem
L∇µu = λ|u|p−2uf in Ω
u = 0 on ∂Ω
A. L. Detalla Missing Terms in Classical Inequalities
![Page 83: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/83.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
Consider the weighted eigenvalue problem
L∇µu = λ|u|p−2uf in Ω
u = 0 on ∂Ω
A. L. Detalla Missing Terms in Classical Inequalities
![Page 84: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/84.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
Consider the weighted eigenvalue problem
L∇µu = λ|u|p−2uf in Ω
u = 0 on ∂Ω
µ →(
n−p
p
)p
A. L. Detalla Missing Terms in Classical Inequalities
![Page 85: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/85.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
Consider the weighted eigenvalue problem
L∇µu = λ|u|p−2uf in Ω
u = 0 on ∂Ω
µ →(
n−p
p
)p
The expression L∇µu = λ|u|p−2uf is related to the
improved Hardy-Sobolev inequality.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 86: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/86.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (6)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 87: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/87.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (6)
∫
Ω
|∆u(x)|2dx ≥ n2(n− 4)2
16
∫
Ω
|u(x)|2|x|4 dx, ∀u ∈ W 2,2
0 (Ω) (6)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 88: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/88.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (6)
∫
Ω
|∆u(x)|2dx ≥ n2(n− 4)2
16
∫
Ω
|u(x)|2|x|4 dx, ∀u ∈ W 2,2
0 (Ω) (6)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 89: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/89.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (6)
∫
Ω
|∆u(x)|2dx ≥ n2(n− 4)2
16
∫
Ω
|u(x)|2|x|4 dx, ∀u ∈ W 2,2
0 (Ω) (6)
ON GOING RESEARCH
A. L. Detalla Missing Terms in Classical Inequalities
![Page 90: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/90.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u(x)|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx (7)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 91: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/91.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u(x)|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx (7)
Here the best constant(
n−2pp
)p (np−n
p
)p
is given by the
infimum of
A. L. Detalla Missing Terms in Classical Inequalities
![Page 92: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/92.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u(x)|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx (7)
Here the best constant(
n−2pp
)p (np−n
p
)p
is given by the
infimum of I(u) =∫Ω |∆u|pdx
∫Ω
|u(x)|p
|x|2pdx.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 93: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/93.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u(x)|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx (7)
Here the best constant(
n−2pp
)p (np−n
p
)p
is given by the
infimum of I(u) =∫Ω |∆u|pdx
∫Ω
|u(x)|p
|x|2pdx.
No extremal function in W 2,p0 (Ω) which attains the infimum
of I(u).
A. L. Detalla Missing Terms in Classical Inequalities
![Page 94: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/94.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u(x)|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx (7)
Here the best constant(
n−2pp
)p (np−n
p
)p
is given by the
infimum of I(u) =∫Ω |∆u|pdx
∫Ω
|u(x)|p
|x|2pdx.
No extremal function in W 2,p0 (Ω) which attains the infimum
of I(u).
A. L. Detalla Missing Terms in Classical Inequalities
![Page 95: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/95.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u(x)|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx (7)
Here the best constant(
n−2pp
)p (np−n
p
)p
is given by the
infimum of I(u) =∫Ω |∆u|pdx
∫Ω
|u(x)|p
|x|2pdx.
No extremal function in W 2,p0 (Ω) which attains the infimum
of I(u).
The candidates of the extremals are singular at the originwhich are not in W 2,p
0 (Ω).
A. L. Detalla Missing Terms in Classical Inequalities
![Page 96: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/96.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u(x)|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx (7)
Here the best constant(
n−2pp
)p (np−n
p
)p
is given by the
infimum of I(u) =∫Ω |∆u|pdx
∫Ω
|u(x)|p
|x|2pdx.
No extremal function in W 2,p0 (Ω) which attains the infimum
of I(u).
The candidates of the extremals are singular at the originwhich are not in W 2,p
0 (Ω).
A. L. Detalla Missing Terms in Classical Inequalities
![Page 97: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/97.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u(x)|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx (7)
Here the best constant(
n−2pp
)p (np−n
p
)p
is given by the
infimum of I(u) =∫Ω |∆u|pdx
∫Ω
|u(x)|p
|x|2pdx.
No extremal function in W 2,p0 (Ω) which attains the infimum
of I(u).
The candidates of the extremals are singular at the originwhich are not in W 2,p
0 (Ω).
Therefore it is natural to consider that there exists amissing terms in the right hand side of (7).
A. L. Detalla Missing Terms in Classical Inequalities
![Page 98: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/98.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OBJECTIVE
A. L. Detalla Missing Terms in Classical Inequalities
![Page 99: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/99.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OBJECTIVE
To achieve an optimal improvement of inequality (7) byadding sharp terms in the right hand side involving a
singular weight of type(
log 1|x|
)−2
A. L. Detalla Missing Terms in Classical Inequalities
![Page 100: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/100.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OBJECTIVE
To achieve an optimal improvement of inequality (7) byadding sharp terms in the right hand side involving a
singular weight of type(
log 1|x|
)−2
A. L. Detalla Missing Terms in Classical Inequalities
![Page 101: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/101.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OBJECTIVE
To achieve an optimal improvement of inequality (7) byadding sharp terms in the right hand side involving a
singular weight of type(
log 1|x|
)−2
Optimal in the sense that the improved inequality holds for
this weight function(
log 1|x|
)−2
but fails for any weight
more singular than this.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 102: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/102.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
A. L. Detalla Missing Terms in Classical Inequalities
![Page 103: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/103.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
THEOREM 3 ( A. Detalla, T. Horiuchi and H. Ando(2004))
Let n ≥ 3, 0 ∈ Ω and Ω is a bounded domain in Rn.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 104: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/104.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
THEOREM 3 ( A. Detalla, T. Horiuchi and H. Ando(2004))
Let n ≥ 3, 0 ∈ Ω and Ω is a bounded domain in Rn.
Noncritical Case(1 < p < n2)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 105: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/105.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
THEOREM 3 ( A. Detalla, T. Horiuchi and H. Ando(2004))
Let n ≥ 3, 0 ∈ Ω and Ω is a bounded domain in Rn.
Noncritical Case(1 < p < n2)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 106: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/106.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
THEOREM 3 ( A. Detalla, T. Horiuchi and H. Ando(2004))
Let n ≥ 3, 0 ∈ Ω and Ω is a bounded domain in Rn.
Noncritical Case(1 < p < n2)
There exists C = C(n,R) > 0 such that if R > supΩ |x| andfor any u ∈ W 2,p
0 (Ω),
A. L. Detalla Missing Terms in Classical Inequalities
![Page 107: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/107.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
THEOREM 3 ( A. Detalla, T. Horiuchi and H. Ando(2004))
Let n ≥ 3, 0 ∈ Ω and Ω is a bounded domain in Rn.
