Perturbation of the boundary in boundary-value problems of partial differential equations / D. Henry ; with editorial assistance from Jack Hale and Antônio Luiz Pereira
Published
Cambridge, UK ; New York : Cambridge University Press, 2005
Geometrical preliminaries -- Differential calculus of boundary perturbations -- Examples using the implicit function theorem -- Bifurcation problems -- The tranversality theorem -- Generic perturbation of the boundary -- Boundary operators for second-order elliptic equations -- The method of rapidly-oscillating solutions
Summary
Perturbation 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. Here 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
