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

Baruch Schneider, Diana Schneiderová, Yifan Zhang

Abstract

We study the small-hole minimization problem for the first Dirichlet eigenvalue in the square \[ Q=(-1,1)^2, \qquad Λ_r(x_1,x_2)=λ_1\Bigl(Q\setminus\bigl(\overline{B_r(x_1)}\cup \overline{B_r(x_2)}\bigr)\Bigr), \] 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.

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

Abstract

We study the small-hole minimization problem for the first Dirichlet eigenvalue in the square where two equal disjoint hard circular obstacles of radius move inside . We prove that, as , 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 -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.

Paper Structure

This paper contains 16 sections, 34 theorems, 285 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The side-tangent one-hole branch
  3. The unperturbed square
  4. Odd reflection and the away-from-corners regime
  5. Endpoint scaling and double reflection
  6. Reduction to the quadrant cell
  7. Far-field coefficients and exact identities
  8. A shifted comparison and monotonicity of the endpoint coefficient
  9. Adjacent corner pairs beat opposite corner pairs
  10. Same-corner branch: one-hole comparison, scalar reduction, and exclusion
  11. A supplementary coercive estimate
  12. A monotone one-hole corner-cell family
  13. Same-corner small-hole asymptotic reduction
  14. Numerical validation
  15. Distinct-corner asymptotics and proof of the main theorem
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(x_{1,r},x_{2,r})\in\mathcal{C}_r$ minimize $\Lambda_r$ over $\mathcal{C}_r$. Then, after relabeling the two centers and applying a symmetry of the square if needed, Equivalently, as $r\to0$, every asymptotically minimizing configuration consists of two equal holes tangent to two adjacent corners of the square; among all corner pairs, the adjacent true-corner configuration is the unique asym

Figures (3)

  • Figure 1: Representative two-hole geometries used in the FEM validation package: adjacent corners, opposite corners, opposite sides at the center, and a same-corner clustered contact-like configuration.
  • Figure 2: First FEM eigenfunctions for the four representative benchmark geometries at $r=0.08$.
  • Figure 3: Left: FEM comparison of the four representative two-hole branches. Right: the isolated positive gap $\lambda_{\mathrm{opp}}-\lambda_{\mathrm{adj}}$, showing that the adjacent-corner branch stays below the opposite-corner branch across the sampled radii.

Theorems & Definitions (67)

  • Theorem 1.1: Main theorem
  • Lemma 2.1
  • proof
  • Proposition 2.2: Odd reflection
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • ...and 57 more