Noncritical Case(1 < p < n2)
There exists C = C(n,R) > 0 such that if R > supΩ |x| andfor any u ∈ W 2,p
0 (Ω),
∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx(11)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 108: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/108.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
A. L. Detalla Missing Terms in Classical Inequalities
![Page 109: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/109.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
THEOREM 3 ( A. Detalla, T. Horiuchi and H. Ando(2004))
Let n ≥ 3, 0 ∈ Ω and Ω is a bounded domain in Rn.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 110: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/110.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
THEOREM 3 ( A. Detalla, T. Horiuchi and H. Ando(2004))
Let n ≥ 3, 0 ∈ Ω and Ω is a bounded domain in Rn.
Critical Case(p = n2)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 111: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/111.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
THEOREM 3 ( A. Detalla, T. Horiuchi and H. Ando(2004))
Let n ≥ 3, 0 ∈ Ω and Ω is a bounded domain in Rn.
Critical Case(p = n2)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 112: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/112.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
THEOREM 3 ( A. Detalla, T. Horiuchi and H. Ando(2004))
Let n ≥ 3, 0 ∈ Ω and Ω is a bounded domain in Rn.
Critical Case(p = n2)
There exists K∗ = K∗(n) > 0 and C∗ = C∗(n) > 0 such
that if R > K∗ supΩ |x| and for any u ∈ W2,n
20 (Ω),
A. L. Detalla Missing Terms in Classical Inequalities
![Page 113: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/113.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
THEOREM 3 ( A. Detalla, T. Horiuchi and H. Ando(2004))
Let n ≥ 3, 0 ∈ Ω and Ω is a bounded domain in Rn.
Critical Case(p = n2)
There exists K∗ = K∗(n) > 0 and C∗ = C∗(n) > 0 such
that if R > K∗ supΩ |x| and for any u ∈ W2,n
20 (Ω),
∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx(12)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 114: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/114.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 115: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/115.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
REMARK 1
In inequality (11) the exponent 2 of the weight function isoptimal.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 116: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/116.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 117: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/117.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
REMARK 2
In inequality (12) the exponent n2of the weight function is
optimal and the constant(
n−2√n
)n
is sharp.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 118: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/118.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 119: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/119.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
REMARK 3
C and C∗ depends on R in a weak sense since(
log R|x|
)−2
and(
log R|x|
)−n2−1
tends to zero as R → ∞.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 120: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/120.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Noncritical Case(1 < p < n2)
There exists C = C(n,R) > 0 such that if R > supΩ |x| andfor any u ∈ W 2,p
0 (Ω),
∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 121: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/121.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Noncritical Case(1 < p < n2)
There exists C = C(n,R) > 0 such that if R > supΩ |x| andfor any u ∈ W 2,p
0 (Ω),
∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 122: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/122.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Noncritical Case(1 < p < n2)
There exists C = C(n,R) > 0 such that if R > supΩ |x| andfor any u ∈ W 2,p
0 (Ω),
∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
REMARK 4
At first we prove inequality (11) assuming that
R > e1p supΩ |x| because of technical reason. Namely, the
weight function g(r) = r−2p(
log Rr
)−2should be monotone
decreasing. Then we can extend (11) for any R > supΩ |x|
A. L. Detalla Missing Terms in Classical Inequalities
![Page 123: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/123.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Critical Case(p = n2): There exists K∗ = K∗(n) > 0 and
C∗ = C∗(n) > 0 such that if R > K∗ supΩ |x|,∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 124: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/124.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Critical Case(p = n2): There exists K∗ = K∗(n) > 0 and
C∗ = C∗(n) > 0 such that if R > K∗ supΩ |x|,∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 125: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/125.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Critical Case(p = n2): There exists K∗ = K∗(n) > 0 and
C∗ = C∗(n) > 0 such that if R > K∗ supΩ |x|,∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
REMARK 4
In the proof of the critical case (12), we used decreasing
rearrangement argument, hence g∗(r) = r−n(
log Rr
)−n2−1
should be monotone decreasing and R ≥ re12+ 1
n .
A. L. Detalla Missing Terms in Classical Inequalities
![Page 126: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/126.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Critical Case(p = n2): There exists K∗ = K∗(n) > 0 and
C∗ = C∗(n) > 0 such that if R > K∗ supΩ |x|,∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
REMARK 4
In the proof of the critical case (12), we used decreasing
rearrangement argument, hence g∗(r) = r−n(
log Rr
)−n2−1
should be monotone decreasing and R ≥ re12+ 1
n .
A. L. Detalla Missing Terms in Classical Inequalities
![Page 127: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/127.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Critical Case(p = n2): There exists K∗ = K∗(n) > 0 and
C∗ = C∗(n) > 0 such that if R > K∗ supΩ |x|,∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
REMARK 4
In the proof of the critical case (12), we used decreasing
rearrangement argument, hence g∗(r) = r−n(
log Rr
)−n2−1
should be monotone decreasing and R ≥ re12+ 1
n .Moreoverwe need the condition to absorb the error terms in the righthand side of (12) with C∗ > 0, hence K∗ ≥ e
12+ 1
n
A. L. Detalla Missing Terms in Classical Inequalities
![Page 128: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/128.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Corollary
Let 1 < p < n2, and let
Fp =
f : Ω → R+|f ∈ L∞
loc(Ω \ 0) with
lim sup|x|→0
|x|2pf(x)(
log1
|x|
)2
< ∞
A. L. Detalla Missing Terms in Classical Inequalities
![Page 129: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/129.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Corollary
Let 1 < p < n2, and let
Fp =
f : Ω → R+|f ∈ L∞
loc(Ω \ 0) with
lim sup|x|→0
|x|2pf(x)(
log1
|x|
)2
< ∞
A. L. Detalla Missing Terms in Classical Inequalities
![Page 130: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/130.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Corollary
