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 3 Definition f P L 1,loc p R q is weakly differentiable if ż ż D g P L 1,loc p R q @ φ P C 8 φ 1 f d λ c p R q : φ g d λ “ ´ g is then called a weak derivative of f . Definition If f P L 1,loc p R d q for d ě 2 and i P t 1, . . . , d u then f is weakly differentiable in the i -th coordinate if ż ż D g i P L 1,loc p R d q @ φ P C 8 c p R d q : φ g i d λ “ ´ pB i φ q f d λ , where B i φ is the partial derivative of φ in the i -th coordinate.

Motivation 4 Integration by parts: if f P C 1 p R q , then for supp p φ q Ď r a , b s , ż b ż b ż b ż ˇ φ 1 f d λ “ φ 1 f d x “ φ f φ f 1 d x “ ´ φ f 1 d λ a ´ ˇ ˇ a a so weak derivative coincides with ordinary derivative. Q: Is weak derivative even unique? Proposition (Functions in C 8 c p R d q separate) For each f P L 1,loc p R d q , ż ´ ¯ @ φ P C 8 c p R d q : φ f d λ “ 0 ñ f “ 0

Proof of Proposition 5 First we need: Lemma C 8 c p R d q is dense in C c p R d q Proof: Let χ p x q : “ a e ´p 1 ´| x |q ´ 1 with a s.t. ş χ d λ “ 1 and denote χ ǫ p x q : “ ǫ ´ d χ p x { ǫ q For φ P C c p R d q set φ ǫ : “ φ ‹ χ ǫ . Then φ ǫ P C 8 c p R d q and ż ˇ ˇ ď ˇ ˇ ˇ ˇ φ p x q ´ φ ǫ p x q ˇ φ p x ` z q ´ φ p x q ˇ χ ǫ p z q d z By uniform continuity of φ , we get } φ ´ φ ǫ } 8 Ñ 0 as ǫ Ó 0.

Proof of Proposition 6 f φ d λ “ 0 for all φ P C c p R d q implies ş Claim: ż @ E P L p R d q : E bounded ñ f d λ “ 0 E First take O Ď R d bounded and open. Setting φ δ p x q : “ p δ ´ 1 dist p x , O c qq ^ 1 we get φ δ Ò 1 O . Dominated Convergence ñ Claim for E : “ O . Dynkin’s π { λ -theorem on Borel subsets of r´ r , r s d extends (after r Ñ 8 ) the above to all bounded Lebesgue-measurable sets. Taking E : “ r´ r , r s d X t f ą ǫ u forces f “ 0.

7 Corollary If f P L 1,loc p R d q weakly differentiable in the i-th coordinate, then its weak derivative B i f is unique λ -a.e. Proof: If g , h P L 1,loc p R d q are two weak derivatives of f , then ż @ φ P C 8 c p R d q : φ p g ´ h q d λ “ 0 Lemma: g ´ h “ 0. Notation: f 1 or B i f denotes the weak derivative (if exists). Distinction weak vs ordinary derivative to be made clear.

Weak vs ordinary derivative 8 Q: How much does weak derivative extend the ordinary one? A: Not at all in d “ 1! Lemma We have f P L 1,loc p R q 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 9 ş r a , x s f 1 p y q d y and so Suppose f AC on r a , b s . Then f p x q ´ f p a q “ ż ż ż φ 1 f d λ “ φ 1 p x q f p a q d x ` φ 1 p x q “ ‰ f p x q ´ f p a q d x r a , b s ż ż φ 1 p x q f 1 p y q 1 t y ă x u d x d y “ ´ φ p y q f 1 p y q d y “ 0 ` for all φ P C 8 c p R q . Conversely: Let g : “ f 1 (weak). Take φ : “ 1 on r a , b s , zero on r a ´ ǫ , b ` ǫ s c and linear otherwise. Then φ ‹ χ δ Ñ φ and p φ ‹ χ δ q 1 Ñ φ 1 pointwise. By Dominated Convergence, ˇ ˇ ż g d λ ´ 1 ´ż ż ż ż ¯ ˇ ˇ ´ f d λ ˇ ď | g | d λ ` | g | d λ ˇ ˇ ǫ ˇ r a , b s r b , b ` ǫ s r a ´ ǫ , a s r a ´ ǫ , a s r b , b ` ǫ s and so, by Lebesgue differentiation, ż g d λ “ f p b q ´ f p a q r a , b s meaning that f is AC with g “ f 1 in ordinary sense.

Gains in d ě 2 10 | x | a is in L 1,loc p R d q for a ă d . Weakly differentiable for 1 f p x q : “ a ă d ´ 1 with x i @ i “ 1, . . . , d : B i f p x q “ ´ a | x | a ` 2 Extends to ˜ f p x q : “ ř i ě 1 b i f p x ´ x i q with t b i u i ě 1 summable and t x i u i ě 1 Ď R d by: Lemma Let t f n u n ě 1 Ď L 1,loc p R d q be weakly differentiable in the i-th coordinate and assume D f , g P L 1,loc p R d q s.t., for all C Ď R d compact, ż ż lim | f n ´ f | d λ “ 0 ^ lim |B i f n ´ g | d λ “ 0 n Ñ8 n Ñ8 C C Then f is weakly differentiable in the i-th coordinate and B i f “ g. Proof: L 1 -convergence. Upshot: Weakly differentiable functions in R d with d ě 2 can have dense singularities!

