Notes on Partial Differential Equations John K. Hunter Department of Mathematics, University of California at Davis Contents Chapter 1. Preliminaries 1 1.1. Euclidean space 1 1.2. Spaces of continuous functions 1 1.3. H¨olderspaces 2 1.4. Lp spaces 3 1.5. Compactness 6 1.6. Averages 7 1.7. Convolutions 7 1.8. Derivatives and multi-index notation 8 1.9. Mollifiers 10 1.10. Boundaries of open sets 12 1.11. Change of variables 16 1.12. Divergence theorem 16 Chapter 2. Laplace's equation 19 2.1. Mean value theorem 20 2.2. Derivative estimates and analyticity 23 2.3. Maximum principle 26 2.4. Harnack's inequality 31 2.5. Green's identities 32 2.6. Fundamental solution 33 2.7. The Newtonian potential 34 2.8. Singular integral operators 43 Chapter 3. Sobolev spaces 47 3.1. Weak derivatives 47 3.2. Examples 47 3.3. Distributions 50 3.4. Properties of weak derivatives 53 3.5. Sobolev spaces 56 3.6. Approximation of Sobolev functions 57 3.7. Sobolev embedding: p < n 57 3.8. Sobolev embedding: p > n 66 3.9. Boundary values of Sobolev functions 69 3.10. Compactness results 71 3.11. Sobolev functions on Ω ⊂ Rn 73 3.A. Lipschitz functions 75 3.B. Absolutely continuous functions 76 3.C. Functions of bounded variation 78 3.D. Borel measures on R 80 v vi CONTENTS 3.E. Radon measures on R 82 3.F. Lebesgue-Stieltjes measures 83 3.G. Integration 84 3.H. Summary 86 Chapter 4. Elliptic PDEs 87 4.1. Weak formulation of the Dirichlet problem 87 4.2. Variational formulation 89 4.3. The space H−1(Ω) 90 1 4.4. The Poincar´einequality for H0 (Ω) 93 4.5. Existence of weak solutions of the Dirichlet problem 94 4.6. General linear, second order elliptic PDEs 96 4.7. The Lax-Milgram theorem and general elliptic PDEs 98 4.8. Compactness of the resolvent 101 4.9. The Fredholm alternative 102 4.10. The spectrum of a self-adjoint elliptic operator 104 4.11. Interior regularity 105 4.12. Boundary regularity 109 4.13. Some further perspectives 112 4.A. Heat flow 114 4.B. Operators on Hilbert spaces 116 4.C. Difference quotients and weak derivatives 119 Chapter 5. The Heat Equation 123 5.1. The initial value problem 123 5.A. The Schwartz space and the Fourier transform 132 Bibliography 139 CHAPTER 1 Preliminaries In this chapter, we collect various definitions and theorems for future use. Proofs may be found in the references e.g. [7, 13, 16, 17, 18]. 1.1. Euclidean space Let Rn be n-dimensional Euclidean space. We denote the Euclidean norm of a n vector x = (x1; x2; : : : ; xn) 2 R by 2 2 2 1=2 jxj = x1 + x2 + ··· + xn and the inner product of vectors x = (x1; x2; : : : ; xn), y = (y1; y2; : : : ; yn) by x · y = x1y1 + x2y2 + ··· + xnyn: We denote Lebesgue measure on Rn by dx, and the Lebesgue measure of a set E ⊂ Rn by jEj. If E is a subset of Rn, we denote the complement by Ec = Rn n E, the closure by E, the interior by E◦ and the boundary by @E = E n E◦. The characteristic n function χE : R ! R of E is defined by 1 if x 2 E, χ (x) = E 0 if x2 = E. A set E is bounded if fjxj : x 2 Eg is bounded in R. A set is connected if it is not the disjoint union of two nonempty relatively open subsets. We sometimes refer to a connected open set as a domain. We say that an open set Ω0 in Rn is compactly contained in an open set Ω, 0 written Ω b Ω, if Ω0 ⊂ Ω and Ω0 is compact. If Ω0 ⊂ Ω, then dist (Ω0;@Ω) = inf fjx − yj : x 2 Ω0; y 2 @Ωg > 0: This distance is finite provided that Ω0 6= ; and Ω 6= Rn. 1.2. Spaces of continuous functions Let Ω be an open set in Rn. We denote the space of continuous functions u :Ω ! R by C(Ω); the space of functions with continuous partial derivatives in Ω of order less than or equal to k 2 N by Ck(Ω); and the space of functions with continuous derivatives of all orders by C1(Ω). Functions in these spaces need not be bounded even if Ω is bounded; for example, (1=x) 2 C1(0; 1). If Ω is a bounded open set in Rn, we denote by C(Ω) the space of continuous functions u : Ω ! R. This is a Banach space with respect to the maximum, or supremum, norm kuk1 = sup ju(x)j: x2Ω 1 2 1. PRELIMINARIES We denote the support of a continuous function u :Ω ! Rn by spt u = fx 2 Ω: u(x) 6= 0g: We denote