Showalter R.E. Hilbert Space Methods for Partial Differential Equations

Austin, Texas, 1977. — 219 p. OCR.
Elements of Hilbert Space.
Linear Algebra.
Convergence and Continuity.
Completeness.
Hilbert Space.
Dual Operators; Identifications.
Uniform Boundedness; Weak Compactness.
Expansion in Eigenfunctions.
Distributions and Sobolev Spaces.
Distributions.
Sobolev Spaces.
Trace.
Sobolev's Lemma and Imbedding.
Density and Compactness.
Boundary Value Problems.Forms, Operators and Green's Formula.
Abstract Boundary Value Problems.
Examples.
Coercivity; Elliptic Forms.
Regularity.
Closed operators, adjoints and eigenfunction expansions.
First Order Evolution Equations.The Cauchy Problem.
Generation of Semigroups.
Accretive Operators; two examples.
Generation of Groups; a wave equation.
Analytic Semigroups.
Parabolic Equations.
Implicit Evolution Equations.Regular Equations.
Pseudoparabolic Equations.
Degenerate Equations.
Examples.
Second Order Evolution Equations.Regular Equations.
Sobolev Equations.
Degenerate Equations.
Examples.
Optimization and Approximation Topics.
Dirichlet's Principle.
Minimization of Convex Functions.
Variational Inequalities.
Optimal Control of Boundary Value Problems.
Approximation of Elliptic Problems.
Approximation of Evolution Equations.
Stuggested Readings.