notation. the coarea formula for wrwalker/research/coarea.pdf · we will prove the coarea formula...

IntroductionThe classical case.

Modifying the classical proofThe general coarea formula.

References

The Coarea formula for W 1,p functionsA Masters Exam Presentation

Ryan Walker

April 21, 2009

Ryan Walker The Coarea formula for W 1,p functions



References

NotationDefinitionsThe classical coarea formulaExtensions of the coarea formula.

Notation.

Let π : Rn+m → Rn be the projection

π(x , xn+1, ..., xn+m) = x .

For a map f : Ω ⊂ Rn → Rm, let f : Ω → Rn+m be the graphmapping:

f (x) = (x , f (x)).

Let Γf = f (Ω) be the graph of f .




References


Notation.

The Jacobian of f : Rn → Rm, denoted Jf (x), is obtained via thepolar decomposition theorem for linear maps and is given by:

Jf (x) =√

det (Df )TDf .

For f : Ω → Rm, we say f ∈ W 1,p(Ω, Rm) provided∫Ω(|u|p + |∇u|p)dx < ∞.




References


The Lorentz spaces: a “refined scale” of Lp spaces.

Definition

If f is a measurable function on Ω, given the distribution function

µf (s) = Ln(x ∈ Ω : |f (x)| > s)

set

‖f ‖Lm,1(Ω) =

∫ ∞

0µf (s)

1/mds.

Then f belongs to the Lorentz space Lm,1(Ω) if ‖f ‖Lm,1(Ω) < ∞.




References


The Lorentz spaces: a “refined scale” of Lp spaces.

L1,1(Ω) = L1(Ω).

If p > m > 1, Lp(Ω) ( Lm,1(Ω) ( Lm(Ω)




References


Hausdorff measure

Definition

Let 0 ≤ δ ≤ ∞ and 0 ≤ q < ∞. The q dimensional Hausdorff-δ measureof the set E , Hq

δ (E ), is the infimum of the sums:

∞∑j=1

αq

(diamEj

2

)q

over all countable coverings of E by sets Ej∞j=1 with diam(Ej) ≤ δ, andαq the ball constant.




References


Hausdorff content and measure

Definition

We call Hq∞(E ) the q dimensional Hausdorff content of E .

Definition

The q dimensional Hausdorff measure of E , denoted Hq(E ), is defined as

limδ→0

Hqδ (E ).




References


Hausdorff countable recitifiability

Definition

For integer q a set E ⊂ Rn is said to be countably Hq rectifiable if thereexist subsets Ek ⊂ Rq and Lipschitz mappings gk : Ek → Rn with theproperty that

Hq (E\ ∪∞k=1 gk(Ek)) = 0.




References


The classical coarea formula

Theorem

For Lipschitz f : Rn → Rm with n ≥ m and A ⊂ Rm measurable:∫A

Jf (x)dx =

∫Rm

Hn−m(A ∩ f −1(y))dy .




References


Curvilinear Fubini

The coarea formula is a generalization of Fubini’s theorem.To see this, let Π : Rm ×Rm−n → Rm be the projection map onto the firstm-components. Then using the polar decomposition theorem, Jf (x) = 1and for a fixed y ∈ Rm,

Hn−m(A ∩ f −1(y)) =

∫Rn−m

χAy (x)dx

where Ay = x ∈ A|x = (y , x) where x ∈ Rn−m and so

Ln(A) =

∫Rm

∫Rn−m

χAy (x)dxdy .




References





References


Spherical coordinates integration formula.

For continuous, integrable functions f : Rm → R, the spherical coordinatesintegration formula is a special case of the coarea formula:∫

Rn

f dx =

∫ ∞

0

(∫∂B(x0,r)

f ds

)dr .

This works by setting u(x) = |x − x0| in the equivalent coarea formulation∫Rn

f (x)Ju(x)dx =

∫Rm

(∫u−1(y)

f (x)dHn−m

)dy .




References


Main Question.

The purpose of this talk is to establish the coarea formula for functionsmore general than Lipschitz functions.

In particular, we’ll investigate the possibility of taking f ∈ W 1,ploc (Ω, Rm).




References


Possible extensions.

The classical coarea formula is easily extend forf ∈ W 1,p(Ω ⊂ Rn, Rm) when p > n.

In general, n may be much larger than m. Can this estimate besharpened?




References


Cases.

Let f ∈ Lp(Ω, Rm):

1 |∇f | ∈ Lp p > m coarea works.