Let 1 < p < n2, and let
Fp =
f : Ω → R+|f ∈ L∞
loc(Ω \ 0) with
lim sup|x|→0
|x|2pf(x)(
log1
|x|
)2
< ∞
If f ∈ Fp, ∃ λ(f) > 0 such that for u ∈ W 2,p0 (Ω)
∫
Ω
|∆u|pdx ≥ Λn,p
∫
Ω
|u(x)|p|x|2p dx+ λ(f)
∫
Ω
|u(x)|pf(x)dx(13)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 131: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/131.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Corollary
Let 1 < p < n2, and let
Fp =
f : Ω → R+|f ∈ L∞
loc(Ω \ 0) with
lim sup|x|→0
|x|2pf(x)(
log1
|x|
)2
< ∞
If f ∈ Fp, ∃ λ(f) > 0 such that for u ∈ W 2,p0 (Ω)
∫
Ω
|∆u|pdx ≥ Λn,p
∫
Ω
|u(x)|p|x|2p dx+ λ(f)
∫
Ω
|u(x)|pf(x)dx(13)
If f /∈ Fp and if |x|2pf(x)(
log 1|x|
)2
tends to ∞ as |x| → 0,
then no inequality of type (13) can hold.A. L. Detalla Missing Terms in Classical Inequalities
![Page 132: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/132.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
A. L. Detalla Missing Terms in Classical Inequalities
![Page 133: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/133.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
APPLICATION
We use the results to determine exactly when the firsteigenvalue of the weighted eigenvalue problem for theoperator
A. L. Detalla Missing Terms in Classical Inequalities
![Page 134: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/134.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
APPLICATION
We use the results to determine exactly when the firsteigenvalue of the weighted eigenvalue problem for theoperator
A. L. Detalla Missing Terms in Classical Inequalities
![Page 135: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/135.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
APPLICATION
We use the results to determine exactly when the firsteigenvalue of the weighted eigenvalue problem for theoperator
L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
A. L. Detalla Missing Terms in Classical Inequalities
![Page 136: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/136.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
APPLICATION
We use the results to determine exactly when the firsteigenvalue of the weighted eigenvalue problem for theoperator
L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
will tend to 0 as µ →(
n−2pp
)p (np−n
p
)p
.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 137: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/137.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
A. L. Detalla Missing Terms in Classical Inequalities
![Page 138: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/138.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
L∆µis related to the variational problem
A. L. Detalla Missing Terms in Classical Inequalities
![Page 139: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/139.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
L∆µis related to the variational problem
A. L. Detalla Missing Terms in Classical Inequalities
![Page 140: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/140.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
L∆µis related to the variational problem
infu∈K
∫
Ω
(
|∆u|p − µ|u|p|x|2p
)
dx
A. L. Detalla Missing Terms in Classical Inequalities
![Page 141: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/141.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
L∆µis related to the variational problem
infu∈K
∫
Ω
(
|∆u|p − µ|u|p|x|2p
)
dx
where K is given by
A. L. Detalla Missing Terms in Classical Inequalities
![Page 142: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/142.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
L∆µis related to the variational problem
infu∈K
∫
Ω
(
|∆u|p − µ|u|p|x|2p
)
dx
where K is given by
K =
u ∈ W 2,p(Ω) ∩W 1,p0 (Ω) :
∫
Ω
|u(x)|pf(x)dx = 1
,
f : is a weight function which will be specified later.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 143: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/143.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
A. L. Detalla Missing Terms in Classical Inequalities
![Page 144: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/144.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
Consider the weighted eigenvalue problem
L∆µu = λ|u|p−2uf in Ω
u = ∆u = 0 on ∂Ω
A. L. Detalla Missing Terms in Classical Inequalities
![Page 145: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/145.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
Consider the weighted eigenvalue problem
L∆µu = λ|u|p−2uf in Ω
u = ∆u = 0 on ∂Ω
A. L. Detalla Missing Terms in Classical Inequalities
![Page 146: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/146.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
Consider the weighted eigenvalue problem
L∆µu = λ|u|p−2uf in Ω
u = ∆u = 0 on ∂Ω
µ →(
n−2pp
)p (np−n
p
)p
A. L. Detalla Missing Terms in Classical Inequalities
![Page 147: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/147.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
Consider the weighted eigenvalue problem
L∆µu = λ|u|p−2uf in Ω
u = ∆u = 0 on ∂Ω
µ →(
n−2pp
)p (np−n
p
)p
A. L. Detalla Missing Terms in Classical Inequalities
![Page 148: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/148.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
Consider the weighted eigenvalue problem
L∆µu = λ|u|p−2uf in Ω
u = ∆u = 0 on ∂Ω
µ →(
n−2pp
)p (np−n
p
)p
The expression L∆µu = λ|u|p−2uf is related to the
improved Rellich inequality.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 149: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/149.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
REARRANGEMENT
For a domain Ω we define a ball Ω∗ such that |Ω∗| = |Ω|with center at the origin.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 150: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/150.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
REARRANGEMENT
For a domain Ω we define a ball Ω∗ such that |Ω∗| = |Ω|with center at the origin.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 151: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/151.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
REARRANGEMENT
For a domain Ω we define a ball Ω∗ such that |Ω∗| = |Ω|with center at the origin.
u∗(x) : spherically symmetric decreasing rearrangement offunction u(x)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 152: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/152.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
REARRANGEMENT
For a domain Ω we define a ball Ω∗ such that |Ω∗| = |Ω|with center at the origin.
u∗(x) : spherically symmetric decreasing rearrangement offunction u(x)
u∗(x) = inf
t ≥ 0 : µ(t) < B|x|n
in Ω∗
µ(t) = |x ∈ Ω : |u(x)| > t|
B: Volume of unit ball
A. L. Detalla Missing Terms in Classical Inequalities
![Page 153: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/153.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
LEMMA (Talente)
Let f ∈ C∞0 (Ω). If u is a solution of
−∆u = f in Ω
u = 0 on ∂Ω
and v is a solution of
−∆v = f ∗ in Ω∗
v = 0 on ∂Ω∗
⇒v≥ u∗.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 154: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/154.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
From this lemma we can show that∫
Ω
|∆u|pdx =
∫
Ω∗
|∆v|pdx
∫
Ω
|u|p|x|2pdx ≤
∫
Ω∗
|v|p|x|2pdx
