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Sobolev Spaces JUHA KINNUNEN Sobolev spaces Department of Mathematics, Aalto University 2021 Contents 1 SOBOLEV SPACES1 1.1 Weak derivatives .............................. 1 1.2 Sobolev spaces ............................... 4 1.3 Properties of weak derivatives ....................... 8 1.4 Completeness of Sobolev spaces .................... 9 1.5 Hilbert space structure ........................... 11 1.6 Approximation by smooth functions ................... 12 1.7 Local approximation in Sobolev spaces ................. 16 1.8 Global approximation in Sobolev spaces ................ 17 1.9 Sobolev spaces with zero boundary values ............... 18 1.10 Chain rule ................................... 21 1.11 Truncation ................................... 23 1.12 Weak convergence methods for Sobolev spaces ........... 25 1.13 Difference quotients ............................. 33 1.14 Absolute continuity on lines ........................ 36 2 SOBOLEV INEQUALITIES 42 2.1 Gagliardo-Nirenberg-Sobolev inequality ................ 43 2.2 Sobolev-Poincaré inequalities ....................... 49 2.3 Morrey’s inequality ............................. 55 1, 2.4 Lipschitz functions and W 1 ........................ 59 2.5 Summary of the Sobolev embeddings .................. 62 2.6 Direct methods in the calculus of variations .............. 62 3 MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 70 3.1 Representation formulas and Riesz potentials ............. 71 3.2 Sobolev-Poincaré inequalities ....................... 78 3.3 Sobolev inequalities on domains ..................... 86 3.4 A maximal function characterization of Sobolev spaces ....... 89 3.5 Pointwise estimates ............................. 92 3.6 Approximation by Lipschitz functions ................... 97 3.7 Maximal operator on Sobolev spaces .................. 102 CONTENTS ii 4 POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 106 4.1 Sobolev capacity .............................. 106 4.2 Capacity and measure .......................... 109 4.3 Quasicontinuity ............................... 116 4.4 Lebesgue points of Sobolev functions .................. 119 4.5 Sobolev spaces with zero boundary values ............... 124 1 Sobolev spaces In this chapter we begin our study of Sobolev spaces. The Sobolev space is a vector space of functions that have weak derivatives. Motivation for studying these spaces is that solutions of partial differential equations, when they exist, belong naturally to Sobolev spaces. 1.1 Weak derivatives Notation. Let ­ Rn be open, f : ­ R and k 1,2,.... Then we use the following ½ ! Æ notations: C(­) {f : f continuous in ­} Æ supp f {x ­ : f (x) 0} the support of f Æ 2 6Æ Æ C0(­) {f C(­) : supp f is a compact subset of ­} Æ 2 Ck(­) {f C(­) : f is k times continuously diferentiable} Æ 2 k k C (­) C (­) C0(­) 0 Æ \ \1 k C1 C (­) smooth functions Æ k 1 Æ Æ C1(­) C1(­) C0(­) 0 Æ \ compactly supported smooth functions Æ test functions Æ ■ W A R N I N G : In general, supp f * ­. Examples 1.1: (1) Let u : B(0,1) R, u(x) 1 x . Then supp u B(0,1). ! Æ ¡ j j Æ 1 CHAPTER 1. SOBOLEV SPACES 2 (2) Let f : R R be ! 8 <x2, x 0, f (x) Ê Æ : x2, x 0. ¡ Ç Now f C1(R) \ C2(R) although the graph looks smooth. 2 (3) Let us define ' : Rn R, ! 8 1 <e x 2 1 , x B(0,1), '(x) j j ¡ 2 Æ :0, x Rn \ B(0,1). 2 n Now ' C1(R ) and supp' B(0,1) (exercise). 2 0 Æ Let us start with a motivation for definition of weak derivatives. Let ­ Rn 1 ½ be open, u C (­) and ' C1(­). Integration by parts gives 2 2 0 Ç' Çu u dx ' dx. ˆ­ Çx j Æ¡ˆ­ Çx j There is no boundary term, since ' has a compact support in ­ and thus vanishes near Ç­. k n Let then u C (­), k 1,2,..., and let ® (®1,®2,...,®n) N (we use the 2 Æ Æ 2 convention that 0 N) be a multi-index such that the order of multi-index ® 2 j j Æ ®1 ... ®n is at most k. We denote Å Å Ç ® u Ç®1 Ç®n D®u j j ... u. ®1 ®n ®1 ®n Æ Çx1 ...Çxn Æ Çx1 Çxn T HEMORAL : A coordinate of a multi-index indicates how many times a function is differentiated with respect to the corresponding variable. The order of a multi-index tells the total number of differentiations. Successive integration by parts gives ® ® ® uD ' dx ( 1)j j D u' dx. ˆ­ Æ ¡ ˆ­ Notice that the left-hand side makes sense even under the assumption u L1 (­). 