2 |∇f | ∈ Lm fails.: Possible to construct a continuous W 1,m(Rn)function with Jf = 0 a.e. which maps every set of the form I × Rn−m

onto an m-cube. This breaks the coarea formula.

3 But if |∇f | ∈ Lm,1 (Lorentz space) the coarea formula holds.




References





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The goal.

Theorem (Main Result)

Suppose that 1 ≤ m ≤ n and that f ∈ W 1,1loc (Ω; Rm), and that

|∇f | ∈ Lm,1(Ω). Then f −1(y) is countably Hn−m rectifiable for a.e.y ∈ Rm, Γf is countably Hn rectifiable and for every measurable E ⊂ Ω::∫

AJf (x)dx =

∫Rm

Hn−m(A ∩ f −1(y))dy .

The proof and development of this theorem is primarily from the paper:Maly, J., Swanson, D., and Ziemer, W. The coarea formula for Sobolevmappings, Trans. Amer. Math. Soc. 355 (2003), 477-492.




References


Outline for the rest of the talk.

We will prove the coarea formula for f ∈ W 1,p(Ω, Rm) with p > m andthen show how the same proof can be extended for this more general case.

1 Sketch the classical proof.

2 Generalize the classical proof.

3 Prove the case for p > m.

4 Indicate how to modify our proof for the f ∈ W 1,1,∇f ∈ Lm,1 case.




References

Idea of the proofProof sketchProof of the Eilenberg inequality.

How do we prove the classical coarea formula?

Use differentiability to decompose the set A = Z ∪ A+ ∪ A0 where

1 Z is the Ln measure zero set where Jf does not exist.

2 A+ is the set where Jf > 0.

3 A0 is the set where Jf = 0.

Then:

1 Use a Lusin type property to show that the coarea formula holds onthe zero measure set Z .

2 Use an affine approximation to show that the coarea formula holds onthe positive level set A+.

3 Use an enlargement procedure to reduce the A0 case to the A+ case.




References


Proof sketch.

Proof. (Sketch.) By Rademacher, since f is Lipschitz, it is differentiablea.e.. Hence the set Z where Jf (x) is undefined has Ln measure zero.We note that the coarea formula will hold on Z if f satisfies the followingLusin property:

Definition (Lusin Property (N))

A mapping f : Rn → Rm with m ≤ n will be said to satisfy the Lusin (N)property if

(N) Ln(B) = 0 ⇒∫

Rm

Hn−m(B ∩ f −1(y))dHm(y) = 0

holds for every measurable B ⊂ Rn.




References


Proof sketch (cont.)

Now recall the famous Eilenberg inequality:

Theorem (Eilenberg Inequality)

For m ≤ d ≤ n, A ⊂ Rn and f ∈ W 1,p(A, Rn) with p > n we have:

(E )

∫ ∗

Rm

Hd−m(A ∩ f −1(y))dHm(y) ≤ cHd(A)1−mp · (

∫Rn

|∇f |p)mp .

Note: For Lipschitz f , (∫

Rn |∇f |p)m/p = (Lip(f ))m.




References



The Eilenberg inequality shows that f satisfies (N) and hence that bothsides of the coarea formula will be zero on the set Z .

For the case of the positive level set A+, we may use the fact that Jf (x) isinvertible and the following lemma to produce an arbitrarily close affineapproximation to f :




References



Lemma

Let t > 1. Then there is a countable collection of Borel sets Ak∞k=1 suchthat

1 A+ = ∪∞k=1Ak

2 f |Akis one-to-one

3 For all k there is a symmetric nonsingular linear map Tk : Rn → Rm

so that

Lip(f |Ek f −1

k ) ≤ t

Lip(fk (f |Ek)−1) ≤ t

| det Tk |tn

≤ (Jf )|Ek≤ tn| det Tk |.




References



For the case of the zero level set A0, we can show that∫Rm

Hn−m(A ∩ f −1(y))dy = 0

by considering the map fε(x) = f (x) + εx , noting that the coarea formulaapplies to Jfε by the previous case, and taking a limit.




References


Proof of the Eilenberg inequality.

Theorem (Eilenberg Inequality)

For m ≤ d ≤ n, A ⊂ Rn and f ∈ W 1,p(A, Rn) with p > n we have:

(E )

∫ ∗

Rm

Hd−m(A ∩ f −1(y))dHm(y) ≤ cHd(A)1−mp · (

∫Rn

|∇f |p)mp .