A. L. Detalla Missing Terms in Classical Inequalities
![Page 155: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/155.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
From this lemma we can show that∫
Ω
|∆u|pdx =
∫
Ω∗
|∆v|pdx
∫
Ω
|u|p|x|2pdx ≤
∫
Ω∗
|v|p|x|2pdx
A. L. Detalla Missing Terms in Classical Inequalities
![Page 156: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/156.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
From this lemma we can show that∫
Ω
|∆u|pdx =
∫
Ω∗
|∆v|pdx
∫
Ω
|u|p|x|2pdx ≤
∫
Ω∗
|v|p|x|2pdx
Hence we have∫
Ω|∆u|pdx
∫
Ω|u|p|x|2pdx
≥∫
Ω∗ |∆v|pdx∫
Ω∗
|v|p|x|2pdx
A. L. Detalla Missing Terms in Classical Inequalities
![Page 157: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/157.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
From this lemma we can show that∫
Ω
|∆u|pdx =
∫
Ω∗
|∆v|pdx
∫
Ω
|u|p|x|2pdx ≤
∫
Ω∗
|v|p|x|2pdx
Hence we have∫
Ω|∆u|pdx
∫
Ω|u|p|x|2pdx
≥∫
Ω∗ |∆v|pdx∫
Ω∗
|v|p|x|2pdx
From this we can assume that u is radial and Ω is a ball.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 158: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/158.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
ASSUMPTIONS
Ω : unit ball B1
u : radially nonincreasing,
−∆u > 0
u > 0
A. L. Detalla Missing Terms in Classical Inequalities
![Page 159: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/159.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OUTLINE OF THE PROOF
1. Noncritical Case(1 < p < n2)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 160: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/160.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OUTLINE OF THE PROOF
1. Noncritical Case(1 < p < n2)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 161: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/161.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OUTLINE OF THE PROOF
1. Noncritical Case(1 < p < n2)
1a) We prove inequality (11)
∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 162: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/162.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OUTLINE OF THE PROOF
1. Noncritical Case(1 < p < n2)
1a) We prove inequality (11)
∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
1b) We show the sharpness of(
n−2pp
)p (np−n
p
)p
A. L. Detalla Missing Terms in Classical Inequalities
![Page 163: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/163.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OUTLINE OF THE PROOF
1. Noncritical Case(1 < p < n2)
1a) We prove inequality (11)
∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
1b) We show the sharpness of(
n−2pp
)p (np−n
p
)p
1c) We show the optimality of the exponent 2.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 164: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/164.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OUTLINE OF THE PROOF
2. Critical Case(p = n2)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 165: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/165.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OUTLINE OF THE PROOF
2. Critical Case(p = n2)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 166: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/166.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OUTLINE OF THE PROOF
2. Critical Case(p = n2)
2a) We prove inequality (12)
∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 167: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/167.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OUTLINE OF THE PROOF
2. Critical Case(p = n2)
2a) We prove inequality (12)
∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
2b) We show the sharpness of(
n−2√n
)n
A. L. Detalla Missing Terms in Classical Inequalities
![Page 168: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/168.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OUTLINE OF THE PROOF
2. Critical Case(p = n2)
2a) We prove inequality (12)
∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
2b) We show the sharpness of(
n−2√n
)n
2c) We show the optimality of the exponent n2.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 169: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/169.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
LEMMA 2
For any R > 1, q ≤ 0, ν ∈ (0, 1) satisfying 2ν − 1 + q = 0
∫ 1
0
|h′(r)|2(
logR
r
)q
rdr ≥ ν2
∫ 1
0
|h(r)|2(
logR
r
)q−2dr
r(14)
holds for any h ∈ C([0, 1]) ∩ C1(0, 1), with h(0) = h(1) = 0.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 170: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/170.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1a) PROOF OF INEQUALITY (11)
For u ∈ C20(B1), u > 0, radially nonincreasing , we define
v(r) = u(r)rnp−2 r = |x|.
then apply this to
A. L. Detalla Missing Terms in Classical Inequalities
![Page 171: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/171.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1a) PROOF OF INEQUALITY (11)
For u ∈ C20(B1), u > 0, radially nonincreasing , we define
v(r) = u(r)rnp−2 r = |x|.
then apply this to
A. L. Detalla Missing Terms in Classical Inequalities
![Page 172: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/172.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1a) PROOF OF INEQUALITY (11)
For u ∈ C20(B1), u > 0, radially nonincreasing , we define
v(r) = u(r)rnp−2 r = |x|.
then apply this to
∫
B1
|∆u|pdx−(
n− 2p
p
)p(np− n
p
)p ∫
B1
|u(x)|p|x|2p dx
we get
A. L. Detalla Missing Terms in Classical Inequalities
![Page 173: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/173.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1a) PROOF OF INEQUALITY (11)
∫
B1
|∆u|pdx−(
n− 2p
p
)p(np− n
p
)p ∫
B1
|u(x)|p|x|2p dx
≥ 4ωn
α
(
n− 2p
p
)p(np− n
p
)p(p− 1
p
)∫ 1
0
|(
vn2 (r)
)′ |2rdr
By Lemma 2(
v = 12, q = 0
)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 174: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/174.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1a) PROOF OF INEQUALITY (11)
∫
B1
|∆u|pdx−(
n− 2p
p
)p(np− n
p
)p ∫
B1
|u(x)|p|x|2p dx
≥ 4ωn
α
(
n− 2p
p
)p(np− n
p
)p(p− 1
p
)∫ 1
0
|(
vn2 (r)
)′ |2rdr
By Lemma 2(
v = 12, q = 0
)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 175: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/175.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1a) PROOF OF INEQUALITY (11)
∫
B1
|∆u|pdx−(
n− 2p
p
)p(np− n
p
)p ∫
B1
|u(x)|p|x|2p dx
≥ 4ωn
α
(
n− 2p
p
)p(np− n
p
)p(p− 1
p
)∫ 1
0
|(
vn2 (r)
)′ |2rdr
By Lemma 2(
v = 12, q = 0
)
LEMMA 2
∫ 1
0
|h′(r)|2(
logR
r
)q
rdr ≥ ν2
∫ 1
0
|h(r)|2(
logR
r
)q−2dr
r
A. L. Detalla Missing Terms in Classical Inequalities
![Page 176: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/176.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1a) PROOF OF INEQUALITY (11)
we get
∫
B1
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
B1
|u(x)|p|x|2p dx
+ C
∫
B1
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 177: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/177.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1a) PROOF OF INEQUALITY (11)
we get
∫
B1
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
B1
|u(x)|p|x|2p dx
+ C
∫
B1
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 178: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/178.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1a) PROOF OF INEQUALITY (11)
