Symmetry Breaking and Other Phenomena in the Optimization of Eigenvalues for Composite Membranes

Abstract

We consider the following eigenvalue optimization problem: Given a bounded domain ⊂ R and numbers α > 0, A ∈ [0, | |], find a subset D ⊂ of area A for which the first Dirichlet eigenvalue of the operator − + αχ D is as small as possible.We prove existence of solutions and investigate their qualitative properties. For example, we show that for some symmetric domains (thin annuli and dumbbells with narrow handle) optimal solutions must possess fewer symmetries than ; on the other hand, for convex reflection symmetries are preserved.Also, we present numerical results and formulate some conjectures suggested by them.

Finite element approximation to the extremal eigenvalue problem for inhomogenous materials

“…5. It has been discussed in [6], when the annulus becomes "thinner", the minimize density function of Problem 2.2 loses rotational symmetry, which is verified numerically in Fig. 5.…”

“…Problem 3.1 with two different materials over the disk has been considered in [6,24]. It was shown that in order to obtain the minimum, the material with high density has to be placed in the center of the disk and the remaining annulus is filled by the low density material.…”

“…6 We consider two cases of the annulus which has rotational symmetry. The inner radius are r = 1 and r = 0.2, respectively.…”

“…The mathematical theory of extremal eigenvalue problems have been discussed in a lot of papers and monographs, see [6,[9][10][11]16] and further references cited there. In [12,17,20], the generalization to the p-Laplacian operator, to the Robin boundary condition and the general (not necessarily positive) density function ρ are also considered.…”

Monotonicity and stability of optimal solutions of a minimization problem

“…Surely, this method can also be applied to many other optimization problems, however, we should warn the readers that the method has its limitations. For example, in [3,4], the authors explore the possibility of designing a membrane, fixed at the boundary, and made out of two materials, so that the corresponding frequency is maximal. This is shape optimization problem to which the method of tangent cones can certainly be applied.…”