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

TL;DR

This paper investigates the convergence of nonlinear Neumann-to-Steklov eigenvalues and trace constants on irregular domains as interior weights concentrate at the boundary, establishing limits and convergence rates.

Contribution

It introduces a new limit process for weighted nonlinear eigenvalues on irregular domains and proves convergence of eigenvalues and minimizers with quantitative estimates.

Findings

Eigenvalues converge to Steklov eigenvalues as weights concentrate.

Normalized minimizers strongly converge to Steklov minimizers.

Quantitative convergence estimates are obtained in the subcritical trace range.

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.