improved poincaré and other classic inequalities: a new

Improved Poincaré and other classicinequalities: a new approach to prove them

and some generalizations

Ricardo G. Durán

Departamento de MatemáticaFacultad de Ciencias Exactas y Naturales

Universidad de Buenos Airesand IMAS, CONICET

Martin Stynes 60th Birthday CelebrationDresden

November 18, 2011

Ricardo G. Durán Poincaré and related inequalities

COWORKERS

G. Acosta

I. Drelichman

A. L. Lombardi

F. López

M. A. Muschietti

M. I. Prieto

E. Russ

P. Tchamitchian

This talk is about several classic inequalities(Korn, Poincaré, Improved Poincaré, etc.).

Questions we are interested in:

What are the relations between them?

For which domains are these inequalitiesvalid?

When they are not valid: Are there weakerestimates still useful for variational analysis?

More generally: in some applications it isinteresting to have “uniformly valid inequalities”for a sequence of domains.

For example: Domain decomposition methodsfor finite elements.

Dohrmann, Klawonn, and Widlund, Domaindecomposition for less regular subdomains:overlapping Schwarz in two dimensions. SIAMJ. Numer. Anal. 46 (2008), no. 4, 2153–2168.

More generally: in some applications it isinteresting to have “uniformly valid inequalities”for a sequence of domains.

For example: Domain decomposition methodsfor finite elements.

Dohrmann, Klawonn, and Widlund, Domaindecomposition for less regular subdomains:overlapping Schwarz in two dimensions. SIAMJ. Numer. Anal. 46 (2008), no. 4, 2153–2168.

MAIN RESULT

A methodology to prove the inequalities ingeneral domains using analogous inequalities incubes.

Main tool:

Whitney decomposition into cubes!

NOTATIONS

Ω ⊂ Rn bounded domain 1 ≤ p ≤ ∞

W 1,p(Ω) =

v ∈ Lp(Ω) :∂v∂xj∈ Lp(Ω) ,∀j = 1, · · · ,n

0 (Ω) = C∞0 (Ω) ⊂W 1,p(Ω)

H1(Ω) = W 1,2(Ω) , H10 (Ω) = W 1,2

0 (Ω)

NOTATIONS

Lp0(Ω) =

f ∈ Lp(Ω) :

W−1,p′(Ω) = W 1,p0 (Ω)′

where p′ is the dual exponent of p

C = C(· · · ) constant depending on · · ·

KORN INEQUALITY

v ∈W 1,p(Ω)n , Dv =(∂vi

ε(v) symmetric part of Dv, i. e.,

εij(v) =12

(∂vi

∂xj+∂vj

KORN INEQUALITY

v ∈W 1,p(Ω)n , Dv =(∂vi

)ε(v) symmetric part of Dv, i. e.,

εij(v) =12

(∂vi

∂xj+∂vj

KORN INEQUALITY

1 < p <∞, there exists C = C(Ω,p) such that

‖v‖W 1,p(Ω)n ≤ C‖v‖Lp(Ω)n + ‖ε(v)‖Lp(Ω)n×n

RIGHT INVERSES OF THE DIVERGENCE

f ∈ Lp0(Ω),1 < p <∞

∃u ∈W 1,p0 (Ω)n such that

div u = f , ‖u‖W 1,p(Ω)n ≤ C‖f‖Lp(Ω)

C = C(Ω,p)

RIGHT INVERSES OF THE DIVERGENCE

f ∈ Lp0(Ω),1 < p <∞

C = C(Ω,p)

DUAL VERSION

Called Lions Lemma (when p = 2)

∃ C = C(Ω,p) such that∫Ω

f = 0 =⇒ ‖f‖Lp′(Ω) ≤ C‖∇f‖W−1,p′(Ω)n

or equivalently (inf-sup condition for Stokes)

inff∈Lp′

0 (Ω)

supv∈W 1,p

0 (Ω)n

∫Ω f div v

‖f‖Lp′‖v‖W 1,p0 (Ω)n

≥ α > 0

with α = C−1.

DUAL VERSION

Called Lions Lemma (when p = 2)∃ C = C(Ω,p) such that∫

f = 0 =⇒ ‖f‖Lp′(Ω) ≤ C‖∇f‖W−1,p′(Ω)n

inff∈Lp′

0 (Ω)

supv∈W 1,p

0 (Ω)n

∫Ω f div v

≥ α > 0

with α = C−1.

