Shape optimization problems for eigenvalues of elliptic operators

Abstract

We consider a general formulation for shape optimization problems involving the eigenvalues of the Laplace operator. Both the cases of Dirichlet and Neumann conditions on the free boundary are studied. We survey the most recent results concerning the existence of optimal domains, and list some conjectures and open problems. Some open problems are supported by efficient numerical computations.

On a shape design problem for one spectral functional

“…The numerical study of this problem has been performed in [15]. Roughly speaking, existence results for such a kind of problems are usually obtained and the problem is well posed in the cases when the admissible domains satisfy some geometrical restrictions, the functional satisfies some monotonicity assumptions, or it depends only on the lower part of the spectrum [1].…”

“…This constraint often produces additional difficulties that lead to a lack of existence of a solution of the shape optimization problem and to the introduction of suitable relaxed formulation of the problem. However, in some cases an optimal solution exists, due to the special form of the cost functional and geometrical restrictions on the class of admissible domains [1]. Another difficulty is related mainly to the mathematical definition of the variation of the domain, characterized by the variation of its boundary.…”

“…1 .D// D F . 1 .D// C G.mes D/; (3.1)where F and G are differentiable functions,1 .D/ is the first eigenvalue of problem (2.1)-(2.2) in the domain D 2 K. Note that considering the relation of the eigenvalues with the applications, various functionals involving eigenvalues are of interest for research. For instance, for the problem of the form min¹ˆ.…”

Spectral optimization problems

“…Some more issues on spectral optimization problems for the Neumann-Laplacian, together with related numerical computations, can be found in [15]. One could also consider mixed boundary conditions: Dirichlet on one part and Neumann on the remaining part.…”

Exponential decay phenomenon of the principal eigenvalue of an elliptic operator with a large drift term of gradient type

“…For the proof of this theorem, see Section 8.12 of [8] and [1], etc. From the last assertion of this theorem, without loss of generality, we can assume that…”