we get
∫
B1
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
B1
|u(x)|p|x|2p dx
+ C
∫
B1
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
where C =(
n−2pp
)p (np−n
p
)p (p−1p
)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 179: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/179.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
For ǫ > 0 sufficiently small, let us define
uǫ =
0, 0 < r < ǫ2
log r
ǫ2
rn−pp log 1
ǫ
, ǫ2 < r < ǫ
log 1r
rn−pp log 1
ǫ
, ǫ < r < 1
A. L. Detalla Missing Terms in Classical Inequalities
![Page 180: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/180.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Let wǫ =∫ 1
ruǫ(ρ)dρ. Direct calculation gives
A. L. Detalla Missing Terms in Classical Inequalities
![Page 181: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/181.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Let wǫ =∫ 1
ruǫ(ρ)dρ. Direct calculation gives
A. L. Detalla Missing Terms in Classical Inequalities
![Page 182: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/182.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Let wǫ =∫ 1
ruǫ(ρ)dρ. Direct calculation gives
wǫ =
(
p
n−2p
)21−2ǫ
2−np +ǫ
2(2−np )
log 1ǫ
, 0 < r < ǫ2
p
n−2p
r2−n
p log r
ǫ2
log 1ǫ
+(
p
n−2p
)21−2ǫ
2−np +r
2−np
log 1ǫ
, ǫ2 < r < ǫ
p
n−2p
r2−n
p log 1r
log 1ǫ
+(
p
n−2p
)21−r
2−np
log 1ǫ
, ǫ < r < 1
A. L. Detalla Missing Terms in Classical Inequalities
![Page 183: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/183.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∆wǫ =
0, 0 < r < ǫ2
1p
r−np
(
−p+ n(1− p) log rǫ2
)
(
log 1ǫ
)−1, ǫ2 < r < ǫ
1p
r−np
(
p+ n(1− p) log 1r
)
(
log 1ǫ
)−1, ǫ < r < 1
A. L. Detalla Missing Terms in Classical Inequalities
![Page 184: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/184.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Direct calculation gives
A. L. Detalla Missing Terms in Classical Inequalities
![Page 185: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/185.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Direct calculation gives
A. L. Detalla Missing Terms in Classical Inequalities
![Page 186: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/186.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Direct calculation gives
∫
B1
|∆wǫ|pdx =2
p+ 1
(
n(p− 1)
p
)p
ωn log1
ǫ+O
(
(
log1
ǫ
)−1)
∫
B1
|wǫ|p|x|2p dx ≥ 2
p+ 1
(
p
n− 2p
)p
ωn log1
ǫ+O
(
(
log1
ǫ
)−1)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 187: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/187.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
B1
|∆wǫ|pdx−(
n− 2p
p
)p(np− n
p
)p ∫
B
|wǫ|p|x|2p dx
≤ O
(
(
log1
ǫ
)−1)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 188: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/188.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
B1
|∆wǫ|pdx−(
n− 2p
p
)p(np− n
p
)p ∫
B
|wǫ|p|x|2p dx
≤ O
(
(
log1
ǫ
)−1)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 189: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/189.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
B1
|∆wǫ|pdx−(
n− 2p
p
)p(np− n
p
)p ∫
B
|wǫ|p|x|2p dx
≤ O
(
(
log1
ǫ
)−1)
and
A. L. Detalla Missing Terms in Classical Inequalities
![Page 190: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/190.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
B1
|∆wǫ|pdx−(
n− 2p
p
)p(np− n
p
)p ∫
B
|wǫ|p|x|2p dx
≤ O
(
(
log1
ǫ
)−1)
and∫
B1
|wǫ|p|x|2p
(
logR
|x|
)−γ
≥ O
(
(
log1
ǫ
)1−γ)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 191: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/191.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1b. SHARPNESS OF(
n−2pp
)p (np−n
p
)p
limǫ→0
I(wǫ) = limǫ→0
∫
B1|∆wǫ|pdx
∫
B1
|wǫ|p|x|2p dx
≤(
n− 2p
p
)p(np− n
p
)p
A. L. Detalla Missing Terms in Classical Inequalities
![Page 192: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/192.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1b. SHARPNESS OF(
n−2pp
)p (np−n
p
)p
limǫ→0
I(wǫ) = limǫ→0
∫
B1|∆wǫ|pdx
∫
B1
|wǫ|p|x|2p dx
≤(
n− 2p
p
)p(np− n
p
)p
A. L. Detalla Missing Terms in Classical Inequalities
![Page 193: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/193.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1b. SHARPNESS OF(
n−2pp
)p (np−n
p
)p
limǫ→0
I(wǫ) = limǫ→0
∫
B1|∆wǫ|pdx
∫
B1
|wǫ|p|x|2p dx
≤(
n− 2p
p
)p(np− n
p
)p
But by Rellich inequality
limǫ→0
I(wǫ) ≥(
n− 2p
p
)p(np− n
p
)p
A. L. Detalla Missing Terms in Classical Inequalities
![Page 194: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/194.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1b. SHARPNESS OF(
n−2pp
)p (np−n
p
)p
limǫ→0
I(wǫ) = limǫ→0
∫
B1|∆wǫ|pdx
∫
B1
|wǫ|p|x|2p dx
≤(
n− 2p
p
)p(np− n
p
)p
But by Rellich inequality
limǫ→0
I(wǫ) ≥(
n− 2p
p
)p(np− n
p
)p
hence
limǫ→0
I(wǫ) =
(
n− 2p
p
)p(np− n
p
)p
A. L. Detalla Missing Terms in Classical Inequalities
![Page 195: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/195.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1c. OPTIMALITY OF THE EXPONENT 2
Assume 0 < γ < 2.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 196: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/196.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1c. OPTIMALITY OF THE EXPONENT 2
Assume 0 < γ < 2.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 197: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/197.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1c. OPTIMALITY OF THE EXPONENT 2
Assume 0 < γ < 2. Optimality will follow if we can showfor a unit ball B1 that inf
u∈W 2,p0 (B1)\0
Iγ(u) = 0
A. L. Detalla Missing Terms in Classical Inequalities
![Page 198: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/198.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1c. OPTIMALITY OF THE EXPONENT 2
Assume 0 < γ < 2. Optimality will follow if we can showfor a unit ball B1 that inf
u∈W 2,p0 (B1)\0
Iγ(u) = 0 where
Iγ(u) =
∫
B1|∆u|pdx−
(
n−2pp
)p (np−n
p
)p∫
B1
|u|p|x|2pdx
∫
B1
|u|p|x|2p
(
log R|x|
)−γ
dx
A. L. Detalla Missing Terms in Classical Inequalities
![Page 199: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/199.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1c. OPTIMALITY OF THE EXPONENT 2
Assume 0 < γ < 2. Optimality will follow if we can showfor a unit ball B1 that inf
u∈W 2,p0 (B1)\0
Iγ(u) = 0 where
Iγ(u) =
∫
B1|∆u|pdx−
(
n−2pp
)p (np−n
p
)p∫
B1
|u|p|x|2pdx
∫
B1
|u|p|x|2p
(
log R|x|
)−γ
dx
Hence using a family wǫ ∈ W 2,p0 (B1)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 200: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/200.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1c. OPTIMALITY OF THE EXPONENT 2
Assume 0 < γ < 2. Optimality will follow if we can showfor a unit ball B1 that inf
u∈W 2,p0 (B1)\0
Iγ(u) = 0 where
Iγ(u) =
∫
B1|∆u|pdx−
(
n−2pp
)p (np−n
p
)p∫
B1
|u|p|x|2pdx
∫
B1
|u|p|x|2p
(
log R|x|
)−γ
dx
Hence using a family wǫ ∈ W 2,p0 (B1) we have
limǫ→0 Iγ(wǫ) = 0.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 201: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/201.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1c. OPTIMALITY OF THE EXPONENT 2
Assume 0 < γ < 2. Optimality will follow if we can showfor a unit ball B1 that inf
u∈W 2,p0 (B1)\0
Iγ(u) = 0 where
Iγ(u) =
∫
B1|∆u|pdx−
(
n−2pp