DUAL VERSION

Called Lions Lemma (when p = 2)∃ C = C(Ω,p) such that∫

f = 0 =⇒ ‖f‖Lp′(Ω) ≤ C‖∇f‖W−1,p′(Ω)n

inff∈Lp′

0 (Ω)

supv∈W 1,p

0 (Ω)n

∫Ω f div v

≥ α > 0

with α = C−1.

IMPROVED POINCARÉ INEQUALITY

d(x) = distance to ∂Ω

∃C = C(Ω,p) such that

f = 0 =⇒ ‖f‖Lp(Ω) ≤ C‖d∇f‖Lp(Ω)n

DUAL VERSION

f ∈ Lp′0 (Ω) ⇒ ∃u ∈ Lp′(Ω)n such that

div u = f ,∥∥∥u

∥∥∥Lp′(Ω)n

≤ C‖f‖Lp′(Ω)

u · n = 0 en ∂Ω

DUAL VERSION

f ∈ Lp′0 (Ω) ⇒ ∃u ∈ Lp′(Ω)n such that

div u = f ,∥∥∥u

∥∥∥Lp′(Ω)n

≤ C‖f‖Lp′(Ω)

u · n = 0 en ∂Ω

RELATIONS

Several connections between these inequalitiesare known:

Right Inverse of Div ⇒ Korn

Improved Poincaré ⇒ Korn(Kondratiev-Oleinik)

Explicit relation between constants in 2d for C1

domains (Horgan-Payne)

RELATIONS

A SIMPLE EXAMPLE

THE DOMAIN CANNOT BE ARBITRARY

EXAMPLE (G. ACOSTA):

Ω = (x , y) ∈ R2 : 0 < x < 1 , 0 < y < x2

A SIMPLE EXAMPLE

THE DOMAIN CANNOT BE ARBITRARY

EXAMPLE (G. ACOSTA):

Ω = (x , y) ∈ R2 : 0 < x < 1 , 0 < y < x2

f (x , y) =1x2 − 3⇒

∫ x2

0f 2 dydx '

1x2 ⇒ f /∈ L2(Ω)

−2yx−3 ∈ L2(Ω)⇒ ∂f∂x

=∂(−2yx−3)

∂y∈ H−1(Ω)

A SIMPLE EXAMPLE

‖f‖L2(Ω) C‖∇f‖H−1(Ω)n

A SIMPLE EXAMPLE

SAME EXAMPLE ⇒

‖f‖L2(Ω) C‖d∇f‖L2(Ω)n

OUR MAIN RESULT

“THEOREM”

‖f‖L1(Ω) ≤ C‖d∇f‖L1(Ω)n ∀f ∈ L10(Ω)

⇓“All” the inequalities valid in cubes are valid in Ω(∀p) !!

MOTIVATION

Why are relations between inequalitiesinteresting?

We could think that, if we are not interested instrange domains, we already know that all theseinequalities are valid (for example for Ω aLipschitz domain)But, in many situations it is important to haveinformation of the constants in terms of thegeometry of the domainThen, we can translate information from oneinequality to other

MOTIVATION

Why are relations between inequalitiesinteresting?We could think that, if we are not interested instrange domains, we already know that all theseinequalities are valid (for example for Ω aLipschitz domain)

But, in many situations it is important to haveinformation of the constants in terms of thegeometry of the domainThen, we can translate information from oneinequality to other

MOTIVATION

Why are relations between inequalitiesinteresting?We could think that, if we are not interested instrange domains, we already know that all theseinequalities are valid (for example for Ω aLipschitz domain)But, in many situations it is important to haveinformation of the constants in terms of thegeometry of the domain

Then, we can translate information from oneinequality to other

MOTIVATION

Why are relations between inequalitiesinteresting?We could think that, if we are not interested instrange domains, we already know that all theseinequalities are valid (for example for Ω aLipschitz domain)But, in many situations it is important to haveinformation of the constants in terms of thegeometry of the domainThen, we can translate information from oneinequality to other

ESTIMATES IN TERMS OF GEOMETRICPROPERTIES

For example:

Let Ω be a convex domain with diameter D andinner diameter ρ.Then, it was recently proved by M. Barchiesi, F.Cagnetti, and N. Fusco that

‖f‖L1(Ω) ≤ CDρ‖d∇f‖L1(Ω)n ∀f ∈ L1

For example:

Let Ω be a convex domain with diameter D andinner diameter ρ.