2 loc Definition 1.2. Assume that u L1 (­) and let ® Nn be a multi-index. Then 2 loc 2 v L1 (­) is the ®th weak partial derivative of u, written D®u v, if 2 loc Æ ® ® u D ' dx ( 1)j j v' dx ˆ­ Æ ¡ ˆ­ 0 (0,...,0) for every test function ' C1(­). We denote D u D u. If ® 1, then 2 0 Æ Æ j j Æ Du (D1u,D2u...,Dnu) Æ is the weak gradient of u. Here Çu (0,...,1,...,0) D j u D u, j 1,..., n, Æ Çx j Æ Æ (the jth component is 1). CHAPTER 1. SOBOLEV SPACES 3 T HEMORAL : Classical derivatives are defined as pointwise limits of differ- ence quotients, but the weak derivatives are defined as a functions satisfying the integration by parts formula. Observe, that changing the function on a set of measure zero does not affect its weak derivatives. W ARNING We use the same notation for the weak and classical derivatives. It should be clear from the context which interpretation is used. Remarks 1.3: (1) If u Ck(­), then the classical partial derivatives up to order k are also 2 the corresponding weak derivatives of u. In this sense, weak derivatives generalize classical derivatives. (2) If u 0 almost everywhere in an open set, then D®u 0 almost everywhere Æ Æ in the same set. Lemma 1.4. A weak ®th partial derivative of u, if it exists, is uniquely defined up to a set of measure zero. Proof. Assume that v,v L1 (­) are both weak ®th partial derivatives of u, that e 2 loc is, ® ® ® uD ' dx ( 1)j j v' dx ( 1)j j ve' dx ˆ­ Æ ¡ ˆ­ Æ ¡ ˆ­ for every ' C1(­). This implies that 2 0 (v ve)' dx 0 for every ' C01(­). (1.1) ˆ­ ¡ Æ 2 Claim: v v almost everywhere in ­. Æ e Reason. Let ­0 b ­ (i.e. ­0 is open and ­0 is a compact subset of ­). The p space C01(­0) is dense in L (­0) (we shall return to this later). There exists a sequence of functions 'i C1(­0) such that 'i 2 in ­0 and 'i sgn(v v) 2 0 j j É ! ¡ e almost everywhere in ­0 as i . Here sgn is the signum function. ! 1 Identity (1.1) and the dominated convergence theorem, with the majorant 1 (v v)'i 2( v v ) L (­0), give j ¡ e j É j j Å jej 2 0 lim (v v)'i dx lim (v v)'i dx Æ i ˆ ¡ e Æ ˆ i ¡ e !1 ­0 ­0 !1 (v v)sgn(v v) dx v v dx Æ ˆ ¡ e ¡ e Æ ˆ j ¡ ej ­0 ­0 This implies that v v almost everywhere in ­0 for every ­0 ­. Thus v v Æ e b Æ e almost everywhere in ­. ■ From the proof we obtain a very useful corollary. CHAPTER 1. SOBOLEV SPACES 4 Corollary 1.5 (Fundamental lemma of the calculus of variations). If f L1 (­) 2 loc satisfies f ' dx 0 ˆ­ Æ for every ' C1(­), then f 0 almost everywhere in ­. 2 0 Æ T HEMORAL : This is an integral way to say that a function is zero almost everywhere. Example 1.6. Let n 1 and ­ (0,2). Consider Æ Æ 8 <x, 0 x 1, u(x) Ç Ç Æ :1, 1 x 2, É Ç and 8 <1, 0 x 1, v(x) Ç Ç Æ :0, 1 x 2. É Ç We claim that u0 v in the weak sense. To see this, we show that Æ 2 2 u'0 dx v' dx ˆ0 Æ¡ˆ0 for every ' C1((0,2)). 2 0 Reason. An integration by parts and the fundamental theorem of calculus give 2 1 2 u(x)'0(x) dx x'0(x) dx '0(x) dx ˆ0 Æ ˆ0 Å ˆ1 ¯1 1 ¯ x'(x)¯ '(x) dx '(2) '(1) Æ ¯0 ¡ˆ0 Å |{z}¡ | {z } 0 '(1) Æ Æ 1 2 '(x) dx v'(x) dx Æ¡ˆ0 Æ¡ˆ0 for every ' C01((0,2)). 2 ■ 1.2 Sobolev spaces Definition 1.7. Assume that ­ is an open subset of Rn. The Sobolev space W k,p(­) consists of functions u Lp(­) such that for every multi-index ® with 2 ® k, the weak derivative D®u exists and D®u Lp(­). Thus j j É 2 W k,p(­) {u Lp(­) : D®u Lp(­), ® k}. Æ 2 2 j j É CHAPTER 1. SOBOLEV SPACES 5 If u W k,p(­), we define its norm 2 1 à ! p X ® p u W k,p(­) D u dx , 1 p , k k Æ ® k ˆ­ j j É Ç 1 j |É and X ® u k, esssup D u . W 1(­) k k Æ ® k ­ j j j |É 0 (0,...,0) Notice that D u D u u. Assume that ­0 is an open subset of ­. We say Æ Æ that ­0 is compactly contained in ­, denoted ­0 b ­, if ­0 is a compact subset of k,p k,p ­. A function u W (­), if u W (­0) for every ­0 ­. 2 loc 2 b T HEMORAL : Thus Sobolev space W k,p(­) consists of functions in Lp(­) that have weak partial derivatives up to order k and they belong to Lp(­).
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