)p (np−n
p
)p∫
B1
|u|p|x|2pdx
∫
B1
|u|p|x|2p
(
log R|x|
)−γ
dx
Hence using a family wǫ ∈ W 2,p0 (B1) we have
limǫ→0 Iγ(wǫ) = 0. Thus optimality follow. i.e. γ ≥ 2.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 202: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/202.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
LEMMA 3
Assume f ∈ C2(B1) and u ∈ C20(B1) are radial satisfying
f(r) > 0,∆f(r) ≤ 0, u(r) > 0, and −∆u > 0 where r = |x|.Set u(r) = f(r)v(r), then for any u ∈ C2
0(B1)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 203: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/203.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
LEMMA 3
∫
B1
|∆u|n2 dx ≥
n(n− 2)
4ωn
∫ 1
0
(v′(r))2v
n−42 (r)rn−1|∆f(r)|n−2
2 f(r)dr
+ ωn
∫ 1
0
vn2 (r)
rn−1|∆f(r)|n2 +
∂r
[
rn−1
(
|∆f(r)|n−22 f ′(r)− ∂r|∆f(r)|n−2
2 f(r)
)]
dr
A. L. Detalla Missing Terms in Classical Inequalities
![Page 204: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/204.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
From Lemma 3, we denote
S1 =n(n− 2)
4ωn
∫ 1
0
(v′(r))2v
n−42 (r)rn−1|∆f(r)|n−2
2 f(r)dr
and
S2 =ωn
∫ 1
0
vn2 (r)
rn−1|∆f(r)|n2 +
∂r
[
rn−1
(
|∆f(r)|n−22 f ′(r)− ∂r|∆f(r)|n−2
2 f(r)
)]
dr
A. L. Detalla Missing Terms in Classical Inequalities
![Page 205: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/205.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
Then∫
B1|∆u|n2 dx ≥ S1 + S2.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 206: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/206.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
Then∫
B1|∆u|n2 dx ≥ S1 + S2.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 207: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/207.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
Then∫
B1|∆u|n2 dx ≥ S1 + S2. For f(r) =
(
log Rr
)a,
0 < a < 1
A. L. Detalla Missing Terms in Classical Inequalities
![Page 208: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/208.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
Then∫
B1|∆u|n2 dx ≥ S1 + S2. For f(r) =
(
log Rr
)a,
0 < a < 1
S1 ≥a
n−22 (1− a)2(n− 2)
n2
4ωn
∫ 1
0
un2
(
logR
r
)−n2−1
dr
r
S2 ≥a
n−22 (1− a)(n− 2)
n2+1
2ωn
∫ 1
0
un2
(
logR
r
)−n2
(
1 +O
(
logR
r
)−2)
dr
r
A. L. Detalla Missing Terms in Classical Inequalities
![Page 209: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/209.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
Then∫
B1|∆u|n2 dx ≥ S1 + S2. For f(r) =
(
log Rr
)a,
0 < a < 1
S1 ≥a
n−22 (1− a)2(n− 2)
n2
4ωn
∫ 1
0
un2
(
logR
r
)−n2−1
dr
r
S2 ≥a
n−22 (1− a)(n− 2)
n2+1
2ωn
∫ 1
0
un2
(
logR
r
)−n2
(
1 +O
(
logR
r
)−2)
dr
r
set Q(a) = an−22 (1− a),
A. L. Detalla Missing Terms in Classical Inequalities
![Page 210: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/210.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
Then∫
B1|∆u|n2 dx ≥ S1 + S2. For f(r) =
(
log Rr
)a,
0 < a < 1
S1 ≥a
n−22 (1− a)2(n− 2)
n2
4ωn
∫ 1
0
un2
(
logR
r
)−n2−1
dr
r
S2 ≥a
n−22 (1− a)(n− 2)
n2+1
2ωn
∫ 1
0
un2
(
logR
r
)−n2
(
1 +O
(
logR
r
)−2)
dr
r
set Q(a) = an−22 (1− a), Q takes its maximum at a = n−2
2
A. L. Detalla Missing Terms in Classical Inequalities
![Page 211: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/211.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
∫
B1
|∆u|n2 dx ≥(
n− 2√n
)n ∫
B1
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
B1
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 212: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/212.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
∫
B1
|∆u|n2 dx ≥(
n− 2√n
)n ∫
B1
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
B1
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 213: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/213.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
∫
B1
|∆u|n2 dx ≥(
n− 2√n
)n ∫
B1
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
B1
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
where C∗ =(
n−2√n
)n (
(n− 2)−1 − k(logR)−1)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 214: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/214.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
∫
B1
|∆u|n2 dx ≥(
n− 2√n
)n ∫
B1
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
B1
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
where C∗ =(
n−2√n
)n (
(n− 2)−1 − k(logR)−1)
> 0
A. L. Detalla Missing Terms in Classical Inequalities
![Page 215: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/215.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
∫
B1
|∆u|n2 dx ≥(
n− 2√n
)n ∫
B1
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
B1
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
where C∗ =(
n−2√n
)n (
(n− 2)−1 − k(logR)−1)
> 0 if
R > e(n−2)k, k = k(n)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 216: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/216.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2b. SHARPNESS OF(
n−2√n
)n
To show sharpness, we use the test function
zǫ =
(
log1
r + ǫ
)n−2n
−(
log1
1 + ǫ
)n−2n
A. L. Detalla Missing Terms in Classical Inequalities
![Page 217: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/217.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2b. SHARPNESS OF(
n−2√n
)n
To show sharpness, we use the test function
zǫ =
(
log1
r + ǫ
)n−2n
−(
log1
1 + ǫ
)n−2n
A. L. Detalla Missing Terms in Classical Inequalities
![Page 218: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/218.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2b. SHARPNESS OF(
n−2√n
)n
To show sharpness, we use the test function
zǫ =
(
log1
r + ǫ
)n−2n
−(
log1
1 + ǫ
)n−2n
then we can show
limǫ→0
∫
B1|∆zǫ|
n2 dx
∫
B1
|zǫ|n2
|x|n
(
log R|x|
)−n2dx
=
(
n− 2√n
)n
A. L. Detalla Missing Terms in Classical Inequalities
![Page 219: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/219.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2c. OPTIMALITY OF THE EXPONENT n2
We use the same test function uǫ with p = n2, and
wǫ =∫ 1
ruǫ(ρ)dρ. Then for 0 < γ < n
2
limǫ→0
∫
B1|∆wǫ|
n2 dx
∫
B1
|wǫ|n2
|x|n
(
log R|x|
)γ
dx= 0
Thus optimality follow. i.e. γ ≥ n2
A. L. Detalla Missing Terms in Classical Inequalities
![Page 220: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/220.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
APPLICATION
Consider the weighted eigenvalue problem with a singularweight
∆(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u = λ|u|p−2uf in Ω
u = ∆u = 0 on ∂Ω (15)
Here f ∈ Fp
Fp =
f : Ω → R+| lim
|x|→0|x|2pf(x) = 0, f ∈ L∞
loc
(
Ω \ 0)
,
1 < p < n2, 0 ≤ µ <
(
n−2pp
)p (np−n
p
)p
and λ ∈ R.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 221: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/221.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
We look for a weak solution
u ∈ W = W 2,p(Ω) ∩W 1,p0 (Ω)
of problem ♯15 and study the asymptotic behaviour of thefirst eigenvalues for different singular weights as µ increases
to(
n−2pp
)p (np−n
p
)p
.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 222: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/222.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
We look for a weak solution
u ∈ W = W 2,p(Ω) ∩W 1,p0 (Ω)
of problem ♯15 and study the asymptotic behaviour of thefirst eigenvalues for different singular weights as µ increases
to(
n−2pp
)p (np−n
p
)p
.