Then, it was recently proved by M. Barchiesi, F.Cagnetti, and N. Fusco that

‖f‖L1(Ω) ≤ CDρ‖d∇f‖L1(Ω)n ∀f ∈ L1

For example:

‖f‖L1(Ω) ≤ CDρ‖d∇f‖L1(Ω)n ∀f ∈ L1

For example:

‖f‖L1(Ω) ≤ CDρ‖d∇f‖L1(Ω)n ∀f ∈ L1

Consequently, we obtain from our results that,for Ω convex,

‖f‖Lp(Ω) ≤ CDρ‖∇f‖W−1,p(Ω)n ∀f ∈ Lp

1 < p <∞

C = C(p,n)

1 < p <∞

C = C(p,n)

1 < p <∞

C = C(p,n)

OPTIMALITY OF THIS ESTIMATE

The case of a rectangular domain shows thatthe dependence of the constant in terms of theeccentricity D/ρ cannot be improved. Indeed(let us consider p = 2 for simplicity)

Ωε = (x , y) ∈ R2 : −1 < x < 1 , −ε < y < ε

‖f‖L2(Ωε) ≤ Cε‖∇f‖H−1(Ωε)n

⇓‖x‖L2 ≤ Cε‖∇x‖H−1 = Cε‖∇y‖H−1 ≤ Cε‖y‖L2

Ωε = (x , y) ∈ R2 : −1 < x < 1 , −ε < y < ε

⇓‖x‖L2 ≤ Cε‖∇x‖H−1 = Cε‖∇y‖H−1 ≤ Cε‖y‖L2

Ωε = (x , y) ∈ R2 : −1 < x < 1 , −ε < y < ε

⇓‖x‖L2 ≤ Cε‖∇x‖H−1 = Cε‖∇y‖H−1 ≤ Cε‖y‖L2

‖x‖L2(Ωε) ∼ ε12 , ‖y‖L2(Ωε) ∼ ε

Cε ≥Cε∼ Dρ

In many cases it is also possible to go in theopposite way:

For example, if Ω ⊂ R2 hasdiameter D and is star-shaped with respect to aball of radius ρ, we could prove that

‖f‖L2(Ω) ≤ CDρ

)‖∇f‖H−1(Ω)n ∀f ∈ L2

And from this estimate it can be deduced that

)‖d∇f‖L2(Ω)n ∀f ∈ L2

In many cases it is also possible to go in theopposite way: For example, if Ω ⊂ R2 hasdiameter D and is star-shaped with respect to aball of radius ρ, we could prove that

)‖∇f‖H−1(Ω)n ∀f ∈ L2

)‖d∇f‖L2(Ω)n ∀f ∈ L2

In many cases it is also possible to go in theopposite way: For example, if Ω ⊂ R2 hasdiameter D and is star-shaped with respect to aball of radius ρ, we could prove that

)‖∇f‖H−1(Ω)n ∀f ∈ L2

)‖d∇f‖L2(Ω)n ∀f ∈ L2

MAIN RESULT

Proof of

‖f‖L1(Ω) ≤ C1‖d∇f‖L1(Ω)n ∀f ∈ L10(Ω)

⇓∀f ∈ Lp

0(Ω),1 < p <∞

C = C(n,p,C1)

MAIN RESULT

Proof of

‖f‖L1(Ω) ≤ C1‖d∇f‖L1(Ω)n ∀f ∈ L10(Ω)

⇓∀f ∈ Lp

0(Ω),1 < p <∞

C = C(n,p,C1)

MAIN RESULT

Proof of

‖f‖L1(Ω) ≤ C1‖d∇f‖L1(Ω)n ∀f ∈ L10(Ω)

⇓∀f ∈ Lp

0(Ω),1 < p <∞

C = C(n,p,C1)

THREE STEPS

Improved Poincaré for p = 1⇒ ImprovedPoincaré for 1 < p <∞

Improved Poincaré for p′ allows us to reducethe problem to “local problems” in cubes

