Sobolev spaces Updated June 1, 2020 Plan 2 Outline: Weak - - PowerPoint PPT Presentation

Sobolev spaces

Updated June 1, 2020

Plan

2

Outline: Weak derivative Relation to ordinary derivative: Nikodym’s theorem Sobolev spaces Morrey’s inequalities and regularity Rademacher’s differentiation theorem

Weak derivative

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Definition f P L1,locpRq is weakly differentiable if Dg P L1,locpRq @φ P C8

c pRq:

ż φg dλ “ ´ ż φ1f dλ g is then called a weak derivative of f. Definition If f P L1,locpRdq for d ě 2 and i P t1, . . . , du then f is weakly differentiable in the i-th coordinate if Dgi P L1,locpRdq @φ P C8

c pRdq:

ż φgi dλ “ ´ ż pBiφqf dλ, where Biφ is the partial derivative of φ in the i-th coordinate.

Motivation

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Integration by parts: if f P C1pRq, then for supppφq Ď ra, bs, ż φ1fdλ “ ż b

a

φ1fdx “ φf ˇ ˇ ˇ

b a ´

ż b

a

φf 1dx “ ´ ż φf 1dλ so weak derivative coincides with ordinary derivative. Q: Is weak derivative even unique? Proposition (Functions in C8

c pRdq separate)

For each f P L1,locpRdq, ´ @φ P C8

c pRdq:

ż φfdλ “ 0 ¯ ñ f “ 0

Proof of Proposition

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First we need: Lemma C8

c pRdq is dense in CcpRdq

Proof: Let χpxq :“ ae´p1´|x|q´1 with a s.t. ş χdλ “ 1 and denote χǫpxq :“ ǫ´dχpx{ǫq For φ P CcpRdq set φǫ :“ φ ‹ χǫ. Then φǫ P C8

c pRdq and

ˇ ˇφpxq ´ φǫpxq ˇ ˇ ď ż ˇ ˇφpx ` zq ´ φpxq ˇ ˇχǫpzqdz By uniform continuity of φ, we get }φ ´ φǫ}8 Ñ 0 as ǫ Ó 0.

Proof of Proposition

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Claim: ş fφdλ “ 0 for all φ P CcpRdq implies @E P LpRdq: E bounded ñ ż

E

fdλ “ 0 First take O Ď Rd bounded and open. Setting φδpxq :“ pδ´1 distpx, Ocqq ^ 1 we get φδ Ò 1O. Dominated Convergence ñ Claim for E :“ O. Dynkin’s π{λ-theorem on Borel subsets of r´r, rsd extends (after r Ñ 8) the above to all bounded Lebesgue-measurable sets. Taking E :“ r´r, rsd X tf ą ǫu forces f “ 0.

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Corollary If f P L1,locpRdq weakly differentiable in the i-th coordinate, then its weak derivative Bif is unique λ-a.e. Proof: If g, h P L1,locpRdq are two weak derivatives of f, then @φ P C8

c pRdq:

ż φpg ´ hqdλ “ 0 Lemma: g ´ h “ 0. Notation: f 1 or Bif denotes the weak derivative (if exists). Distinction weak vs ordinary derivative to be made clear.

Weak vs ordinary derivative

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Q: How much does weak derivative extend the ordinary one? A: Not at all in d “ 1! Lemma We have f P L1,locpRq weakly differentiable ô f is AC on any bounded interval of R The weak derivative then coincides with the ordinary derivative λ-a.e.

Proof of Lemma

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Suppose f AC on ra, bs. Then fpxq ´ fpaq “ ş

ra,xs f 1pyqdy and so

ż φ1fdλ “ ż φ1pxqfpaqdx ` ż

ra,bs

φ1pxq “ fpxq ´ fpaq ‰ dx “ 0 ` ż φ1pxqf 1pyq1tyăxudxdy “ ´ ż φpyqf 1pyqdy for all φ P C8

c pRq.