Proof. Cover A by closed balls Bij so that A ⊂ ∪i ,jBij , diam(Bij) ≤ 1/j ,and

∑i Ln(A) + 1/j .




References


Eilenberg inequality proof. (cont.)

Let B be a ball in Rn. A version of the Sobolev embedding theorem gives:

oscB(f ) ≤ C (diam(B))1−np

(∫B|∇f |p

) 1p

.

For each i , j , the isodiametric inequality implies:

Ld(f (Bij)) ≤ C [diam(f (B))]d

and so

Ld(f (Bij)) ≤ C [r1−n/pij ]d

(∫Bij

|∇f |p) d

p




References



Let gij(y) = Cdiam(Bij)χf (Bij )(y).∫Rm

Hn−d(A ∩ f −1(y))dy =

∫Rm

limj→∞

Hn−d1/j (A ∩ f −1(y))dy

≤∫

Rm

lim infj→∞

∞∑i=1

gij(y)dy

≤ lim infj→∞

∞∑i=1

∫Rm

gij(y)dy

= C lim infj→∞

∞∑i=1

[diam(Bij)]n−dLd(f (Bij))




References



≤ C lim infj→∞

∞∑i=1

[diam(Bij)]n−d · [diam(Bij)]

d−nd/p ·

(∫Bij

|∇f |p) d

m

= C lim infj→∞

∞∑i=1

[diam(Bij)n]1−d/p

(∫Bij

|∇f |p) d

p

≤ CLn(A)1−d/p ·(∫

Rn

|∇f |p) d

p

.




References


What changes when the class of functions is more generalthan L∞?

Differentiability. If f ∈ W 1,ploc (Rn, Rm) for p > n, the Calderon

theorem says that f is still differentiable a.e.. Beyond this class, wemust modify our notion of differentiability.

Lusin. Without the Eilenberg inequality (which holds only whenp > n) property (N) may fail to hold.




References

DifferentiabilityA new Lusin property.Content and capacity.

Approximate differentiability

In the general context, we can replace a.e. differentiability with the weakernotion of approximate differentiability:

Definition

A measurable function f : Rn → Rm is said to be approximatelydifferentiable at a point x if for all ε > 0

limr↓0

|y ∈ Rn : |f (y)− f (x)− Df (x) · (y − x)| > ε|y − x | ∩ Br (x)|rn

= 0.




References


A new decomposition result.

Now we will argue that we can decompose A as

A = A0 ∪ (∪∞i=1Ai )

where |A0| = 0 and f |Aiis Lipschitz for each i .

We’ll need two lemmas first.




References


A lemma.

Lemma

If f ∈ W 1,1(Rn, Rm), then there is F with |F | = 0 so that for allx ∈ Rn\F ,

1

rn

∫Br (x)

|u(y)− u(x)− Du(x) · (y − x)|dy ≤ o(r)




References


Proof of lemma.

Proof. Since Du ∈ L1(Rn) by the Lebesgue differentiation theorem, thereexists an F with |F | = 0 so that

limr↓0−∫

Br (x)|Du(y)− Du(x)|dy = 0

and

limr↓0−∫

Br (x)|u(y)− uBr (x)|dy = 0

for all x ∈ Rn\F .




References


Proof of lemma (cont.)

By a variant of the Poincare inequality:

−∫

Br (x)|u(y)−u(x)−Du(x)·(y−x)|dy ≤ Cr−

∫Br (x)

|Du(y)−Du(x)|dy = o(r)




References


Another lemma.

Lemma

If f ∈ W 1,1(Rn, Rm) then there exists F with |F | = 0 such that f isapproximately differentiable for all x ∈ Rn\F .




References


Proof of lemma.

Proof. Let F be the measure zero set of the previous lemma. Then for allx ∈ Rn\F

1

rn

∫Br (x)

|u(y)− u(x)− Du(x) · (y − x)|dy = o(1)r .

Now let ε > 0 and

Eε = y : |u(y)− u(x)− Du(x)(y − x)| > ε|y − x |




References


Proof of lemma.

o(1)r =1

rn

∫Br (x)

|u(y)− u(x)− Du(x) · (y − x)|dy

≥ 1

rn

∫Eε∩Br (x)

ε|y − x |dy

≥ Cr|Eε ∩ Br (x)|

rn

Hence|Eε ∩ Br (x)|

rn→ 0 as r → 0

and so f is approximately differentiable.




References


The new decomposition theorem.