Definition
u ∈ W is said to be a weak solution of (15) iff for anyφ ∈ C2(Ω) with φ = 0 on ∂Ω
∫
Ω
(
|∆u|p−2∆u∆φ− µ
|x|2p |u|p−2uφ
)
dx = λ
∫
Ω
|u|p−2ufφdx.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 223: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/223.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
LEMMA
For u ∈ W ∃ v ∈ W such that v > 0 and satisfies
∫
Ω|∆u|pdx− λ
∫
Ω|u|p|x|2pdx
∫
Ω|u|pfdx ≥
∫
Ω|∆v|pdx− λ
∫
Ω|v|p|x|2pdx
∫
Ω|v|pfdx .
A. L. Detalla Missing Terms in Classical Inequalities
![Page 224: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/224.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
LEMMA
For u ∈ W ∃ v ∈ W such that v > 0 and satisfies
∫
Ω|∆u|pdx− λ
∫
Ω|u|p|x|2pdx
∫
Ω|u|pfdx ≥
∫
Ω|∆v|pdx− λ
∫
Ω|v|p|x|2pdx
∫
Ω|v|pfdx .
REMARK
Since λ is first eigenvalue and u is the correspondingeigenfunction, by using the above lemma, we can assumeu > 0 in Ω. Then by the elliptic regularity theory, u issmooth near the boundary. From the definition of weaksolution one can derive the boundary condition of (15) byusing integration by parts.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 225: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/225.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
THEOREM
For all 1 < p < n2, 0 ≤ µ <
(
n−2pp
)p (np−n
p
)p
, the above
problem ♯15 admits a positive weak solution u ∈ Wcorresponding to the first eigenvalue λ = λ1
µ(f) > 0.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 226: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/226.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
THEOREM
For all 1 < p < n2, 0 ≤ µ <
(
n−2pp
)p (np−n
p
)p
, the above
problem ♯15 admits a positive weak solution u ∈ Wcorresponding to the first eigenvalue λ = λ1
µ(f) > 0.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 227: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/227.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
THEOREM
For all 1 < p < n2, 0 ≤ µ <
(
n−2pp
)p (np−n
p
)p
, the above
problem ♯15 admits a positive weak solution u ∈ Wcorresponding to the first eigenvalue λ = λ1
µ(f) > 0.
Moreover, as µ →(
n−2pp
)p (np−n
p
)p
,
A. L. Detalla Missing Terms in Classical Inequalities
![Page 228: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/228.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
THEOREM
For all 1 < p < n2, 0 ≤ µ <
(
n−2pp
)p (np−n
p
)p
, the above
problem ♯15 admits a positive weak solution u ∈ Wcorresponding to the first eigenvalue λ = λ1
µ(f) > 0.
Moreover, as µ →(
n−2pp
)p (np−n
p
)p
,
If lim sup|x|→0
|x|2pf(x) = 0, ⇒ λ1µ(f) → λ(f) ≥ 0
A. L. Detalla Missing Terms in Classical Inequalities
![Page 229: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/229.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
THEOREM
For all 1 < p < n2, 0 ≤ µ <
(
n−2pp
)p (np−n
p
)p
, the above
problem ♯15 admits a positive weak solution u ∈ Wcorresponding to the first eigenvalue λ = λ1
µ(f) > 0.
Moreover, as µ →(
n−2pp
)p (np−n
p
)p
,
If lim sup|x|→0
|x|2pf(x) = 0, ⇒ λ1µ(f) → λ(f) ≥ 0
If lim sup|x|→0
|x|2pf(x)(
log1
|x|
)2
< ∞, ⇒ λ(f) > 0
A. L. Detalla Missing Terms in Classical Inequalities
![Page 230: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/230.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
THEOREM
For all 1 < p < n2, 0 ≤ µ <
(
n−2pp
)p (np−n
p
)p
, the above
problem ♯15 admits a positive weak solution u ∈ Wcorresponding to the first eigenvalue λ = λ1
µ(f) > 0.
Moreover, as µ →(
n−2pp
)p (np−n
p
)p
,
If lim sup|x|→0
|x|2pf(x) = 0, ⇒ λ1µ(f) → λ(f) ≥ 0
If lim sup|x|→0
|x|2pf(x)(
log1
|x|
)2
< ∞, ⇒ λ(f) > 0
If lim sup|x|→0
|x|2pf(x)(
log1
|x|
)2
= ∞, ⇒ λ(f) = 0
A. L. Detalla Missing Terms in Classical Inequalities
![Page 231: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/231.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
REMARK
In the proof of the theorem u will be characterize as asolution of variational problem defined by
Jµ(u) =
∫
Ω
(
|∆u|p − µ|u|p|x|2p
)
dx
and the problem (♯15) stated earlier becomes Euler-Lagrange equation of this variational problem.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 232: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/232.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
A. L. Detalla Missing Terms in Classical Inequalities
![Page 233: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/233.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
A. L. Detalla Missing Terms in Classical Inequalities
![Page 234: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/234.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
We define Jµ(u) =∫
Ω
(
|∆u|p − µ |u|p|x|2p
)
dx
A. L. Detalla Missing Terms in Classical Inequalities
![Page 235: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/235.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
We define Jµ(u) =∫
Ω
(
|∆u|p − µ |u|p|x|2p
)
dx
A. L. Detalla Missing Terms in Classical Inequalities
![Page 236: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/236.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
We define Jµ(u) =∫
Ω
(
|∆u|p − µ |u|p|x|2p
)
dx
We minimize Jµ over M = u ∈ W |∫
Ω|u(x)|pf(x)dx = 1
and let λ1µ > 0 be the infimum.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 237: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/237.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
We define Jµ(u) =∫
Ω
(
|∆u|p − µ |u|p|x|2p
)
dx
We minimize Jµ over M = u ∈ W |∫
Ω|u(x)|pf(x)dx = 1
and let λ1µ > 0 be the infimum.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 238: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/238.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
We define Jµ(u) =∫
Ω
(
|∆u|p − µ |u|p|x|2p
)
dx
We minimize Jµ over M = u ∈ W |∫
Ω|u(x)|pf(x)dx = 1
and let λ1µ > 0 be the infimum.
