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arXiv · 2605.09759

Weighted Neumann-to-Steklov limits for nonlinear eigenvalues and trace constants

Alexander Menovschikov
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Abstract

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 \(\gamma_a\). We prove that, as \(a\to0\), these eigenvalues converge to the corresponding weighted \((p,q)\)-Steklov eigenvalue with boundary weight \(\beta\). Moreover, normalized minimizers converge, up to subsequences, strongly in \(W^{1,p}\) to Steklov minimizers. Equivalently, the best constants in the weighted Poincar\'e inequalities converge to the best constants in the weighted trace inequalities; in fact, a quantitative convergence estimate is obtained in the subcritical trace range.

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Alexander Menovschikov. 2026-05-10. Weighted Neumann-to-Steklov limits for nonlinear eigenvalues and trace constants. https://arxiv.org/abs/2605.09759

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