Conversely: Let g :“ f 1 (weak). Take φ :“ 1 on ra, bs, zero on ra ´ ǫ, b ` ǫsc and linear otherwise. Then φ ‹ χδ Ñ φ and pφ ‹ χδq1 Ñ φ1 pointwise. By Dominated Convergence, ˇ ˇ ˇ ˇ ż

ra,bs

gdλ ´ 1 ǫ ´ż

rb,b`ǫs

´ ż

ra´ǫ,as

¯ fdλ ˇ ˇ ˇ ˇ ď ż

ra´ǫ,as

|g|dλ ` ż

rb,b`ǫs

|g|dλ and so, by Lebesgue differentiation, ż

ra,bs

gdλ “ fpbq ´ fpaq meaning that f is AC with g “ f 1 in ordinary sense.

Gains in d ě 2

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fpxq :“

1 |x|a is in L1,locpRdq for a ă d. Weakly differentiable for

a ă d ´ 1 with @i “ 1, . . . , d: Bifpxq “ ´a xi |x|a`2 Extends to ˜ fpxq :“ ř

iě1 bifpx ´ xiq with tbiuiě1 summable and

txiuiě1 Ď Rd by: Lemma Let tfnuně1 Ď L1,locpRdq be weakly differentiable in the i-th coordinate and assume Df, g P L1,locpRdq s.t., for all C Ď Rd compact, lim

nÑ8

ż

C

|fn ´ f|dλ “ 0 ^ lim

nÑ8

ż

C

|Bifn ´ g|dλ “ 0 Then f is weakly differentiable in the i-th coordinate and Bif “ g. Proof: L1-convergence. Upshot: Weakly differentiable functions in Rd with d ě 2 can have dense singularities!

ACL characterization of weakly-differentiable functions

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Theorem (O. Nikodym 1933) If f P L1,locpRdq is weakly differentiable in the i-th coordinate, then there exists a Borel function g: Rd Ñ R such that f “ g λ-a.e. xi ÞÑ gpx1, . . . , xdq is (locally) AC for all xj P R with j ‰ i Big “ Bif λ-a.e. Note: B. Levi (“Sul principio di Dirichlet,” Rendiconti del Circolo Matematico di Palermo 22 (1906), no. 1, 293–359) defined weak differentiability by above properties.

Vanishing weak derivative

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Lemma Let f P L1,locpRdq and i P t1, . . . , du be such that @φ P C8

c pRdq:

ż pBiφqfdλ “ 0 Then Dh: Rd Ñ R Borel that does not depend on the i-th coordinate and f “ h λ-a.e.

Proof of Lemma

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Assume f Borel (WLOG), i “ 1. Pick η P C8

c pRq with

ş ηdλ “ 1. Using x P R and z P Rd´1, set aφpzq :“ ż φpx, zqdx and ψpx, zq :“ ż

p´8,xs

“ φpy, zq ´ ηpyqaφpzq ‰ dy Then ψ P C8

c pRdq and φpx, zq “ ηpxqaφpzq ` B1ψpx, zq. So

ż φfdλ “ ż ηpxqaφpzqfpx, zqdxdz “ ż φhdλ where hpzq :“ ż ηpyqfpy, zqdy. Proposition above: f “ h λ-a.e.

Proof of Nikodym’s Theorem

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WLOG i “ 1 again, f Borel. Since B1f P L1,locpRdq, E :“ č

ně1

! z P Rd´1 : ż

r´n,ns

ˇ ˇB1fpx, zq ˇ ˇdx ă 8 ) is Borel with λpRd´1 Eq “ 0. Define gpx, zq :“ $ ’ & ’ % ş

r0,xs B1fpy, zqdy,

if z P E ^ x ě 0 ´ ş

rx,0s B1fpy, zqdy,

if z P E ^ x ă 0 0, if z R E Then g Borel with x ÞÑ gpx, zq (locally) AC and g P L1,locpRdq. Also B1g “ B1f λ-a.e. and so ż pB1φqf dλ “ ´ ż φpB1fq dλ “ ´ ż φpB1gq dλ “ ż pB1φqgdλ Lemma: f ´ g “ h λ-a.e. where h does not depend on x1.