Solve for the divergence in cubes and sum

THREE STEPS

STEP 2

Assume improved Poincaré for p′

Take a Whitney decomposition of Ω, i. e.,

Ω = ∪jQj , Q0j ∩Q0

i = ∅

diam Qj ∼ dist (Qj , ∂Ω) =: dj

For f ∈ Lp0(Ω), there exists a decomposition

f =∑

STEP 2

Ω = ∪jQj , Q0j ∩Q0

i = ∅

f =∑

STEP 2

Ω = ∪jQj , Q0j ∩Q0

i = ∅

f =∑

STEP 2

Ω = ∪jQj , Q0j ∩Q0

i = ∅

f =∑

STEP 2

such that

fj ∈ Lp0(Qj) , Qj :=

‖f‖pLp(Ω) ∼

‖fj‖pLp(Qj )

STEP 2

such that

‖f‖pLp(Ω) ∼

‖fj‖pLp(Qj )

STEP 2

such that

‖f‖pLp(Ω) ∼

‖fj‖pLp(Qj )

DECOMPOSITION OF FUNCTIONS

Proof of the existence of this decomposition:

Take a partition of unity associated with theWhitney decomposition

φj = 1 , sopφj ⊂ Qj =98

‖φj‖L∞ ≤ 1 , ‖∇φj‖L∞ ≤ C/dj

Proof of the existence of this decomposition:

Take a partition of unity associated with theWhitney decomposition

φj = 1 , sopφj ⊂ Qj =98

‖φj‖L∞ ≤ 1 , ‖∇φj‖L∞ ≤ C/dj

Assuming the improved Poincaré for p′, weobtain by duality:

For f ∈ Lp0(Ω) there exists u ∈ Lp(Ω)n such that

div u = f in Ω u · n = 0 on ∂Ω

∥∥∥ud

∥∥∥Lp(Ω)

≤ C‖f‖Lp(Ω)

∥∥∥ud

∥∥∥Lp(Ω)

≤ C‖f‖Lp(Ω)

∥∥∥ud

∥∥∥Lp(Ω)

≤ C‖f‖Lp(Ω)

Then, we define

fj = div (φju)

and so

f = div u = div u∑

φj =∑

div (φju) =∑

sopφj ⊂ Qj ⇒ sop fj ⊂ Qj ,

∫fj = 0

Then, we define

fj = div (φju)

and so

φj =∑

div (φju) =∑

∫fj = 0

Then, we define

fj = div (φju)

and so

φj =∑

div (φju) =∑

∫fj = 0

Then, we define

fj = div (φju)

and so

φj =∑

div (φju) =∑

∫fj = 0

From finite superposition:

|f (x)|p ≤ C∑

|fj(x)|p

and therefore

‖f‖pLp(Ω) ≤ C

‖fj‖pLp(Qj )

From finite superposition:

|f (x)|p ≤ C∑

|fj(x)|p

and therefore

‖f‖pLp(Ω) ≤ C

‖fj‖pLp(Qj )

To prove the other estimate we use

‖φj‖L∞ ≤ 1 , ‖∇φj‖L∞ ≤ C/dj

fj = div (φju) = div uφj +∇φj · u = fφj +∇φj · u

To prove the other estimate we use

‖φj‖L∞ ≤ 1 , ‖∇φj‖L∞ ≤ C/dj

fj = div (φju) = div uφj +∇φj · u = fφj +∇φj · u

‖fj‖pLp(Qj )

≤ C‖f‖p

Lp(Qj )+∥∥∥u

∥∥∥p

Lp(Qj )

and therefore, using∥∥∥ud

∥∥∥Lp(Ω)

≤ C‖f‖Lp(Ω)

we obtain, ∑j

‖fj‖pLp(Qj )

≤ C‖f‖pLp(Ω)

‖fj‖pLp(Qj )

≤ C‖f‖p

Lp(Qj )+∥∥∥u

∥∥∥p

Lp(Qj )

and therefore, using∥∥∥u

∥∥∥Lp(Ω)

≤ C‖f‖Lp(Ω)

we obtain, ∑j

‖fj‖pLp(Qj )

≤ C‖f‖pLp(Ω)

