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Elementary Theory and Methods for Elliptic Partial Differential Equations

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Elementary Theory and Methods for Elliptic Partial Differential Equations Elementary Theory and Methods for Elliptic Partial Differential Equations John Villavert Contents 1 Introduction and Basic Theory 4 1.1 Harmonic Functions . 5 1.1.1 Mean Value Properties . 5 1.1.2 Sub-harmonic and Super-harmonic Functions . 8 1.1.3 Further Properties of Harmonic Functions . 11 1.1.4 Energy and Comparison Methods for Harmonic Functions . 14 1.2 Classical Maximum Principles . 17 1.2.1 The Weak Maximum Principle . 17 1.2.2 The Strong Maximum Principle . 18 1.3 Newtonian and Riesz Potentials . 21 1.3.1 The Newtonian Potential and Green's Formula . 21 1.3.2 Riesz Potentials and the Hardy-Littlewood-Sobolev Inequalitiy . 23 1.3.3 Green's Function and Representation Formulas of Solutions . 25 1.3.4 Green's Function for a Half-Space . 26 1.3.5 Green's Function for a Ball . 28 1.4 H¨olderRegularity for Poisson's Equation . 31 1.4.1 The Dirichlet Problem for Poisson's Equation . 33 1.4.2 Interior H¨olderEstimates for Second Derivatives . 36 1.4.3 Boundary H¨olderEstimates for Second Derivatives . 40 2 Existence Theory 43 2.1 The Lax-Milgram Theorem . 43 2.1.1 Existence of Weak Solutions . 44 2.2 The Fredholm Alternative . 47 2.2.1 Existence of Weak Solutions . 48 2.3 Eigenvalues and Eigenfunctions . 52 1 2.4 Topological Fixed Point Theorems . 53 2.4.1 Brouwer's Fixed Point Theorem . 54 2.4.2 Schauder's Fixed Point Theorem . 55 2.4.3 Schaefer's Fixed Point Theorem . 56 2.4.4 Application to Nonlinear Elliptic Boundary Value Problems . 57 2.5 Perron Method . 59 2.6 Continuity Method . 65 2.7 Calculus of Variations I: Minimizers and Weak Solutions . 68 2.7.1 Existence of Weak Solutions . 71 2.7.2 Existence of Minimizers Under Constraints . 73 2.8 Calculus of Variations II: Critical Points and the Mountain Pass Theorem . 75 2.8.1 The Deformation and Mountain Pass Theorems . 75 2.8.2 Application of the Mountain Pass Theorem . 79 2.9 Calculus of Variations III: Concentration Compactness . 83 2.10 Sharp Existence Results for Semilinear Equations . 87 3 Regularity Theory for Second-order Elliptic Equations 92 3.1 Preliminaries . 92 3.1.1 Flattening out the Boudary . 92 3.1.2 Weak Lebesgue Spaces and Lorentz Spaces . 93 3.1.3 The Marcinkiewicz Interpolation Inequalities . 96 3.1.4 Calder´on{Zygmund and the John-Nirenberg Lemmas . 96 3.1.5 Lp Boundedness of Integral Operators . 97 3.2 W 2;p Regularity for Weak Solutions . 105 3.2.1 W 2;p A Priori Estimates . 105 3.2.2 Regularity of Solutions and A Priori Estimates . 110 3.3 Bootstraping: Two Basic Examples . 114 3.4 Regularity in the Sobolev Spaces Hk . 115 3.4.1 Interior regularity . 116 3.4.2 Higher interior regularity . 119 3.4.3 Global regularity . 121 3.4.4 Higher global regularity . 125 3.5 The Schauder Estimates and C2,α Regularity . 127 3.6 H¨olderContinuity for Weak Solutions: A Perturbation Approach . 129 3.6.1 Morrey{Campanato Spaces . 130 3.6.2 Preliminary Estimates . 132 3.6.3 H¨olderContinuity of Weak Solutions . 134 3.6.4 H¨olderContinuity of the Gradient . 141 3.7 De Giorgi{Nash{Moser Regularity Theory . 141 3.7.1 Motivation . 141 2 3.7.2 Local Boundedness and Preliminary Lemmas . 142 3.7.3 Proof of Local Boundedness: Moser Iteration . 145 3.7.4 H¨olderRegularity: De Giorgi's Approach . 150 3.7.5 H¨olderRegularity: the Weak Harnack Inequality . 155 3.7.6 Further Applications of the Weak Harnack Inequality . 160 4 Viscosity Solutions and Fully Nonlinear Equations 163 4.1 Introduction . 163 4.2 A Harnack Inequality . 169 4.3 Schauder Estimates . 174 4.4 W 2;p Estimates . 178 5 The Method of Moving Planes and Its Variants 180 5.1 Preliminaries . 180 5.2 The Proof of Theorem 5.1 . 183 5.3 The Method of Moving Spheres . 186 6 Concentration and Non-compactness of Critical Sobolev Embeddings 191 6.1 Introduction . 191 6.2 Concentration and Sobolev Inequalities . 193 6.3 Minimizers for Critical Sobolev Inequalities . 195 A Basic Inequalities, Embeddings and Convergence Theorems 199 A.1 Basic Inequalities . 199 A.2 Sobolev Inequalities . 202 A.2.1 Extension and Trace Operators . 205 A.2.2 Sobolev Embeddings and Poincar´eInequalities . 207 A.3 Convergence Theorems . 