Henry D. Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations

Cambridge University Press, 2005. — 216 p. — (London Mathematical Society Lecture Note Series 318) — ISBN: 0521574919Perturbation of the boundary is a rather neglected topic in the study of PDEs for two main reasons. First, on the surface it appears trivial, merely a change of variables and an application of the chain rule. Second, carrying out such a change of variables frequently results in long and difficult calculations. In this book, first published in 2005, the author carefully discusses a calculus that allows the computational morass to be bypassed, and he goes on to develop more general forms of standard theorems, which help answer a wide range of problems involving boundary perturbations. Many examples are presented to demonstrate the usefulness of the author's approach, while on the other hand many tantalizing open questions remain. Anyone whose research involves PDEs will find something of interest in this book.Geometrical Preliminaries
Differential Calculus of Boundary Perturbations
Examples Using the Implicit Function Theorem
Bifurcation Problems
The Transversality Theorem
Generic Perturbation of the Boundary
Boundary Operators for Second-Order Elliptic Equations
The Method of Rapidly-Oscillating Solutions
Appendix 1 Eigenvalues of the Laplacian in the Presence of Symmetry
Appendix 2 On Micheletti’s Metric Space