Methods of Applied Mathematics Todd Arbogast and Jerry L. Bona Department of Mathematics, and Institute for Computational Engineering and Sciences The University of Texas at Austin Copyright 1999{2001, 2004{2005, 2007{2008 (corrected version) by T. Arbogast and J. Bona. Contents Chapter 1. Preliminaries 5 1.1. Elementary Topology 5 1.2. Lebesgue Measure and Integration 13 1.3. Exercises 23 Chapter 2. Normed Linear Spaces and Banach Spaces 27 2.1. Basic Concepts and Definitions. 27 2.2. Some Important Examples 34 2.3. Hahn-Banach Theorems 43 2.4. Applications of Hahn-Banach 48 2.5. The Embedding of X into its Double Dual X∗∗ 52 2.6. The Open Mapping Theorem 53 2.7. Uniform Boundedness Principle 57 2.8. Compactness and Weak Convergence in a NLS 58 2.9. The Dual of an Operator 63 2.10. Exercises 66 Chapter 3. Hilbert Spaces 73 3.1. Basic Properties of Inner-Products 73 3.2. Best Approximation and Orthogonal Projections 75 3.3. The Dual Space 78 3.4. Orthonormal Subsets 79 3.5. Weak Convergence in a Hilbert Space 86 3.6. Exercises 87 Chapter 4. Spectral Theory and Compact Operators 89 4.1. Definitions of the Resolvent and Spectrum 90 4.2. Basic Spectral Theory in Banach Spaces 91 4.3. Compact Operators on a Banach Space 93 4.4. Bounded Self-Adjoint Linear Operators on a Hilbert Space 99 4.5. Compact Self-Adjoint Operators on a Hilbert Space 104 4.6. The Ascoli-Arzel`aTheorem 107 4.7. Sturm Liouville Theory 109 4.8. Exercises 122 Chapter 5. Distributions 125 5.1. The Notion of Generalized Functions 125 5.2. Test Functions 127 5.3. Distributions 129 5.4. Operations with Distributions 133 3 4 CONTENTS 5.5. Convergence of Distributions and Approximations to the Identity 138 5.6. Some Applications to Linear Differential Equations 140 5.7. Local Structure of D0 148 5.8. Exercises 148 Chapter 6. The Fourier Transform 151 d 6.1. The L1(R ) Theory 153 6.2. The Schwartz Space Theory 157 d 6.3. The L2(R ) Theory 162 6.4. The S0 Theory 164 6.5. Some Applications 170 6.6. Exercises 172 Chapter 7. Sobolev Spaces 177 7.1. Definitions and Basic Properties 177 d 7.2. Extensions from Ω to R 181 7.3. The Sobolev Imbedding Theorem 185 7.4. Compactness 191 7.5. The Hs Sobolev Spaces 193 7.6. A Trace Theorem 198 7.7. The W s;p(Ω) Sobolev Spaces 203 7.8. Exercises 204 Chapter 8. Boundary Value Problems 207 8.1. Second Order Elliptic Partial Differential Equations 207 8.2. A Variational Problem and Minimization of Energy 210 8.3. The Closed Range Theorem and Operators Bounded Below 213 8.4. The Lax-Milgram Theorem 215 8.5. Application to Second Order Elliptic Equations 219 8.6. Galerkin Approximations 224 8.7. Green's Functions 226 8.8. Exercises 229 Chapter 9. Differential Calculus in Banach Spaces and the Calculus of Variations 233 9.1. Differentiation 233 9.2. Fixed Points and Contractive Maps 241 9.3. Nonlinear Equations 245 9.4. Higher Derivatives 252 9.5. Extrema 256 9.6. The Euler-Lagrange Equations 259 9.7. Constrained Extrema and Lagrange Multipliers 265 9.8. Lower Semi-Continuity and Existence of Minima 269 9.9. Exercises 273 Bibliography 279 CHAPTER 1 Preliminaries We discuss in this Chapter some of the pertinent aspects of topology and measure theory that are needed in the course of the rest of the book. We treat this material as background, and well prepared students may wish to skip either of both topics. 1.1. Elementary Topology In applied mathematics, we are often faced with analyzing mathematical structures as they might relate to real-world phenomena. In applying mathematics, real phenomena or objects are conceptualized as abstract mathematical objects. Collections of such objects are called sets. The objects in a set of interest may also be related to each other; that is, there is some structure on the set. We call such structured sets spaces. Examples. (1) A vector space (algebraic structure). (2) The set of integers Z (number theoretical structure or arithmetic structure). (3) The set of real numbers R or the set of complex numbers C (algebraic and topological structure). We start the discussion of spaces by putting forward sets of \points" on which we can talk about the notions of convergence or