Theorem

If f : A ⊂ Rn → Rm satisfies

ap limx→a

|f (x)− f (a)||x − a|

< ∞

for all a ∈ A then A = ∪∞i=1Ai of measurable sets such that f |Aiis

Lipschitz.




References


Proof of theorem.

Proof. Set

θ(u, v) :=|B(u, |u − v |) ∩ B(v , |u − v |)|

|u − v |n

for u 6= v ∈ Rn and note that 0 < θ < 1.




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Proof of theorem.

Then set:

Q(u, i , j) : = B(u, r) ∩ x : x /∈ A or |f (x)− f (u)| > j |x − y |

Bj : = A ∩ u : |Q(u, r , j)| < θrn

2, 0 < r <

1

j.

Then Bj is measurable and A = ∪∞j=1Bj .




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Proof of theorem.

Now we claim f |Bjis locally Lipschitz. To see why, let u 6= v ∈ Bj ,

r = |u − v | < 1/j . Note that

|Q(u, r , j) ∪ Q(v , r , j)| < θrn

2+

θrn

2= θrn = |B(u, r) ∩ B(v , r)|.

Hence, there is x ∈ [B(u, r) ∩ B(v , r)]\[Q(u, r , j) ∪ Q(v , r , j)] so that

|f (x)− f (u)| ≤ j |x − y | ≤ jr

|f (x)− f (v)| ≤ j |x − y | ≤ jr .




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Proof of theorem.

By the triangle inequality:

|f (u)− f (v)| ≤ |f (x)− f (u)|+ |f (x)− f (v)| ≤ 2jr = 2j |u − v |.

So if we further decompose each Bj into sets of diameter less than 1/j wehave a decomposition of A so that f is locally Lipschitz on each setBij .




References


The W 1,1 decomposition corollary.

Since the approximate differentiability of f ∈ W 1,1(Ω, Rm) implies that forall x :

ap limx→a

|f (x)− f (a)||x − a|

< ∞

we have:

Corollary

If f ∈ W 1,1loc (Rn, Rm) there exists disjoint sets Ek in Rn so that

Ln(Rn\ ∪ Ek) = 0

fk = f |Ekis Lipschitz on Ek .




References


Lusin property (N).

It turns out that a slightly different Lusin condition will facilitate a proofof the more general case of the coarea formula.

Definition

A function f : Rn → Rm will be said to satisfy the Lusin condition (N) if

(N) Hn(E ) = 0 ⇒ Hn(f (E )) = 0




References


Application of (N).

Theorem

Let f ∈ W 1,1loc (Rn; Rm) with 1 ≤ m ≤ n and suppose that f satisfies the

Lusin condition (N). Then f −1(y) is countably Hn−m rectifiable foralmost all y ∈ Rm, the graph Γf is countably Hn rectifiable and the coareaformula holds for all measurable A ⊂ Ω.

Proof. Decompose A = F ∪ (∪Ak) where fk = f |Akis Lipschitz as in the

W 1,1 decomposition corollary and F is Ln measure zero. Then the coareaformula will certainly hold for every E ⊂ Ak since this is just the classicalcase.




References


Proof of the theorem.

Now we must argue that the coarea formula will hold on the Ln measurezero set F where Jf (x) does not exist.

Suppose that Ln(E ) = 0. Let π : Rn+m → Rn and ρ : Rn+m → Rm be theprojections:

π(x , xn+1, ..., xn+m) = x and ρ(x , xn+1m, ..., xn+m) = (xn+1, ..., xn+m).

Since projections are clearly Lipschitz, we apply the Eilenberg inequality:∫Rm

Hn−m(f (E ) ∩ ρ−1(y))dHm(y) ≤ C · (Lip(ρ))mHn(f (E )) = 0




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Proof of the theorem (cont.)

Claim:π(f (E ) ∩ ρ−1(y)) = E ∩ f −1(y).

“⊃” x ∈ E ∩ f −1(y) ⇒ f (x) = y ⇒ ρ(x , y) = x ⇒ x ∈ ρ−1(y).

“⊂” x ∈ π(f (E ) ∩ ρ−1(y)) ⇒ ρ(x) = y = f (x) ⇒ x ∈ f −1(y) ∩ E .




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Proof of the theorem (cont.)

Since Hausdorff measures do not increase on projection:∫Rm

Hn−m(E ∩ f −1(y))dHm(y) =

∫Rm

Hn−m(π(f (E ) ∩ ρ−1(y)))dHm(y)

≤∫

Rm

Hn−m(f (E ) ∩ ρ−1(y))dHm(y)

= 0.

