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MAT201C Lecture Notes: Introduction to Sobolev Spaces

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MAT201C Lecture Notes: Introduction to Sobolev Spaces MAT201C Lecture Notes: Introduction to Sobolev Spaces Steve Shkoller Department of Mathematics University of California at Davis Davis, CA 95616 USA email: [email protected] May 26, 2011 These notes, intended for the third quarter of the graduate Analysis sequence at UC Davis, should be viewed as a very short introduction to Sobolev space theory, and the rather large collection of topics which are foundational for its development. This includes the theory of Lp spaces, the Fourier series and the Fourier transform, the notion of weak derivatives and distributions, and a fair amount of differential analysis (the theory of dif- ferential operators). Sobolev spaces and other very closely related functional frameworks have proved to be indispensable topologies for answering very basic questions in the fields of partial differential equations, mathematical physics, differential geometry, harmonic anal- ysis, scientific computation, and a host of other mathematical specialities. These notes provide only a brief introduction to the material, essentially just enough to get going with the basics of Sobolev spaces. As the course progresses, I will add some additional topics and/or details to these notes. In the meantime, a good reference is Analysis by Lieb and Loss, and of course Applied Analysis by Hunter and Nachtergaele, particularly Chapter 12, which serves as a nice compendium of the material to be presented. If only I had the theorems! Then I should find the proofs easily enough. –Bernhard Riemann (1826-1866) Facts are many, but the truth is one. –Rabindranath Tagore (1861-1941) 1 Shkoller CONTENTS Contents 1 Lp spaces 4 1.1 Notation . 4 1.2 Definitions and basic properties . 4 1.3 Basic inequalities . 5 p 1.4 The space (L (X), k · kLp (X) is complete . 7 1.5 Convergence criteria for Lp functions . 8 1.6 The space L∞(X)................................ 10 1.7 Approximation of Lp(X) by simple functions . 11 1.8 Approximation of Lp(Ω) by continuous functions . 11 1.9 Approximation of Lp(Ω) by smooth functions . 12 1.10 Continuous linear functionals on Lp(X).................... 14 1.11 A theorem of F. Riesz . 15 1.12 Weak convergence . 18 1.13 Integral operators . 20 1.14 Appendix 1: The monotone and dominated convergence theorems and Fa- tou’s lemma . 22 1.15 Appendix 2: The Fubini and Tonelli Theorems . 23 1.16 Exercises . 24 2 The Sobolev spaces Hk(Ω) for integers k ≥ 0 27 2.1 Weak derivatives . 27 2.2 Definition of Sobolev Spaces . 29 2.3 A simple version of the Sobolev embedding theorem . 30 2.4 Approximation of W k,p(Ω) by smooth functions . 32 2.5 H¨older Spaces . 33 2.6 Morrey’s inequality . 33 2.7 The Gagliardo-Nirenberg-Sobolev inequality . 38 2.8 Local coordinates near ∂Ω............................ 43 2.9 Sobolev extensions and traces. 43 1,p 2.10 The subspace W0 (Ω).............................. 45 2.11 Weak solutions to Dirichlet’s problem . 47 2.12 Strong compactness . 48 2.13 Exercises . 51 3 The Fourier Transform 55 1 n n 3.1 Fourier transform on L (R ) and the space S(R ).............. 55 n 3.2 The topology on S(R ) and tempered distributions . 59 0 n 3.3 Fourier transform on S (R )........................... 60 2 Shkoller CONTENTS 2 n 3.4 The Fourier transform on L (R )........................ 61 p n 3.5 Bounds for the Fourier transform on L (R ).................. 62 3.6 Convolution and the Fourier transform . 63 3.7 An explicit computation with the Fourier Transform . 63 3.8 Applications to the Poisson, Heat, and Wave equations . 65 3.9 Exercises . 70 s n 4 The Sobolev Spaces H (R ), s ∈ R 74 s n 4.1 H (R ) via the Fourier Transform . 74 4.2 Fractional-order Sobolev spaces via difference quotient norms . 80 5 Fractional-order Sobolev spaces on domains with boundary 84 s n 5.1 The space H (R+)................................ 84 5.2 The Sobolev space Hs(Ω) ............................ 85 s n 6 The Sobolev Spaces H (T ), s ∈ R 87 6.1 The Fourier Series: Revisited . 87 6.2 The Poisson Integral Formula and the Laplace operator . 89 6.3 Exercises . 