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 (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.

• The paper establishes the rigorous convergence of weighted (p,q)-Neumann eigenvalues to Steklov eigenvalues as the interior weight concentrates on the boundary.
• It employs advanced variational methods, composition operators, and change-of-variable techniques to address irregular domains and singular measures.
• Quantitative estimates provide explicit asymptotic bounds that support applications in spectral optimization and boundary control problems.

Weighted Neumann-to-Steklov Limits for Nonlinear Eigenvalues and Trace Constants

Problem Formulation and Analytic Framework

The paper investigates the asymptotic behavior of weighted nonlinear spectral problems as interior weights in the domain concentrate at the boundary, focusing on the convergence of (p,q)-Neumann eigenvalues to corresponding Steklov eigenvalues for the p-Laplace operator. The analysis is performed in the context of admissible, potentially irregular domains, defined as images of the unit ball through trace-compatible Sobolev homeomorphisms with weak p-quasiconformality and other geometric properties. The function spaces, weighted Lebesgue and Sobolev spaces, trace spaces, and composition operators induced by these homeomorphisms are developed rigorously, accommodating the transfer mechanism from the ball to general domains.

Key technical elements include the use of change-of-variable formulae for both the bulk and boundary, Radon measures, volume derivatives, and the doubling property for rectifiable boundaries. The dense embedding and compactness results, and the transfer of Sobolev and trace information from the model domain via composition operators are established, ensuring the validity of variational formulations for weighted spectral problems in irregular geometric settings.

Variational Characterization and Main Results

The principal objects are the first nontrivial weighted (p,q)-Neumann eigenvalue $A_{p,q}(\mathcal{Y}_a)$ associated with an interior weight ${\mathcal Y}_a$ concentrating at the boundary, and its boundary limit, the weighted (p,q)-Steklov eigenvalue $A^{St}_{p,q}(\mathcal{B})$ involving the induced boundary weight $\mathcal{B}$. Both eigenvalues are variationally characterized via minimization over Sobolev spaces with suitable orthogonality or nontriviality conditions, leading to sharp constants for weighted Poincaré and trace inequalities.

Strong numerical results are demonstrated: quantitative estimates provide rates for the convergence of best constants $$C^N_{p,q}(\mathcal{Y}_a) \to C^{St}_{p,q}(\mathcal{B})$$ and eigenvalues $$A_{p,q}(\mathcal{Y}_a) \to A^{St}_{p,q}(\mathcal{B})$$ as $a \to 0$, with explicit asymptotic bounds of the form $$|C^N_{p,q}(\mathcal{Y}_a) - C^{St}_{p,q}(\mathcal{B})| \leq C_s a^s$$ for $s$ below a threshold determined by embedding and concentration exponents. Normalized minimizers for the Neumann problem are shown to converge strongly in $W^{1,p}$ to Steklov minimizers, up to subsequences. Equivalent convergence is established for the best constants in Poincaré and trace inequalities.

The proofs leverage detailed analysis of concentration mechanisms on the unit ball, including Poisson extensions, boundary layer estimates for zero-trace functions, and the convergence of weighted moments. These are transferred to general domains via composition operators, mapping Sobolev and trace data under weak p-quasiconformal homeomorphisms, and the induced measures. Quantitative comparisons of quotient seminorms between bulk and boundary representations play a critical role in bridging variational levels.

Practical and Theoretical Implications

This work establishes a robust nonlinear Neumann-to-Steklov limit for weighted eigenvalue problems on broad classes of domains, including irregular and singular geometries such as bilipschitz images and outward cuspidal domains. The analytic machinery developed—composition operators, induced weighted embeddings, and precise concentration analysis—extends classical spectral geometry into more generalized geometric and measure-theoretic settings.

From a practical standpoint, these results provide rigorous justification for asymptotic spectral reductions in PDEs with mass concentration phenomena and supports numerical approximations for trace inequalities, boundary spectral optimization, and nonlinear transmission problems in domains beyond the reach of classical smooth theory. The quantitative convergence and strong compactness results secure the stability of variational computations under singular perturbations.

Theoretically, this approach unifies disparate research threads in nonlinear spectral theory, measure concentration, and Sobolev/trace embedding analysis. It resolves the limiting behavior of nonlinear eigenvalues and sharp constants, connecting bulk and boundary phenomena in a precise manner. The techniques offer potential for further extension to more general boundary concentration profiles, weights, and fractional or singular operators, possibly intersecting with shape optimization, homogenization, and geometric measure theory.

Future Directions

Several avenues are highlighted for future development. The analytic framework is adaptable to a wider range of concentrating densities, boundary weights, and measure schemes—extending the results to cases with weaker regularity assumptions, more general domains, and possibly fractal boundaries. The compactness and convergence mechanisms could be further refined to include spectral stability under domain perturbation or for higher eigenvalue levels. There is room for generalization to vector-valued or system PDE settings, fractional Laplacians, and coupled boundary-value problems.

Applications in geometric spectral theory, optimal inequalities, and boundary control in nonlinear PDEs can benefit from these methodologies. The quantitative asymptotic estimates enable rigorous error analysis for numerical schemes and optimization routines, particularly in singular or near-boundary concentrated configurations.

Conclusion

This paper provides a technically sophisticated and rigorous treatment of nonlinear Neumann-to-Steklov limits for weighted eigenvalue problems, establishing quantitative convergence between variational levels and minimizers in admissible domains with boundary concentration. The combination of geometric measure theory, composition operator techniques, and variational spectral analysis represents a substantial methodological advance with broad implications for nonlinear spectral geometry and applied analytic PDE theory.