‖fj‖pLp(Qj )

≤ C‖f‖p

Lp(Qj )+∥∥∥u

∥∥∥p

Lp(Qj )

∥∥∥Lp(Ω)

≤ C‖f‖Lp(Ω)

we obtain,

‖fj‖pLp(Qj )

≤ C‖f‖pLp(Ω)

‖fj‖pLp(Qj )

≤ C‖f‖p

Lp(Qj )+∥∥∥u

∥∥∥p

Lp(Qj )

∥∥∥Lp(Ω)

≤ C‖f‖Lp(Ω)

we obtain, ∑j

‖fj‖pLp(Qj )

≤ C‖f‖pLp(Ω)

STEP 3

Then, to solve div u = f

solve div uj = fj

uj ∈W 1,p0 (Qj) , ‖uj‖W 1,p(Qj )

≤ C‖fj‖Lp(Qj )

Observe that C = C(n,p)

andu =

is the required solution!

STEP 3

solve div uj = fj

uj ∈W 1,p0 (Qj) , ‖uj‖W 1,p(Qj )

andu =

STEP 3

solve div uj = fj

uj ∈W 1,p0 (Qj) , ‖uj‖W 1,p(Qj )

andu =

STEP 3

solve div uj = fj

uj ∈W 1,p0 (Qj) , ‖uj‖W 1,p(Qj )

andu =

STEP 3

solve div uj = fj

uj ∈W 1,p0 (Qj) , ‖uj‖W 1,p(Qj )

u =∑

STEP 3

solve div uj = fj

uj ∈W 1,p0 (Qj) , ‖uj‖W 1,p(Qj )

andu =

STEP 1: Poincaré in L1 implies Poincaré in Lp,p <∞

To conclude the proof of our theorem we need:

‖f − fΩ‖L1(Ω) ≤ C‖d∇f‖L1(Ω)

‖f − fΩ‖Lp(Ω) ≤ C‖d∇f‖Lp(Ω)

STEP 1: Poincaré in L1 implies Poincaré in Lp,p <∞

To conclude the proof of our theorem we need:

‖f − fΩ‖L1(Ω) ≤ C‖d∇f‖L1(Ω)

‖f − fΩ‖Lp(Ω) ≤ C‖d∇f‖Lp(Ω)

Poincaré in L1 implies Poincaré in Lp,p <∞

OR MORE GENERALLY

‖f − fΩ‖L1(Ω) ≤ C‖w∇f‖L1(Ω)

‖f − fΩ‖Lp(Ω) ≤ C‖w∇f‖Lp(Ω)

∀p <∞

1)‖f − fΩ‖Lp(Ω) ≤ C‖w∇f‖Lp(Ω)

∀E ⊂ Ω such that |E | ≥ 12|Ω|

f |E = 0⇒ ‖f‖Lp(Ω) ≤ C‖w∇f‖Lp(Ω)

with C independent of E and f .

1)‖f − fΩ‖Lp(Ω) ≤ C‖w∇f‖Lp(Ω)

∀E ⊂ Ω such that |E | ≥ 12|Ω|

f |E = 0⇒ ‖f‖Lp(Ω) ≤ C‖w∇f‖Lp(Ω)

with C independent of E and f .

Case p = 1 :

f = f+ − f−

Suppose |f+ = 0| ≥ 12 |Ω|

⇒ ‖f+‖L1(Ω) ≤ C‖w∇f+‖L1(Ω) ≤ C‖w∇f‖L1(Ω)

Case p = 1 :

f = f+ − f−

Suppose |f+ = 0| ≥ 12 |Ω|

⇒ ‖f+‖L1(Ω) ≤ C‖w∇f+‖L1(Ω) ≤ C‖w∇f‖L1(Ω)

f = 0⇒∫

and therefore,

‖f‖L1(Ω) =

f+ + f− = 2∫

f+ ≤ 2C‖w∇f+‖L1(Ω)

But ∫Ω

f = 0⇒∫

and therefore,

‖f‖L1(Ω) =

f+ + f− = 2∫

f+ ≤ 2C‖w∇f+‖L1(Ω)

But ∫Ω

f = 0⇒∫

and therefore,

‖f‖L1(Ω) =

f+ + f− = 2∫

f+ ≤ 2C‖w∇f+‖L1(Ω)

If ∫Ω

f p+ =

f p−

the same argument than for p = 1 applies!

