دانلود کتاب Hilbert Space Methods for Partial Differential Equations

فهرست مطالب :

Contents

I Elements of Hilbert Space . . . . . . . . . . . 1
1 Linear Algebra . . . . . . . . . . . 1
2 Convergence and Continuity . . . . . . . . . . . 6
3 Completeness . . . . . . . . . . . . . . . . . . . . . 10
4 HilbertSpace. . . . . . . . . . . . . . . . . . . . . 14
5 Dual Operators; Identifications 19
6 Uniform Boundedness; Weak Compactness. . 22
7 Expansion in Eigenfunctions. . . . . . . 24

II Distributions and Sobolev Spaces . . . . . . . . . . . . . . .33
1 Distributions. . . . . . . . . . . . . . . 33
2 SobolevSpaces . . . . . . . . . . . . . . . 42
3 Trace . . . . . . . . . . . . . . 47
4 Sobolev’s Lemma and Imbedding . . . . . . . . . . . . . . . 50
5 Density and Compactness. . . . . . . . . . . . . . . 53

III Boundary Value Problems . . . . . . . . . . . . . . . 61
1 Introduction. . . . . . . . . . . . . . . . . . . . 61
2 Forms, Operators and Green’s Formula. . . . . . . . 63
3 Abstract Boundary Value Problems. . . . . . . . . . . 67
4 Examples . . . . . . . . . . . . . . .. . . . . . . 69
5 Coercivity; Elliptic Forms . . . . . . . . . . . 76
6 Regularityn . . . . . . . . . . . . . . . . 79
7 Closed operators, adjoints and eigenfunction expansions 85

IV First Order Evolution Equations 97
1 Introduction . . . . . . . . . . . 97
2 The Cauchy Problem . . . . . . . . . . . 100
3 Generation of Semigroups . . . . . . . . . . . 102
4 Accretive Operators; two examples . . . . . . . . . . . . . . . 107
5 Generation of Groups; a wave equation. . . . . . . . . . . . . 111
6 Analytic Semigroups . . . . . . . . . . . . . . . 115
7 Parabolic Equations . . . . . . . . . . . . . . . 121

V Implicit Evolution Equations 129
1 Introduction............................129
2 Regular Equations........................130
3 Pseudoparabolic Equations . . . . . . . . . . . . . . . . . . . 134
4 Degenerate Equations. . . . . . . . . . . . . . . . . . . 138
5 Examples.............................140

VI Second Order Evolution Equations 147
1 Introduction............................147
2 Regular Equations........................148
3 SoboleV Equations........................156
4 Degenerate Equations......................158
5 Examples.............................162

VI Optimization and Approximation Topics 171
1 Dirichlet’s Principle.......................171
2 Minimization of Convex Functions . . . . . . . . . . . . . . . 172
3 Variational Inequalities. . . . . . . . . . . . . . . . . . . . .. 178
4 Optimal Control of Boundary Value Problems. . . . . . . . . 182
5 Approximation of Elliptic Problems . . . . . . . . . . . . . . 189
6 Approximation of Evolution Equations . . . . . . . . . . . . . 197

VI Suggested Readings 209