Global small-hole minimization of the first Dirichlet eigenvalue in a square with two hard obstacles

Abstract

We study the global minimization of the first Dirichlet eigenvalue of a square containing two equal non-overlapping circular obstacles as their common radius $r$ tends to zero. A capacitary localization theorem first shows that every obstacle in a configuration whose eigenvalue excess above $\pi^2/2$ is $O(r^4)$ must lie in a fixed $O(r)$ corner layer; this excludes interior, open-side, and intermediate boundary scales. We then analyze the resulting corner cells. The correct leading functional is the full $u$-capacity, containing both the exterior harmonic corrector energy and the polynomial energy inside the reflected obstacles. Its monotonicity forces asymptotic true-corner tangency, while a direct quantitative capacity estimate excludes two-hole clustering at one corner. For holes near distinct corners, an exact polarization argument shows that a diagonally opposite placement is strictly improved by reflecting one obstacle to an adjacent corner. Hence every unrestricted global minimizer is asymptotic, up to the symmetries of the square and interchange of the holes, to a pair of disks tangent at adjacent corners, in the sense that the center errors are $o(r)$. We also obtain the leading $r^4$ expansion of the global minimum. A reproducible finite element study provides a qualitative validation of the representative branch ordering.