But, from Bolzano’s theorem

∃λ ∈ Rsuch that ∫

(f − λ)p+ =

(f − λ)p−

If ∫Ω

f p+ =

f p−

(f − λ)p+ =

(f − λ)p−

If ∫Ω

f p+ =

f p−

(f − λ)p+ =

(f − λ)p−

f |E = 0⇒ |f |p|E = 0

Apply 1−Poincaré to |f |p

f |E = 0⇒ |f |p|E = 0

Apply 1−Poincaré to |f |p

|f |p ≤ Cp∫

|f |p−1|∇f |w

≤(∫

|f |p)1/p′ (∫

|∇f |pwp)1/p

therefore,

‖f‖Lp(Ω) ≤ C‖w∇f‖Lp(Ω)

|f |p ≤ Cp∫

|f |p−1|∇f |w

≤(∫

|f |p)1/p′ (∫

|∇f |pwp)1/p

therefore,

‖f‖Lp(Ω) ≤ C‖w∇f‖Lp(Ω)

Proof of the improved Poincaré in L1

Now, to complete our arguments we have toprove the improved Poincaré in L1.

We will show that it can be proved for a verygeneral class of domains by elementarycalculus arguments

Proof of the improved Poincaré in L1

Now, to complete our arguments we have toprove the improved Poincaré in L1.

We will show that it can be proved for a verygeneral class of domains by elementarycalculus arguments

REPRESENTATION FORMULA

f (y)− f = −∫

G(x , y) · ∇f (x) dx

Assume Ω star-shaped with respect to a ballB(0, δ)

ω ∈ C∞0 (Bδ/2)

∫Bδ/2

ω = 1 f =

y ∈ Ω , z ∈ Bδ/2

f (y)− f (z) =

0(y − z) · ∇f (y + t(z − y)) dt

f (y)− f = −∫

G(x , y) · ∇f (x) dx

ω ∈ C∞0 (Bδ/2)

∫Bδ/2

ω = 1 f =

y ∈ Ω , z ∈ Bδ/2

f (y)− f (z) =

0(y − z) · ∇f (y + t(z − y)) dt

f (y)− f = −∫

G(x , y) · ∇f (x) dx

ω ∈ C∞0 (Bδ/2)

∫Bδ/2

ω = 1 f =

y ∈ Ω , z ∈ Bδ/2

f (y)− f (z) =

0(y − z) · ∇f (y + t(z − y)) dt

Multiplying by ω(z) and integrating:

f (y)− f =

0(y−z) ·∇f (y + t(z−y))ω(z)dtdz

and changing variables

x = y + t(z − y)

f (y)− f =

x = y + t(z − y)

f (y)− f =

x = y + t(z − y)

f (y)−f =

(y − x)

tn+1 ·∇f (x)ω

x − yt

that is,

f (y)− f = −∫

G(x , y) · ∇f (x)dx

f (y)−f =

(y − x)

tn+1 ·∇f (x)ω

x − yt

that is,

f (y)− f = −∫

G(x , y) · ∇f (x)dx

G(x , y) =

(x − y)

x − yt

)1tn dt

PROPERTIES OF G(x , y)

|G(x , y)| ≤ C1

|x − y |n−1

|x − y | > C2d(x) ⇒ G(x , y) = 0

G(x , y) =

(x − y)

x − yt

)1tn dt

|G(x , y)| ≤ C1

|x − y |n−1

|x − y | > C2d(x) ⇒ G(x , y) = 0

G(x , y) =

(x − y)

x − yt

)1tn dt

|G(x , y)| ≤ C1

|x − y |n−1

|x − y | > C2d(x) ⇒ G(x , y) = 0

G(x , y) =

(x − y)

x − yt

)1tn dt

|G(x , y)| ≤ C1

|x − y |n−1

|x − y | > C2d(x) ⇒ G(x , y) = 0

MORE GENERAL DOMAINS

The representation formula can be generalizedto John domains (Acosta-D.-Muschietti)

G(x , y) =

(x − γt

+γ(t , y))ω

(x − γ(t , y)

where γ(t , y) are “ John curves”