Hence the coarea formula holds for each component of the decompositionA = F ∪ (∪kAk).




References


A general criterion for (N).

We now show that f ∈ W 1,ploc (Ω, Rm) with p > m (and more generally

f ∈ W 1,1loc (Ω, Rm) with |∇f | ∈ Lm,1(Ω)) satisfies the Lusin property (N).

To do this, we provide a more general criterion for condition (N).




References


A general criterion for (N).

Theorem

Let 1 ≤ m ≤ n, f : Rn → Rm, and suppose θ ∈ L1loc(Rn) is s.t.

Hn−m∞ (π(Gf ∩ B(z , r))) ≤ r−m

∫π(Γf ∩B(z,4r))

θ(x)dx

∀z ∈ Rn+m, r > 0. Then ∃C = C (n,m) s.t.

Hn(f (E )) ≤ C

∫E

θ(x)dx

for all Ln measurable E ⊂ Rn. In particular, f satisfies condition (N).




References


Content lemma.

Lemma

Suppose m ≤ d ≤ m + n and let E ⊂ Rn+m. Then:

Hd∞(E ) ≤ C (diamE )mHd−m

∞ (π(E ))

where C is a constant depending on m, n and d.




References


Proof of general criterion for condition (N).

Proof. Define a set function σ on Rn+m by

σ(E ) =

∫π(Γf ∩E)

θ(x)dx .

Using the content lemma with d = n :

Hn∞(Γf ∩ B(z , r)) ≤ CrmHn−m

∞ (π(Γf ∩ B(z , r)))

≤ C


θ(x)dx

= Cσ(B(z , 4r))

for any z ∈ Rn+m and r > 0.




References


Proof of general criterion (cont.)

Recall that for E ⊂ Rn of finite Hq measure the q-dimensional upperdensity of x ∈ A with respect to Hq

δ measure is bounded below:

1

2q≤ lim

r→0

Hq∞(B(x , r) ∩ A)

α(q)rq≤ 1.

Hence:limr→0

r−nHn∞(Γf ∩ B(z , r)) ≥ C

for Hn a.e. z ∈ Γf . So

limr→0

r−nσ(B(z , r)) ≥ C

for Hn a.e. z ∈ Γf .




References



Can easily show the projection π(Γf ∩ E ) is Lebesgue measurable forevery Borel E .

⇒ σ is a Borel measure on the Σ-algebra of Rn+m.

⇒ We can then extend it to be a regular Borel outer measure on all ofRn+m:

σ(E ) = infσ(B) : E ⊂ B,B is a borel set.

⇒ Since the integrand function θ is locally integrable, σ will be finite oncompact subsets of Rn+m which makes σ a Radon measure.




References



Use the Besicovitch lemma to cover of E with a family of balls Bi sothat

σ(E\ ∪ Bi ) = 0.

Then:

σ(E ) =∞∑i=1

σ(Bi )

≥∞∑i=1

Crni

≥ CHn(E ).




References



Now let G be a Borel set containing E . Then f (E ) ⊂ G ×Rm is Borel and

Hn(f (E )) ≤ C σ(f (E ))

≤ Cσ(G × Rm)

= C

∫G

θ(x)dx

Take the infimum over all such G to obtain:

Hn∞(E ) ≤ C

∫E

θ(x)dx .




References


Capacity.

To proceed, we need a capacitary lemma.

Definition

For 1 ≤ p < ∞, the p-capacity γp(E ) of a set E ⊂ RN is:

γp(E ) = infu∈W

1,p(Rn)0

∫

Rn

|∇u|pdx : u ≥ 1 on a neighborhood of E

Idea: Gauge the size of a set with test functions rather than set coverings.




References


Content and capacity.

We now relate Hausdorff content and capacity through the followingtheorem.

Theorem

Suppose that 1 ≤ m < p and E ⊂ RN . Then

Hn−m∞ (E ) ≤ Cγp(E )

where C = C (n,m, p).




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A content estimate.

From this result, we get the following corollary.

Corollary

Suppose that 1 ≤ m < p, E ⊂ Rn, and u ∈ W 1,p(Rn) with u ≥ 1 on E.Then

Hn−m∞ (E ) ≤ C

∫Rn

(|∇u|p + |u|p)dx




References

The W 1,p , p > m case.The Lm,1 case.

The W 1,p, p > m case.

Theorem

The coarea formula holds for f ∈ W 1,ploc (Ω, Rm) when p > m.




References


Proof.

