Extremum problems for eigenvalues of elliptic operators.

*(English)*Zbl 1109.35081
Frontiers in Mathematics. Basel: Birkhäuser (ISBN 3-7643-7705-4/pbk). x, 202 p. (2006).

This monograph contains an extensive collection of solved and unsolved extremal problems regarding eigenvalues of elliptic operators. In particular, the author discusses questions of the following type:

(i) Which domain maximizes or minimizes a given eigenvalue (or a given function of the eigenvalues) of the Laplace (or, in a few cases, the bi-Laplace) operator with various boundary conditions and various geometric constraints? For example, on page 51 one finds Open Problem 2 regarding the Laplacian: Prove that the regular \(N\)-gone has the least first (Dirichlet) eigenvalue among all the \(N\)-gones of given area, for \(N \geq 5\) (e.g. the domain must have fixed volume).

(ii) Similar questions regarding other elliptic operators, like Schrödinger operators, vibrating string/membrane operators, as well as vibrating beam/plate operators. Here the unknown is not the shape of the domain, but the coefficient(s) of the operator.

The 1st chapter reviews some basic material on the spectral theory of elliptic operators, while the 2nd chapter presents the main tools used in the rest of the book (e.g. Schwarz rearrangement, Steiner symmetrization, and derivatives of eigenvalues with respect to the domain or the coefficients of the operator). Finally it is worth mentioning that the Bibliography contains 215 references.

The book can be valuable to pure and applied mathematicians, as well as theoretical physicists and applied scientists.

(i) Which domain maximizes or minimizes a given eigenvalue (or a given function of the eigenvalues) of the Laplace (or, in a few cases, the bi-Laplace) operator with various boundary conditions and various geometric constraints? For example, on page 51 one finds Open Problem 2 regarding the Laplacian: Prove that the regular \(N\)-gone has the least first (Dirichlet) eigenvalue among all the \(N\)-gones of given area, for \(N \geq 5\) (e.g. the domain must have fixed volume).

(ii) Similar questions regarding other elliptic operators, like Schrödinger operators, vibrating string/membrane operators, as well as vibrating beam/plate operators. Here the unknown is not the shape of the domain, but the coefficient(s) of the operator.

The 1st chapter reviews some basic material on the spectral theory of elliptic operators, while the 2nd chapter presents the main tools used in the rest of the book (e.g. Schwarz rearrangement, Steiner symmetrization, and derivatives of eigenvalues with respect to the domain or the coefficients of the operator). Finally it is worth mentioning that the Bibliography contains 215 references.

The book can be valuable to pure and applied mathematicians, as well as theoretical physicists and applied scientists.

Reviewer: Vassilis G. Papanicolaou (Athena)

##### MSC:

35P15 | Estimates of eigenvalues in context of PDEs |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35B37 | PDE in connection with control problems (MSC2000) |

49Q10 | Optimization of shapes other than minimal surfaces |

35P05 | General topics in linear spectral theory for PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

35J10 | Schrödinger operator, Schrödinger equation |

49J20 | Existence theories for optimal control problems involving partial differential equations |

49K20 | Optimality conditions for problems involving partial differential equations |