by Cc(Ω) the space of continuous functions whose support is compactly 1 contained in Ω, and by Cc (Ω) the space of functions with continuous derivatives of all orders and compact support in Ω. We will sometimes refer to such functions as test functions. n n The completion of Cc(R ) with respect to the uniform norm is the space C0(R ) of continuous functions that approach zero at infinity. (Note that in many places the 1 notation C0 and C0 is used to denote the spaces of compactly supported functions 1 that we denote by Cc and Cc .) If Ω is bounded, we say that a function u : Ω ! R belongs to Ck(Ω) if it is continuous and its partial derivatives of order less than or equal to k are uniformly continuous in Ω, in which case they extend to continuous functions on Ω. The space Ck(Ω) is a Banach space with respect to the norm X α kukCk(Ω) = sup j@ uj Ω jα|≤k where we use the multi-index notation for partial derivatives explained in Sec- tion 1.8. This norm is finite because the derivatives @αu are continuous functions on the compact set Ω. A vector field X :Ω ! Rm belongs to Ck(Ω) if each of its components belongs to Ck(Ω). 1.3. H¨olderspaces The definition of continuity is not a quantitative one, because it does not say how rapidly the values u(y) of a function approach its value u(x) as y ! x. The modulus of continuity ! : [0; 1] ! [0; 1] of a general continuous function u, satisfying ju(x) − u(y)j ≤ ! (jx − yj) ; may decrease arbitrarily slowly. As a result, despite their simple and natural ap- pearance, spaces of continuous functions are often not suitable for the analysis of PDEs, which is almost always based on quantitative estimates. A straightforward and useful way to strengthen the definition of continuity is to require that the modulus of continuity is proportional to a power jx − yjα for some exponent 0 < α ≤ 1. Such functions are said to be H¨oldercontinuous, or Lip- schitz continuous if α = 1. Roughly speaking, one can think of H¨oldercontinuous functions with exponent α as functions with bounded fractional derivatives of the the order α. Definition 1.1. Suppose that Ω is an open set in Rn and 0 < α ≤ 1. A function u :Ω ! R is uniformly H¨oldercontinuous with exponent α in Ω if the quantity ju(x) − u(y)j (1.1) [u]α,Ω = sup α x; y 2 Ω jx − yj x 6= y is finite. A function u :Ω ! R is locally uniformly H¨oldercontinuous with exponent 0 0,α α in Ω if [u]α,Ω0 is finite for every Ω b Ω. We denote by C (Ω) the space of locally 1.4. Lp SPACES 3 uniformly H¨oldercontinuous functions with exponent α in Ω. If Ω is bounded, we denote by C0,α Ω the space of uniformly H¨oldercontinuous functions with exponent α in Ω. We typically use Greek letters such as α, β both for H¨olderexponents and multi-indices; it should be clear from the context which they denote. When α and Ω are understood, we will abbreviate `u is (locally) uniformly H¨oldercontinuous with exponent α in Ω’ to `u is (locally) H¨oldercontinuous.' If u is H¨oldercontinuous with exponent one, then we say that u is Lipschitz continu- ous. There is no purpose in considering H¨oldercontinuous functions with exponent greater than one, since any such function is differentiable with zero derivative, and is therefore constant. The quantity [u]α,Ω is a semi-norm, but it is not a norm since it is zero for constant functions. The space C0,α Ω, where Ω is bounded, is a Banach space with respect to the norm kukC0,α(Ω) = sup juj + [u]α,Ω : Ω Example 1.2. For 0 < α < 1, define u(x) : (0; 1) ! R by u(x) = jxjα. Then u 2 C0,α ([0; 1]), but u2 = C0,β ([0; 1]) for α < β ≤ 1. Example 1.3. The function u(x):(−1; 1) ! R given by u(x) = jxj is Lipschitz continuous, but not continuously differentiable. Thus, u 2 C0;1 ([−1; 1]), but u2 = C1 ([−1; 1]). We may also define spaces of continuously differentiable functions whose kth derivative is H¨oldercontinuous. Definition 1.4. If Ω is an open set in Rn, k 2 N, and 0 < α ≤ 1, then Ck,α(Ω) consists of all functions u :Ω ! R with continuous partial derivatives in Ω of order less than or equal to k whose kth partial derivatives are locally uniformly H¨oldercontinuous with exponent α in Ω.