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Antoine Henrot, "Extremum Problems for Eigenvalues of Elliptic Operators"

Problems linking the shape of a domain or the coefficients of an elliptic operator to the sequence of its eigenvalues are among the most fascinating of mathematical analysis. In this book, we focus on extremal problems. For instance, we look for a domain which minimizes or maximizes a given eigenvalue of the Laplace operator with various boundary conditions and various geometric constraints. We also consider the case of functions of eigenvalues. We investigate similar questions for other elliptic operators, such as the Schrödinger operator, non homogeneous membranes, or the bi-Laplacian, and we look at optimal composites and optimal insulation problems in terms of eigenvalues.

Providing also a self-contained presentation of classical isoperimetric inequalities for eigenvalues and 30 open problems, this book will be useful for pure and applied mathematicians, particularly those interested in partial differential equations, the calculus of variations, differential geometry, or spectral theory

Review

"The book is a good collection of extremal problems for eigenvalues of elliptic operators and it gives a good account of the present state of research. It presents 30 open problems and is an absolutely necessary starting point for research work in this field. All proofs are strictly rigorous and the author refers for some other proofs to the bibliography, which contains 215 references. The material is interesting for specialists in both pure and applied mathematics, and can also be used in students' work." —Mathematical Reviews

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