Weighted Neumann-to-Steklov limits for nonlinear eigenvalues and trace constants
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| Format: | Preprint |
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2026
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| _version_ | 1866909030956400640 |
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| author | Menovschikov, Alexander |
| author_facet | Menovschikov, Alexander |
| contents | We study a nonlinear Neumann-to-Steklov limit generated by a family of interior weights concentrating at the boundary. On a class of admissible possibly irregular domains obtained from the unit ball by trace-compatible Sobolev homeomorphisms, we consider the first nontrivial weighted \((p,q)\)-Neumann eigenvalue with respect to a concentrating bulk weight \(γ_a\). We prove that, as \(a\to0\), these eigenvalues converge to the corresponding weighted \((p,q)\)-Steklov eigenvalue with boundary weight \(β\). Moreover, normalized minimizers converge, up to subsequences, strongly in \(W^{1,p}\) to Steklov minimizers. Equivalently, the best constants in the weighted Poincaré inequalities converge to the best constants in the weighted trace inequalities; in fact, a quantitative convergence estimate is obtained in the subcritical trace range. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_09759 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weighted Neumann-to-Steklov limits for nonlinear eigenvalues and trace constants Menovschikov, Alexander Analysis of PDEs Functional Analysis Primary 46E35, Secondary 35P30, 35J92, 35B40, 35B25, 30C65 We study a nonlinear Neumann-to-Steklov limit generated by a family of interior weights concentrating at the boundary. On a class of admissible possibly irregular domains obtained from the unit ball by trace-compatible Sobolev homeomorphisms, we consider the first nontrivial weighted \((p,q)\)-Neumann eigenvalue with respect to a concentrating bulk weight \(γ_a\). We prove that, as \(a\to0\), these eigenvalues converge to the corresponding weighted \((p,q)\)-Steklov eigenvalue with boundary weight \(β\). Moreover, normalized minimizers converge, up to subsequences, strongly in \(W^{1,p}\) to Steklov minimizers. Equivalently, the best constants in the weighted Poincaré inequalities converge to the best constants in the weighted trace inequalities; in fact, a quantitative convergence estimate is obtained in the subcritical trace range. |
| title | Weighted Neumann-to-Steklov limits for nonlinear eigenvalues and trace constants |
| topic | Analysis of PDEs Functional Analysis Primary 46E35, Secondary 35P30, 35J92, 35B40, 35B25, 30C65 |
| url | https://arxiv.org/abs/2605.09759 |