Vanishing derivatives implies constancy

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Corollary Let f P L1,locpRdq be weakly differentiable with Bif “ 0 for all i “ 1, . . . , d. Then there is a P R such that f “ a λ-a.e. Proof: If B1f “ 0 then Lemma gave f “ h λ-a.e. where @z P Rd´1 : hpzq :“ ż ηpyqfpy, zqdy If (weak) B2f “ 0 then also (weak) B2h “ 0 by Dominated Convergence and so can iterate. Hence: f is constant λ-a.e.

Rules for weak derivatives

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Weak derivative is a linear operator: Bipaf ` bgq “ apBifq ` bpBigq Product rule: Lemma If f P L1,locpRdq is weakly differentiable and g P C1pRq is bounded, then also fg is weakly differentiable in the i-th coordinate and Bipfgq “ gpBifq ` fpBigq. Chain rule: Need outer function be C1

Higher order derivatives, other domains ...

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Definition Multi-index α “ pα1, . . . , αdq. Then weak derivative of f P L1,locpRdq of order α exists if there is Bαf P L1,locpRdq s.t. @φ P C8

c pRdq:

ż φpBαfqdλ “ p´1q|α| ż pBαφq f dλ where |α| :“ řd

i“1 αi and Bαφ :“ Bα1 1 . . . Bαd d φ.

Lemma (Commutativity of weak derivatives) Given two multi-indices α and β, if any of the weak derivatives Bα`βf, BαBβf

• r

BβBαf exists for f P L1,locpRdq, then all of them exist and are equal. Gradient: ∇f :“ pB1f, . . . , Bdfq Weak derivative in open O Ď Rd: restrict to φ with supppfq Ď O

Sobolev space

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Definition Let k P N0, p P r1, 8s and let O Ď Rd be non-empty open. Then Wk,ppOq :“ č

αPNd 0ď|α|ďk

! f P L1,locpOq: (weak) Bαf exists and Bαf P Lp) is the Sobolev space of k-times weakly-differentiable functions

• n O with p-integrable derivatives.

Note: Linear space, W0,ppOq “ LppOq. Introduced by B. Levi, G. Fubini. Named after S.L. Sobolev.

Sobolev norms

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For f P Wk,ppOq and p P r1, 8q, set }f}Wk,ppOq :“ ˆ ÿ

α: |α|ďk

ż |Bαf|pdλ ˙1{p and (for p “ 8) let }f}Wk,8pOq :“ max

α: |α|ďk inf

• t ě 0: λpO X t|Bαf| ą tu

˘ “ 0 ( Then f ÞÑ }f}Wk,ppOq is a seminorm on Wk,ppOq with }f}Wk,ppOq “ 0 implying f “ 0 λ-a.e. So } ¨ }Wk,ppOq is a norm

• n equivalence classes.

Note: Other equivalent norms are possible, e.g., ř

α: |α|ďk }Bαf}p.

Completeness of Wk,ppOq

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Lemma For all O Ď Rd non-empty and open, all k ě 0 and all p P r1, 8s, the normed space Wk,ppOq is complete. Proof: If tfnu Cauchy in Wk,ppOq, then for all α P Nd

0 with |α| ď k

there is gα such that Bαfn Ñ gα in Lp Lp-convergence implies convergence in L1,locpOq and so ż pBαφqg0dλ “ ż φgαdλ So g0 admits derivatives up to k and gα “ Bαg0.

Approximations by C8

c -functions

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Lemma For all k ě 0 and all p P r1, 8q, C8

c pRdq is dense in Wk,ppRdq

Proof: Pick f P Wk,ppRdq and set ηrpxq :“ e´pr´|x|q´11t|x|ăru. Then ηrf Ñ f in Lp as r Ñ 8 and, since Bαηr Ñ 0, also Bαpηrfq Ñ Bαf. So, may assume supppfq is compact. Next take χ :“ η1 and set χǫpxq :“ ǫ´dχpx{ǫq. Then, for f with compact support, f ‹ χǫ P C8

c pRdq for each ǫ ą 0 and f ‹ χǫ Ñ f

in Wk,ppRdq as ǫ Ó 0.

