Asymptotic minimization of the first Dirichlet eigenvalue in a square with two hard obstacles

Abstract

We study an asymptotic minimization problem for the first Dirichlet eigenvalue in the square $Q=(-1,1)^2$, where two equal hard circular obstacles of radius $r$ move inside $Q$ with disjoint interiors; tangency between the obstacles and with the outer boundary is allowed. We prove that, as $r\to0$, every minimizing configuration approaches, up to the symmetries of the square and interchange of the holes, a pair of corner-tangent obstacles at adjacent corners, with displacement $o(r)$ from the corresponding corner-tangent centers. A uniform $u$-capacity localization argument shows that any obstacle whose individual eigenvalue excess is $O(r^4)$ must lie in a fixed $O(r)$ neighborhood of a corner, thereby excluding interior, open-side, and intermediate approaches to the boundary. At the corner scale, the leading coefficient is the full $u$-capacity. The proof is then organized by geometric branches. The side-tangent one-hole branch is asymptotically driven to the adjacent corner; an exact polarization argument shows that the adjacent corner-tangent pair has smaller first eigenvalue than the opposite pair; and configurations concentrated near a single corner are excluded by a scalar one-hole comparison. Additivity for well-separated corner defects then completes the distinct-corner analysis. We also include reproducible finite element computations that support the branch ordering predicted by the proof.