On domains of Euclidean spaces we consider inequalities for test functions and their Laplacians. We describe a family of domains having vanishing Rellich constants. For the Euclidean space of dimension 4 we present a new version of the Rellich inequality. In addition, we prove new one-dimensional Rellich-type integral inequalities for linear combinations of test functions and their derivatives of orders one and two.

Perturbation theory of eigenvalue problems

On Rellich’s Inequalities in Euclidean Spaces

The paper considers different formulations of inverse eigenvalue problems for matrices whose entries either polynomially or rationally depend on unknown parameters. An approach to solving inverse problems together with numerical algorithms is suggested. The solution of inverse problems is reduced to the problem of finding the so-called discrete solutions of nonlinear algebraic systems. The corresponding systems are constructed using the method of traces, and their discrete roots are found by applying the algorithms for solving nonlinear algebraic systems in several variables previously suggested by the author. Bibliography: 30 titles.

An Approach to Solving Inverse Eigenvalue Problems for Matrices

A sign characteristic for selfadjoint rational matrix functions

We consider the problem of distributing two conducting materials with a prescribed volume ratio in a given domain so as to minimize the first eigenvalue of an elliptic operator with Dirichlet conditions. The gap between the two conductivities is assumed to be small (low contrast regime). For any geometrical configuration of the mixture, we provide a complete asymptotic expansion of the first eigenvalue. We then consider a relaxation approach to minimize the second order approximation with respect to the mixture. We present numerical simulations in dimensions two and three.

Perturbation theory of eigenvalue problems

Citations

On Rellich’s Inequalities in Euclidean Spaces

An Approach to Solving Inverse Eigenvalue Problems for Matrices

A sign characteristic for selfadjoint rational matrix functions

Minimization of the ground state of the mixture of two conducting materials in a small contrast regime

Matrices over differential fields which commute with their derivative

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