Boundary issues ...

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Note: Can define Wk,ppRdq as the closure of C8

c pRdq. Fails for

p “ 8 and also in O Ď Rd where Wk,p

0 pOq :“ C8 c pOq

}¨}Wk,ppRdq

is a proper subset of Wk,ppOq. Alternative notation is used for the special case of p “ 2, HkpOq :“ Wk,2pOq ^ Hk

0pOq :“ Wk,p 0 pOq.

Note: these are Hilbert spaces.

Morrey’s inequalities

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Typically: the more assumed about derivatives, the more regular is the function. Theorem For all p P pd, 8s there is cpd, pq P p0, 8q such that @f P C8

c pRdq:

}f}8 ď cpd, pq}f}W1,ppRdq and, for α :“ 1 ´ d{p, @f P C8

c pRdq:

sup

x,yPRd x‰y

|fpxq ´ fpyq| |x ´ y|α ď cpd, pq}∇f}p. Note: True even for d “ 1 “ p but fails for p “ d ě 2: For fǫpxq :“ logpǫ ` |x|´1q1{3gpxq with g P C8

c pRdq s.t. supppgq Ď Bp0, 1{2q.

Key lemma

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Write Bpx, rq :“ ty P Rd : |x ´ y| ă ru. Then: Lemma For all f P C8

c pRdq, all x P Rd and all r ą 0:

ż

Bpx,rq

ˇ ˇfpyq ´ fpxq ˇ ˇdy ď rd d ż

Bpx,rq

|∇fpyq| |y ´ x|d´1 dy Proof: fpx ` zq ´ fpxq “ ż 1 d dtfpx ` tzqdt “ ż 1 z ¨ ∇fpx ` tzqdt and so ˇ ˇfpx ` zq ´ fpxq ˇ ˇ ď ż 1 |z| ˇ ˇ∇fpx ` tzq ˇ ˇdt. Now integrate over z P Bp0, rq . . .

Proof of Lemma continued ...

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ż

Bp0,rq

ˇ ˇfpx ` zq ´ fpxq ˇ ˇdz ď ż

Bp0,rq

´ż 1 |z| ˇ ˇ∇fpx ` tzq ˇ ˇdt ¯ dz “ ż

t|e|“1u

´ż 1 ´ż r ρ ˇ ˇ∇fpx ` tρeq ˇ ˇ ρd´1dρ ¯ dt ¯ σpdeq, where σ is the surface measure on te P Rd : |e| “ 1u. Substitute ˜ ρ :“ tρ to get ż 1 ´ż r ρ ˇ ˇ∇fpx ` tρeq ˇ ˇρd´1dρ ¯ dt “ ż r ˇ ˇ∇fpx ` ˜ ρeq ˇ ˇ ˜ ρd´ż 1

˜ ρ{r

1 td`1 dt ¯ d ˜ ρ “ ż r ˇ ˇ∇fpx ` ˜ ρeq ˇ ˇ ˜ ρd 1 d ˆ´ r ˜ ̺ ¯d ´ 1 ˙ d ˜ ρ ď rd d ż r |∇fpx ` ˜ ρeq| ˜ ρd´1 ˜ ρd´1d ˜ ρ Now integrate over e.

Proof of Morrey’s inequalities

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Let x, y P Rd and set r :“ |x ´ y| ą 0. Denote Ar :“ Bpx, rq X Bpy, rq. By triangle inequality and averaging

• ver z P Ar:

λpArq ˇ ˇfpxq ´ fpyq ˇ ˇ ď ż

Ar

ˇ ˇfpxq ´ fpzq ˇ ˇdz ` ż

Ar

ˇ ˇfpyq ´ fpzq ˇ ˇdz Pick p P pd, 8s and let q be the H¨

• lder conjugate of p. H¨
• lder

then gives ż

Ar

ˇ ˇfpxq ´ fpzq ˇ ˇdz ď ż

Bpx,rq

ˇ ˇfpxq ´ fpzq ˇ ˇdz ď rd d }∇f}p ´ż

Bp0,rq

1 |z|qpd´1q dz ¯1{q The integral convergent since qpd ´ 1q “ d ´ p ´ d p ´ 1 “ d ´ qα Hereby we get . . .

