Extremum Problems for Eigenvalues of Elliptic Operators

ISBN-13: 9783764377052 / Angielski / Miękka / 2006 / 202 str.

Antoine Henrot

Opis

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 Schrodinger 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."

Recenzje

From the reviews:

"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

“This is the first book devoted mainly to this subject and is therefore highly welcome. The book contains many interesting results, documents some recent progress and presents 30 open problems. … The book will help the readers (pure and applied mathematicians interested in this area) to update their knowledge in this lively field of research.” (M. Hoffmann-Ostenhof, Monatshefte für Mathematik, Vol. 159 (3), February, 2010)

Spis treści

Eigenvalues of elliptic operators.- Tools.- The first eigenvalue of the Laplacian-Dirichlet.- The second eigenvalue of the Laplacian-Dirichlet.- The other Dirichlet eigenvalues.- Functions of Dirichlet eigenvalues.- Other boundary conditions for the Laplacian.- Eigenvalues of Schrödinger operators.- Non-homogeneous strings and membranes.- Optimal conductivity.- The bi-Laplacian operator.