limits and associated continuity of functions. A simple example is a set X with a notion of distance between any two points of X.A 1 sequence fxngn=1 ⊂ X converges to x 2 X if the distance from xn to x tends to 0 as n increases. This definition relies on the following formal concept. Definition. A metric or distance function on a set is a function d : X × X ! R satisfying: (1) (positivity) for any x; y 2 X, d(x; y) ≥ 0, and d(x; y) = 0 if and only if x = y; (2) (symmetry) for any x; y 2 X, d(x; y) = d(y; x); (3) (triangle inequality) for any x; y; z 2 X, d(x; y) ≤ d(x; z) + d(z; y). A metric space (X; d) is a set X together with an associated metric d : X × X ! R. d d Example. (R ; j · j) is a metric space, where for x; y 2 R , the distance from x to y is d 1=2 X 2 jx − yj = (xi − yi) : i=1 It turns out that the notion of distance or metric is sometimes stronger than what actu- ally appears in practice. The more fundamental concept upon which much of the mathematics developed here rests, is that of limits. That is, there are important spaces arising in applied mathematics that have well defined notions of limits, but these limiting processes are not com- patible with any metric. We shall see such examples later; let it suffice for now to motivate a weaker definition of limits. 5 6 1. PRELIMINARIES 1 A sequence of points fxngn=1 can be thought of as converging to x if every \neighborhood" of x contains all but finitely many of the xn, where a neighborhood is a subset of points containing x that we think of as \close" to x. Such a structure is called a topology. It is formalized as follows. Definition. A topological space (X; T ) is a nonempty set X of points with a family T of subsets, called open, with the properties: (1) X 2 T , ; 2 T ; (2) If !1;!2 2 T , then !1 \ !2 2 T ; S (3) If !α 2 T for all α in some index set I, then α2I !α 2 T . The family T is called a topology for X. Given A ⊂ X, we say that A is closed if its complement Ac = X n A = fx 2 X : x 62 Ag is open. Example. If X is any nonempty set, we can always define the two topologies: (1) T1 = f;;Xg, called the trivial topology; (2) T2 consisting of the collection of all subsets of X, called the discrete topology. Proposition 1.1. The sets ; and X are both open and closed. Any finite intersection of open sets is open. Any intersection of closed sets is closed. The union of any finite number of closed sets is closed. Proof. We need only show the last two statements, as the first two follow directly from the definitions. Let Aα ⊂ X be closed for α 2 I. Then one of deMorgan's laws gives that c \ [ c Aα = Aα is open. α2I α2I Finally, if J ⊂ I is finite, the other deMorgan law gives c [ \ c Aα = Aα is open. α2J α2J It is often convenient to define a simpler collection of open sets that immediately generates a topology. Definition. Given a topological space (X; T ) and an x 2 X, a base for the topology at x is a collection Bx of open sets containing x such that for any open E 3 x, there is B 2 Bx such that x 2 B ⊂ E: A base for the topology, B, is a collection of open sets that contains a base at x for all x 2 X. Proposition 1.2. A collection B of subsets of X is a base for a topology T if and only if (1) each x 2 X is contained in some B 2 B and (2) if x 2 B1 \ B2 for B1; B2 2 B, then there is some B3 2 B such that x 2 B3 ⊂ B1 \ B2. If (1) and (2) are valid, then T = fE ⊂ X : E is a union of subsets in Bg : Proof. ()) Since X and B1 \ B2 are open, (1) and (2) follow from the definition of a base at x. (() Let T be defined as above. Then ; 2 T (the vacuous union), X 2 T by (1), and arbitrary unions of sets in T are again in T . It remains to show the intersection property. Let 1.1. ELEMENTARY TOPOLOGY 7 E1;E2 2 T , and x 2 E1 \ E2 (if E1 \ E2 = ;, there is nothing to prove). Then there are sets B1;B2 2 B such that x 2 B1 ⊂ E1 ; x 2 B2 ⊂ E2 ; so x 2 B1 \ B2 ⊂ E1 \ E2 : Now (2) gives B3 2 B such that x 2 B3 ⊂ E1 \ E2 : Thus E1 \ E2 is a union of elements in B, and is thus in T . We remark that instead of using open sets, one can consider neighborhoods of points x 2 X, which are sets N 3 x such that there is an open set E satisfying x 2 E ⊂ N.