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

TL;DR

This paper investigates how two small circular obstacles inside a square influence the first Dirichlet eigenvalue, showing that optimal configurations as the obstacles shrink tend to be corner-tangent pairs at adjacent corners.

Contribution

It characterizes the asymptotically optimal obstacle placements inside a square for minimizing the first eigenvalue as obstacle size approaches zero.

Findings

Optimal configurations are corner-tangent pairs at adjacent corners.

Corner-tangent obstacles asymptotically minimize the eigenvalue.

Finite element validation supports the theoretical results.

Abstract

We study the small-hole minimization problem for the first Dirichlet eigenvalue in the square \[ Q=(-1,1)^2, \qquad \Lambda_r(x_1,x_2)=\lambda_1\Bigl(Q\setminus\bigl(\overline{B_r(x_1)}\cup \overline{B_r(x_2)}\bigr)\Bigr), \] where two equal disjoint hard circular obstacles of radius $r$ move inside $Q$. We prove that, as $r\to0$, every minimizing configuration consists, up to the dihedral symmetries of the square and interchange of the two holes, of two true corner-tangent obstacles located at adjacent corners. The argument is organized by geometric branches. On the side-tangent one-hole branch, odd reflection and simple-eigenvalue $u$-capacity asymptotics show that the true corner is the unique asymptotic minimizer. For configurations with holes near two distinct corners, an exact polarization argument proves that the adjacent true-corner pair strictly beats the opposite pair. For same-corner clusters, a reflected comparison principle reduces the two-hole cell problem to a scalar one-hole inequality, which is then closed by an explicit competitor. We also include a reproducible finite element validation that supports the analytic branch ordering.