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

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.