ACL characterization of weakly-differentiable functions 11 Theorem (O. Nikodym 1933) If f P L 1,loc p R d q is weakly differentiable in the i-th coordinate, then there exists a Borel function g : R d Ñ R such that f “ g λ -a.e. x i ÞÑ g p x 1 , . . . , x d q is (locally) AC for all x j P R with j ‰ i B i g “ B i f 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 12 Lemma Let f P L 1,loc p R d q and i P t 1, . . . , d u be such that ż @ φ P C 8 c p R d q : pB i φ q f d λ “ 0 Then D h : R d Ñ R Borel that does not depend on the i-th coordinate and f “ h λ -a.e.

Proof of Lemma 13 Assume f Borel (WLOG), i “ 1. Pick η P C 8 ş c p R q with η d λ “ 1. Using x P R and z P R d ´ 1 , set ż a φ p z q : “ φ p x , z q d x and ż “ ‰ ψ p x , z q : “ φ p y , z q ´ η p y q a φ p z q d y p´8 , x s Then ψ P C 8 c p R d q and φ p x , z q “ η p x q a φ p z q ` B 1 ψ p x , z q . So ż ż ż φ f d λ “ η p x q a φ p z q f p x , z q d x d z “ φ h d λ where ż h p z q : “ η p y q f p y , z q d y . Proposition above: f “ h λ -a.e.

Proof of Nikodym’s Theorem 14 WLOG i “ 1 again, f Borel. Since B 1 f P L 1,loc p R d q , ż ! ) z P R d ´ 1 : č ˇ ˇ E : “ ˇ B 1 f p x , z q ˇ d x ă 8 r´ n , n s n ě 1 is Borel with λ p R d ´ 1 � E q “ 0. Define $ ş r 0, x s B 1 f p y , z q d y , if z P E ^ x ě 0 ’ & ş g p x , z q : “ ´ r x ,0 s B 1 f p y , z q d y , if z P E ^ x ă 0 ’ if z R E % 0, Then g Borel with x ÞÑ g p x , z q (locally) AC and g P L 1,loc p R d q . Also B 1 g “ B 1 f λ -a.e. and so ż ż ż ż pB 1 φ q f d λ “ ´ φ pB 1 f q d λ “ ´ φ pB 1 g q d λ “ pB 1 φ q g d λ Lemma: f ´ g “ h λ -a.e. where h does not depend on x 1 .

Vanishing derivatives implies constancy 15 Corollary Let f P L 1,loc p R d q be weakly differentiable with B i f “ 0 for all i “ 1, . . . , d. Then there is a P R such that f “ a λ -a.e. Proof: If B 1 f “ 0 then Lemma gave f “ h λ -a.e. where ż @ z P R d ´ 1 : h p z q : “ η p y q f p y , z q d y If (weak) B 2 f “ 0 then also (weak) B 2 h “ 0 by Dominated Convergence and so can iterate. Hence: f is constant λ -a.e.

Rules for weak derivatives 16 Weak derivative is a linear operator: B i p af ` bg q “ a pB i f q ` b pB i g q Product rule: Lemma If f P L 1,loc p R d q is weakly differentiable and g P C 1 p R q is bounded, then also fg is weakly differentiable in the i-th coordinate and B i p fg q “ g pB i f q ` f pB i g q . Chain rule: Need outer function be C 1

Higher order derivatives, other domains ... 17 Definition Multi-index α “ p α 1 , . . . , α d q . Then weak derivative of f P L 1,loc p R d q of order α exists if there is B α f P L 1,loc p R d q s.t. ż ż @ φ P C 8 c p R d q : φ pB α f q d λ “ p´ 1 q | α | pB α φ q f d λ where | α | : “ ř d 1 . . . B α d i “ 1 α i and B α φ : “ B α 1 d φ . Lemma (Commutativity of weak derivatives) Given two multi-indices α and β , if any of the weak derivatives B α ` β f , B α B β f B β B α f or exists for f P L 1,loc p R d q , then all of them exist and are equal. Gradient: ∇ f : “ pB 1 f , . . . , B d f q Weak derivative in open O Ď R d : restrict to φ with supp p f q Ď O

Sobolev space 18 Definition Let k P N 0 , p P r 1, 8s and let O Ď R d be non-empty open. Then ! f P L 1,loc p O q : (weak) B α f exists and B α f P L p ) č W k , p p O q : “ α P N d 0 0 ď| α |ď k is the Sobolev space of k -times weakly-differentiable functions on O with p -integrable derivatives. Note: Linear space, W 0, p p O q “ L p p O q . Introduced by B. Levi, G. Fubini. Named after S.L. Sobolev.

Sobolev norms 19 For f P W k , p p O q and p P r 1, 8q , set ˙ 1 { p ˆ ż ÿ |B α f | p d λ } f } W k , p p O q : “ α : | α |ď k and (for p “ 8 ) let t ě 0: λ p O X t|B α f | ą t u � ˘ ( } f } W k , 8 p O q : “ max “ 0 α : | α |ď k inf Then f ÞÑ } f } W k , p p O q is a seminorm on W k , p p O q with } f } W k , p p O q “ 0 implying f “ 0 λ -a.e. So } ¨ } W k , p p O q is a norm on equivalence classes. Note: Other equivalent norms are possible, e.g., ř α : | α |ď k }B α f } p .

Completeness of W k , p p O q 20 Lemma For all O Ď R d non-empty and open, all k ě 0 and all p P r 1, 8s , the normed space W k , p p O q is complete. Proof: If t f n u Cauchy in W k , p p O q , then for all α P N d 0 with | α | ď k there is g α such that B α f n Ñ g α in L p L p -convergence implies convergence in L 1,loc p O q and so ż ż pB α φ q g 0 d λ “ φ g α d λ So g 0 admits derivatives up to k and g α “ B α g 0 .