92 7 Regularity of the Laplacian on Ω 94 8 Inequalities for the normal and tangential decomposition of vector fields on ∂Ω 100 8.1 The regularity of ∂Ω............................... 100 8.2 Tangential and normal derivatives . 102 8.3 Some useful inequalities . 103 8.4 Elliptic estimates for vector fields . 105 9 The div-curl lemma 106 9.1 Exercises . 108 3 Shkoller 1 LP SPACES 1 Lp spaces 1.1 Notation d We will usually use Ω to denote an open and smooth domain in R , for d = 1, 2, 3, ... In this chapter on Lp spaces, we will sometimes use X to denote a more general measure space, but the reader can usually think of a subset of Euclidean space. Ck(Ω) is the space of functions which are k times differentiable in Ω for integers k ≥ 0. C0(Ω) then coincides with C(Ω), the space of continuous functions on Ω. ∞ k C (Ω) = ∩k≥0C (Ω). spt f denotes the support of a function f, and is the closure of the set {x ∈ Ω | f(x) 6= 0}. C0(Ω) = {u ∈ C(Ω) | spt u compact in Ω}. k k C0 (Ω) = C (Ω) ∩ C0(Ω). ∞ ∞ C0 (Ω) = C (Ω) ∩ C0(Ω). We will also use D(Ω) to denote this space, which is known as the space of test functions in the theory of distributions. 1.2 Definitions and basic properties Definition 1.1. Let 0 < p < ∞ and let (X, M, µ) denote a measure space. If f : X → R is a measurable function, then we define 1 Z p p kfkLp(X) := |f| dx and kfkL∞(X) := ess supx∈X |f(x)| . X Note that kfkLp(X) may take the value ∞. Unless stated otherwise, we will usually consider d X to be a smooth, open subset Ω of R , and we will assume that all functions under consideration are measurable. Definition 1.2. The space Lp(X) is the set p L (X) = {f : X → R | kfkLp(X) < ∞} . The space Lp(X) satisfies the following vector space properties: p p 1. For each α ∈ R, if f ∈ L (X) then αf ∈ L (X); 2. If f, g ∈ Lp(X), then |f + g|p ≤ 2p−1(|f|p + |g|p) , so that f + g ∈ Lp(X). 4 Shkoller 1 LP SPACES 3. The triangle inequality is valid if p ≥ 1. The most interesting cases are p = 1, 2, ∞, while all of the Lp arise often in nonlinear estimates. Definition 1.3. The space lp, called “little Lp”, will be useful when we introduce Sobolev spaces on the torus and the Fourier series. For 1 ≤ p < ∞, we set ( ∞ ) p X p l = {xn}n∈Z | |xn| < ∞ , n=−∞ where Z denotes the integers. 1.3 Basic inequalities Convexity is fundamental to Lp spaces for p ∈ [1, ∞). Lemma 1.4. For λ ∈ (0, 1), xλ ≤ (1 − λ) + λx. Proof. Set f(x) = (1−λ)+λx−xλ; hence, f 0(x) = λ−λxλ−1 = 0 if and only if λ(1−xλ−1) = 0 so that x = 1 is the critical point of f. In particular, the minimum occurs at x = 1 with value f(1) = 0 ≤ (1 − λ) + λx − xλ . Lemma 1.5. For a, b ≥ 0 and λ ∈ (0, 1), aλb1−λ ≤ λa + (1 − λ)b with equality if a = b. Proof. If either a = 0 or b = 0, then this is trivially true, so assume that a, b > 0. Set x = a/b, and apply Lemma 1 to obtain the desired inequality. Theorem 1.6 (H¨older’s inequality). Suppose that 1 ≤ p ≤ ∞ and 1 < q < ∞ with 1 1 p q 1 p + q = 1. If f ∈ L and g ∈ L , then fg ∈ L . Moreover, kfgkL1 ≤ kfkLp kgkLq . Note that if p = q = 2, then this is the Cauchy-Schwarz inequality since kfgkL1 = |(f, g)L2 |. Proof. We use Lemma 1.5. Let λ = 1/p and set |f|p |g|q a = p , and b = q kfkLp kgkLp 5 Shkoller 1 LP SPACES for all x ∈ X. Then aλb1−λ = a1/pb1−1/p = a1/pb1/q so that |f| · |g| 1 |f|p 1 |g|q ≤ p + q . kfkLp kgkLq p kfkLp q kgkLq Integrating this inequality yields Z |f| · |g| Z 1 |f|p 1 |g|q 1 1 dx ≤ p + q dx = + = 1 . X kfkLp kgkLq X p kfkLp q kgkLq p q p 1 1 Definition 1.7. The exponent q = p−1 (or q = 1 − p ) is called the conjugate exponent of p. Theorem 1.8 (Minkowski’s inequality). If 1 ≤ p ≤ ∞ and f, g ∈ Lp then kf + gkLp ≤ kfkLp + kgkLp . Proof. If f + g = 0 a.e., then the statement is trivial. Assume that f + g 6= 0 a.e. Consider the equality |f + g|p = |f + g| · |f + g|p−1 ≤ (|f| + |g|)|f + g|p−1 , and integrate over X to find that Z Z |f + g|pdx ≤ (|f| + |g|)|f + g|p−1 dx X X H¨older’s p−1 p p ≤ (kfkL + kgkL ) |f + g| Lq . p Since q = p−1 , 1 Z q p−1 p |f + g| Lq = |f + g| dx , X from which it follows that 1 Z 1− q p |f + g| dx ≤ kfkLp + kgkLq , X 1 1 which completes the proof, since p = 1 − q . Corollary 1.9. For 1 ≤ p ≤ ∞, Lp(X) is a normed linear space. 6 Shkoller 1 LP SPACES n Example 1.10. Let Ω denote a subset of R whose Lebesgue measure is equal to one.
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