Small-hole minimization of the first Dirichlet eigenvalue in a square with two hard obstacles
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| Format: | Preprint |
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2026
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| author | Schneider, Baruch Schneiderová, Diana Zhang, Yifan |
| author_facet | Schneider, Baruch Schneiderová, Diana Zhang, Yifan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_29015 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Small-hole minimization of the first Dirichlet eigenvalue in a square with two hard obstacles Schneider, Baruch Schneiderová, Diana Zhang, Yifan Analysis of PDEs Spectral Theory 35P15, 49Q10, 35J05, 65N30 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. |
| title | Small-hole minimization of the first Dirichlet eigenvalue in a square with two hard obstacles |
| topic | Analysis of PDEs Spectral Theory 35P15, 49Q10, 35J05, 65N30 |
| url | https://arxiv.org/abs/2603.29015 |