Proof of Morrey’s inequalities

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ˇ ˇfpxq ´ fpyq ˇ ˇ ď 1 λpArq rd d ´ż

Bp0,rq

1 |z|qpd´1q dz ¯1{q looooooooooooooooooomooooooooooooooooooon

ďcrα

}∇f}p Similarly we get ˇ ˇfpxq ˇ ˇ ď 1 λpBpx, 1qq ż

Bpx,1q

ˇ ˇfpxq ´ fpyq ˇ ˇdy looooooooooooooooooomooooooooooooooooooon

ďc}∇f}p

` 1 λpBpx, 1qq ż

Bpx,rq

ˇ ˇfpyq ˇ ˇdy loooooooooooooomoooooooooooooon

ďc}f}p

so }f}8 ď c}f}W1,ppRdq.

An embedding theorem

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Corollary Let p P pd, 8s and α :“ 1 ´ d{p. Then, for a suitable choice of representatives of functions in W1,ppRdq we have W1,ppRdq Ď C0,αpRdq X L8pRdq where C0,αpRdq is the set of all α-H¨

• lder functions on Rd.

Proof: For p ă 8 this follows by density of C8

0 pRdq in W1,ppRdq.

For p “ 8 we only get W1,8

• pRdq. A mollifier argument reduces

f P W1,8pRdq to fηr P W1,8 pRdq.

Rademacher’s differentiation theorem

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For p “ 8 we get α “ 8 so f P W1,8pRdq is globally Lipschitz. Theorem Let f : Rd Ñ R be globally Lipschitz continuous, meaning that M :“ sup

x,yPRd x‰y

|fpxq ´ fpyq| |x ´ y| ă 8. Then f is weakly differentiable with the weak gradient ∇f satisfying }∇f}8 ď M. Moreover, @v P Rd : lim

tÓ0

ˇ ˇ ˇ ˇ fpx ` tvq ´ fpxq t ´ v ¨ ∇fpxq ˇ ˇ ˇ ˇ “ 0 holds for λ-a.e. x P Rd.

Proof of Rademacher’s theorem

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f Lipschitz ñ f AC in each coordinate direction so weak derivative Bif exists for each i. Writing, for v P Rd, Dvφpxq :“ d dtφpx ` vtq ˇ ˇ ˇ

t“0

we get @φ P C8

c pRdq:

ż pDvφqfdλ “ ´ ż φpv ¨ ∇fqdλ So f weakly differentiable in all directions with v ¨ ∇f serving as weak derivative. By the proof of ACL characterization of weak derivative and continuity of f . . .

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. . . we get that @v P Rd DEv P LpRdq with λpEc

vq “ 0 such that

x P Ev implies |∇fpxq| ă 8 and x ` tv P Ev for all t P R and @x P Ev @t P R: fpx ` tvq ´ fpxq “ ż

r0,ts

v ¨ ∇fpx ` svq ds By Lebesgue differentiation theorem, E1

v :“

" x P Ev : lim

tÑ0

ˇ ˇ ˇ fpx ` tvq ´ fpxq t ´ v ¨ ∇fpxq ˇ ˇ ˇ “ 0 *

• beys λpRd E1

vq “ 0.

Now set E :“ Ş

vPQd E1

• v. Then λpEcq “ 0 and, for all x P E, the

limit claim holds. Lipschitz continuity in v extends this to all v P Rd. The Lipschitz condition then ensures @v P Rd : }v ¨ ∇f}8 ď M|v| thus showing }∇f}8 ď M.