213 3 CHAPTER 1 Introduction and Basic Theory In this introductory chapter, we provide some preliminary background which we will use later in establishing various results for general elliptic partial differential equations (PDEs). The material found within these notes aims to compile the fundamental theory for second- order elliptic PDEs and serves as complementary notes to many well-known references on the subject, c.f., [5, 6, 8, 11, 13]. Several recommended resources on basic background that supplement these notes and the aforementioned references are the textbooks [2, 9, 21]. We will mainly focus on the Dirichlet problem, Lu = f in U; (1.1) u = 0 on @U; where U is a bounded open subset of Rn with boundary @U, and u : Rn 7! R is the unknown quantity. For this problem, f : U 7! R is given, and L is a second-order differential operator having either the form n n X ij X i Lu = − Dj a (x)Diu + b (x)Diu + c(x)u; (1.2) i;j=1 i=1 or else n n X ij X i Lu = − a (x)Diju + b (x)Diu + c(x)u; (1.3) i;j=1 i=1 for given coefficient functions aij, bi, and c (i; j = 1; 2; : : : ; n) which are assumed to be measurable in U¯, the closure of the set U. However, in this chapter, we take these coefficients to be continuous in U¯. If L takes the form (1.2), then it is said to be in divergence form, and if it takes the form (1.3), then it is said to be in non-divergence form. 4 Remark 1.1. Here, Dij = DiDj. In practice, (1.2) is natural for energy methods while (1.3) is more appropriate for the maximum principles. In addition, the Dirichlet problem (1.1) + can be extended to systems, i.e., Lui = fi in U, and ui = 0 on @U, for i = 1; 2;:::;L 2 Z . A simple example of a second-order differential operator is the Laplacian, L := −∆, where ij i a = δij, b = c = 0 (i; j = 1; 2 : : : ; n) in either (1.2) or (1.3). Remark 1.2. The elliptic theory for equations in divergence form was developed first as we can easily exploit the distributional framework and energy methods for weak solutions in Sobolev spaces, for example. Much of our focus in these notes will be on establishing the basic elliptic PDE theory for equations in divergence form. Remark 1.3. Extending this theory to elliptic equations in non-divergence form has certain obstacles, and its treatment requires a somewhat different approach. We shall study one way of examining such equations using another concept of a weak solution called a viscosity solution, which are defined with the help of maximum and comparison principles. We shall give a brief introduction to fully nonlinear elliptic equations in non-divergence form and their viscosity solutions in Chapter 4. Unless stated otherwise, we shall always assume that L is uniformly elliptic, i.e., there exist λ, Λ > 0 such that n 2 X ij 2 n λjξj ≤ a (x)ξiξj ≤ Λjξj a.e x 2 U; for all ξ 2 R : i;j=1 1 Moreover, u 2 H0 (U) is said to be a weak solution of (1.1) in divergence form if 1 B[u; v] = (f; v); for all v 2 H0 (U); where B[·; ·] is the associated bilinear form, n n X X B[u; v] := aijD uD v + bi(x)D uv + c(x)uv dx: ˆ i j i U i;j=1 i=1 1.1 Harmonic Functions First we shall introduce the mean-value property, which provides the key ingredient in es- tablishing many important properties for harmonic functions. 1.1.1 Mean Value Properties Definition 1.1. For u 2 C(U) we define 5 (i) u satisfies the first mean value property (in U) if 1 u(x) = u(y) dσy for any Br(x) ⊂ U; j@Br(x)j ˆ@Br(x) (ii) u satisfies the second mean value property if 1 u(x) = u(y) dy for any Br(x) ⊂ U: jBr(x)j ˆBr(x) Remark 1.4. These two definitions are equivalent. To see this, observe that if we rewrite (i) as n−1 1 u(x)r = u(y) dσy; !n ˆ@Br(x) n where !n denotes the surface area of the (n − 1)-dimensional unit sphere S , then integrate with respect to r, we get rn 1 r 1 u(x) = u(y) dσy ds = u(y) dy: n !n ˆ0 ˆ@Bs(x) !n ˆBr(x) If we rewrite (ii) as r n n n u(x)r = u(y) dy = u(y) dσyds !n ˆBr(x) !n ˆ0 ˆ@Bs(x) then differentiate with respect to r, we obtain (i). Remark 1.5. The mean value properties can easily be expressed in the following ways. (i) u 2 C(U) satisfies the first mean value property if 1 u(x) = u(x + r!) dσ! for any Br(x) ⊂ U; !n ˆ@B1(0) (ii) u 2 C(U) satisfies the second mean value property if n u(x) = u(x + ry) dy for any Br(x) ⊂ U; !n ˆB1(0) Theorem 1.1.
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