Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations D. Henry London Mathematical Society Lecture Note Series 318, Cambridge University Press (2005) 214 pp., price USD 60.00 [48.00 electronic] ISBN:0521574919
Reviewer: R. Ghrist, University of Illinois, Urbana-Champaign, USA.

Level: advanced.

Most examples that one sees of boundary values problems for PDEs occur on Euclidean domains with a great deal of symmetry. That this is so does not imply that symmetric domains are prevalent in applications. Rather, this belies the fact that symmetric domains are those for which explicit examples are most easily solved. The irony of this situation is this: PDEs on domains which are not ``generic'' often exhibit degeneracies in their solutions. By introducing perturbations in the domain, one is led to problems for which explicit solutions are extraordinarily difficult to compute, but which nevertheless possess desirable qualitative properties. This assertion is the foundation of this fascinating monograph.

This thesis is not so counterintuitive to anyone who knows dynamical systems. The change in emphasis from explicit solutions of specific simple ODEs to qualitative behavior of generic classes of ODEs is one which has been both insightful and fruitful.

A simple example is to be found in the scalar Laplacian. Anyone working in differential equations has seen solutions to

D u + l u = 0; u: W ® R ; u( ¶ W ) = 0

in the case where Omega is a particularly simple domain, say, the unit square or the round disc. The nodal sets express this symmetry. It was proved in 1972 (by Micheletti and Uhlenbeck independently) that for a generic C² regular boundary, all the nonzero eigenvalues of this equation are simple (the eigenspaces are one dimensional).

There is a small but important history of such genericity theorems for PDEs. The 1976 paper by Uhlenbeck [Amer. J. Math.] proves that eigendecompositions of second-order elliptic operators are generically simple and that the eigenfunctions are Morse in their interior. Thus, nodal sets are neat submanifolds, and do not ``intersect'' as is often seen in pictures of explicit solutions on symmetric domains. This genericity is proved in detail for two specific cases: (1) first order perturbations to the operator, and (2) perturbations to the Riemannian metric. Uhlenbeck states that the proof holds for generic perturbations to the boundary of the domain, but that the computations are too messy to include. The challenges associated with that parameter space is where this monograph begins.

The techniques developed are fairly broad. The first few chapters of the book cover the preliminaries and simple examples of generic solution phenomena using the Implicit Function Theorem. Further results require a deep treatment of transversality theory and the smoothness issues associated with its applications. The last two chapters are particularly deep and difficult. The penultimate is an approach to boundary perturbations using weakly singular integral operators. The final chapter is a parallel approach to this same problem using pseudodifferential operators and rapidly oscillating solutions.

Although the material demands a level of detail that is often stifling in similar monographs, this text resists the temptation to descend into equation soup. In particular, there is a fair amount of explanatory prose in a style which is more chatty than terse. The overall tone of the book cannot be described as grave. This is an attractive feature. The book is exceptional in that it has taken over twenty years to write, and the end result is both careful and reflective.

This monograph is suitable for a special topics graduate course in PDEs or global analysis.

This text is available in electronic format at a twenty percent savings. The downloadable format is pdf.