We choose minimizing sequence (um)m∈N ⊂ M such thatJµ(um) → λ1
µ.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 239: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/239.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
A. L. Detalla Missing Terms in Classical Inequalities
![Page 240: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/240.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
A. L. Detalla Missing Terms in Classical Inequalities
![Page 241: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/241.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
For a subsequence umkof um, umk
u weakly in W whereu ∈ W ∩M and
Jµ(umk) → λ1
µ = λ
J ′µ(umk
) → 0
A. L. Detalla Missing Terms in Classical Inequalities
![Page 242: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/242.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
For a subsequence umkof um, umk
u weakly in W whereu ∈ W ∩M and
Jµ(umk) → λ1
µ = λ
J ′µ(umk
) → 0
A. L. Detalla Missing Terms in Classical Inequalities
![Page 243: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/243.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
For a subsequence umkof um, umk
u weakly in W whereu ∈ W ∩M and
Jµ(umk) → λ1
µ = λ
J ′µ(umk
) → 0
By Fatou’s lemma, we get
umk→ u strongly in W
umk→ u strongly in Lp (Ω, |x|−2p)
A. L. Detalla Missing Terms in Classical Inequalities
![Page 244: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/244.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
A. L. Detalla Missing Terms in Classical Inequalities
![Page 245: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/245.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
A. L. Detalla Missing Terms in Classical Inequalities
![Page 246: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/246.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
Hence
Jµ(umk) → Jµ(u) = λ1
µ = λ
J ′µ(umk
) = 0
A. L. Detalla Missing Terms in Classical Inequalities
![Page 247: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/247.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
Hence
Jµ(umk) → Jµ(u) = λ1
µ = λ
J ′µ(umk
) = 0
A. L. Detalla Missing Terms in Classical Inequalities
![Page 248: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/248.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
Hence
Jµ(umk) → Jµ(u) = λ1
µ = λ
J ′µ(umk
) = 0
u satisfies Euler- Lagrange equation in a distribution sense.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 249: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/249.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
Hence
Jµ(umk) → Jµ(u) = λ1
µ = λ
J ′µ(umk
) = 0
u satisfies Euler- Lagrange equation in a distribution sense.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 250: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/250.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
Hence
Jµ(umk) → Jµ(u) = λ1
µ = λ
J ′µ(umk
) = 0
u satisfies Euler- Lagrange equation in a distribution sense.
Since u ∈ W , it is a weak solution of problem (♯15).
A. L. Detalla Missing Terms in Classical Inequalities
![Page 251: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/251.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
Hence
Jµ(umk) → Jµ(u) = λ1
µ = λ
J ′µ(umk
) = 0
u satisfies Euler- Lagrange equation in a distribution sense.
Since u ∈ W , it is a weak solution of problem (♯15).
A. L. Detalla Missing Terms in Classical Inequalities
![Page 252: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/252.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
Hence
Jµ(umk) → Jµ(u) = λ1
µ = λ
J ′µ(umk
) = 0
u satisfies Euler- Lagrange equation in a distribution sense.
Since u ∈ W , it is a weak solution of problem (♯15).
The remaining part of the proof follows from the corollaryof the main theorem.
A. L. Detalla Missing Terms in Classical Inequalities
![Page 253: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/253.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Corollary
Let 1 < p < n2, and let
Fp =
f : Ω → R+|f ∈ L∞
loc(Ω \ 0) with
lim sup|x|→0
|x|2pf(x)(
log1
|x|
)2
< ∞
If f ∈ Fp, ∃ λ(f) > 0 such that for u ∈ W 2,p0 (Ω)
∫
Ω
|∆u|pdx ≥ Λn,p
∫
Ω
|u(x)|p|x|2p dx+λ(f)
∫
Ω
|u(x)|pf(x)dx (13)
If f /∈ Fp and if |x|2pf(x)(
log 1|x|
)2
tends to ∞ as |x| → 0,
then no inequality of type (13) can hold.A. L. Detalla Missing Terms in Classical Inequalities
![Page 254: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/254.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
As µ →(
n−2pp
)p (np−n
p
)p
, λ1µ → λ(f) where
A. L. Detalla Missing Terms in Classical Inequalities
![Page 255: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/255.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
As µ →(
n−2pp
)p (np−n
p
)p
, λ1µ → λ(f) where
A. L. Detalla Missing Terms in Classical Inequalities
![Page 256: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/256.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
As µ →(
n−2pp
)p (np−n
p
)p
, λ1µ → λ(f) where
λ(f) = infu∈W (Ω\0)
∫
Ω
(
|∆u|p −(
n−2pp
)p (np−n
p
)p |u|p|x|2p
)
dx∫
Ω|u|pfdx ≥ 0
A. L. Detalla Missing Terms in Classical Inequalities
![Page 257: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/257.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
As µ →(
n−2pp
)p (np−n
p
)p
, λ1µ → λ(f) where
λ(f) = infu∈W (Ω\0)
∫
Ω
(
|∆u|p −(
n−2pp
)p (np−n
p
)p |u|p|x|2p
)
dx∫
Ω|u|pfdx ≥ 0
if f ∈ Fp ⇒ λ(f) > 0
A. L. Detalla Missing Terms in Classical Inequalities
![Page 258: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/258.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
As µ →(
n−2pp
)p (np−n
p
)p
, λ1µ → λ(f) where
λ(f) = infu∈W (Ω\0)
∫
Ω
(
|∆u|p −(
n−2pp
)p (np−n
p
)p |u|p|x|2p
)
dx∫
Ω|u|pfdx ≥ 0
if f ∈ Fp ⇒ λ(f) > 0if f 6= Fp ⇒ λ(f) = 0
A. L. Detalla Missing Terms in Classical Inequalities
![Page 259: Al Detalla Theory](https://reader031.vdocument.in/reader031/viewer/2022020219/55cf9694550346d0338c708d/html5/thumbnails/259.jpg)
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
THANK YOU VERY MUCH!
A. L. Detalla Missing Terms in Classical Inequalities