We will show that f ∈ W 1,p(Ω, Rm) satisfies the Lusin (N) property. Ifthis is the case then the coarea formula will hold by our theorem.

We show that f satisfies the Lusin (N) property by verifying that fsatisfies the conditions of the general criterion. Need to prove:∃θ ∈ L1

loc(Rn) so that

Hn−m∞ (π(Γf ∩ B(z , r))) ≤ r−m


θ(x)dx .




References


Proof.

Proof. Let z ∈ Rn+m and r > 0. Write z = (x0, y0). Then:

Γf ∩ B(z , r) ⊂ Γf ∩ [B(x0, r)× B(y0, r)]

andπ(Γf ∩ B(z , r)) ⊂ B(x0, r) ∩ f −1(B(y0, r)).

Let E := B(x0, r) ∩ f −1(B(y0, r)). Define

E =1

2r(E − x0) = x ∈ Rn : x0 + 2rx ∈ E.




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Proof (cont.)

By properties of Hausdorff measure:

Hn−m∞ (E ) = (2r)m−nHn−m

∞ (E )

and soHn−m∞ (π(Γf ∩ B(z , r))) ≤ Crn−mHn−m

∞ (E ).

Now if ξ ∈ E we have:

|ξ| ≤ 1

2and

|f (x0 + 2rξ)− f (x0)|2r

≤ 1

2.




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Proof (cont.)

Consider the test function uη ∈ W 1,p(Rn) where:

u(ξ) = 2

(1− |f (x0 + 2rξ)− f (x0)|

2r

)+

and η is a smooth cutoff function so that

χB(0,1/2) ≤ η ≤ χB(0,1)




References


Proof (cont.)

We notice that uη ≥ 1 on E and so by the content-capacity corollary:

Hn−m∞ (E ) ≤ C

∫Rn

(|uη|p + |∇uη|p)dx

≤ C

∫B(0,1)∩u>0

(1 + |∇u|p)dx

Now perform the change variable x = x0 + 2rξ:

Hn−m∞ (E ) ≤ Cr−n

∫B(x0,2r)∩f −1(B(y0,2r))

(1 + |∇u|p)dx




References


Proof (cont.)

Now notice:

B(x0, 2r) ∩ f −1(B(y0, 2r)) ⊂ π(Γf ∩ B(z , 4r))

So:

Hn−m∞ (π(Γf ∩ B(z , r))) ≤ Crn−mHn−m

∞ (E )

≤ Crn−mr−n

∫B(x0,2r)∩f −1(B(y0,2r))

(1 + |∇u|p)dx

≤ Cr−m


(1 + |∇u|p)dx




References


Proof (cont.)

Now, set θ = C (1 + |∇u|p). Then θ verifies the hypotheses of generalcriterion theorem.




References


The Lm,1 case.

In the content-capacity corollary above, we related the Hausdorff contentof E to the p-power integral of u and ∇u where u ∈ W 1,p, p > m.

It turns out that a more general result will hold but we will need somefacts about Young functions.

Definition

A convex, nonnegative function F : [0,∞) → R is a Young function if itsatisfies

F (t) = 0 ⇔ t = 0.




References


Young functions and capacity.

Theorem

Let m > 1, u ∈ W 1,1loc (Rn, Rm) with |∇u| ∈ Lm,1 and u ≥ 1 on E. Then

there is a unique Young function F so that:

Hn−m∞ (E ) ≤ C

∫Rn

(F (|∇u|) + F (|u|))dx .




References


How to modify the proof of the p > m case.

Remark

For f ∈ W 1,1loc (Ω, Rm) and |∇f | ∈ Lm,1, we can modify the proof above by

simply replacing:

Hn−m∞ (E ) ≤ C

∫Rn(|uη|p + |∇uη|p)dx

withHn−m∞ (E ) ≤ C

∫Rn(F (|∇u|) + F (|u|))dx .

The rest of the proof follows in the same way as before.




References

References:

Fanghua, L. and Xiaoping, Y. Geometric Measure Theory: AnIntroduction. Science Press, Beijing, 2002.

Maly, J., Swanson, D., and Ziemer, W. The coarea formula forSobolev mappings, Trans. Amer. Math. Soc. 355 (2003), 477-492.

Maly, J., Swanson, D., and Ziemer, W. Fine behavior of functionswith gradients in a Lorentz space, Studia Mathematica. 355 (2009),33-71.


notation. the coarea formula for wrwalker/research/coarea.pdf · we will prove the coarea formula...

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