γ(0, y) = y , γ(1, y) = 0

|γ(t , y)| ≤ K , d(γ(t , y)) ≥ δt

G(x , y) =

(x − γt

+γ(t , y))ω

(x − γ(t , y)

γ(0, y) = y , γ(1, y) = 0

|γ(t , y)| ≤ K , d(γ(t , y)) ≥ δt

G(x , y) =

(x − γt

+γ(t , y))ω

(x − γ(t , y)

γ(0, y) = y , γ(1, y) = 0

|γ(t , y)| ≤ K , d(γ(t , y)) ≥ δt

And the same properties for G(x , y) holds.

|G(x , y)| ≤ C1

|x − y |n−1

|x − y | > C2d(x) ⇒ G(x , y) = 0

with constants depending on δ, K y ω

PROOF OF IMPROVED POINCARÉ IN L1

(Drelichman-D.)

|f (y)− f |dy ≤∫

|G(x , y)||∇f (x)|dxdy

|G(x , y)| · |∇f (x)|dydx

≤ C∫

∫|x−y |≤Cd(x)

dy|x − y |n−1 |∇f (x)|dx

≤ C‖d∇f‖L1(Ω)

(Drelichman-D.)

|f (y)− f |dy ≤∫

|G(x , y)| · |∇f (x)|dydx

≤ C∫

∫|x−y |≤Cd(x)

dy|x − y |n−1 |∇f (x)|dx

≤ C‖d∇f‖L1(Ω)

(Drelichman-D.)

|f (y)− f |dy ≤∫

|G(x , y)| · |∇f (x)|dydx

≤ C∫

∫|x−y |≤Cd(x)

dy|x − y |n−1 |∇f (x)|dx

≤ C‖d∇f‖L1(Ω)

GENERALIZATION

If Ω satisfies the weaker improved Poincaréinequality

‖f‖L1(Ω) ≤ C1‖dβ∇f‖L1(Ω)n ∀f ∈ L10(Ω)

for some β < 1 then,

∀f ∈ Lp0(Ω) ∃u ∈W 1,p

0 (Ω)n such that

div u = f , ‖d1−β∇u‖Lp(Ω) ≤ C‖f‖Lp(Ω)

GENERALIZATION

‖f‖L1(Ω) ≤ C1‖dβ∇f‖L1(Ω)n ∀f ∈ L10(Ω)

∀f ∈ Lp0(Ω) ∃u ∈W 1,p

0 (Ω)n such that

GENERALIZATION

‖f‖L1(Ω) ≤ C1‖dβ∇f‖L1(Ω)n ∀f ∈ L10(Ω)

for some β < 1

∀f ∈ Lp0(Ω) ∃u ∈W 1,p

0 (Ω)n such that

GENERALIZATION

‖f‖L1(Ω) ≤ C1‖dβ∇f‖L1(Ω)n ∀f ∈ L10(Ω)

∀f ∈ Lp0(Ω) ∃u ∈W 1,p

0 (Ω)n such that

APPLICATION TO Hölder α DOMAINS

In this case β = α

that is,

∀f ∈ Lp0(Ω) ∃u ∈W 1,p

0 (Ω)n such that

div u = f , ‖d1−α∇u‖Lp(Ω) ≤ C‖f‖Lp(Ω)

The case p = 2 can be used to prove existenceand uniqueness for the Stokes equations.

In this case β = αthat is,

∀f ∈ Lp0(Ω) ∃u ∈W 1,p

0 (Ω)n such that

∀f ∈ Lp0(Ω) ∃u ∈W 1,p

0 (Ω)n such that

∀f ∈ Lp0(Ω) ∃u ∈W 1,p

0 (Ω)n such that

FINAL COMMENTS

We have a general methodology to proveinequalities once they are known in cubes.

The decomposition of functions of zero integralis interesting in itself because it might haveother applications (we are working in theapplication of this decomposition to weighted apriori estimates).

FINAL COMMENTS

We have a general methodology to proveinequalities once they are known in cubes.

The decomposition of functions of zero integralis interesting in itself because it might haveother applications (we are working in theapplication of this decomposition to weighted apriori estimates).

THANK YOU FOR YOUR ATTENTION!

HAPPY BIRTHDAY MARTIN!!

improved poincaré and other classic inequalities: a new

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