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The nonlinear Steklov problem in outward cuspidal domains

Pier Domenico Lamberti, Alexander Ukhlov

TL;DR

This paper investigates the nonlinear weighted Steklov problem in outward $γ$-cuspidal domains, addressing the challenge of noncompact traces in non-Lipschitz geometries. By leveraging the compactness of the weighted trace embedding, it proves a Friedrichs–Poincaré inequality and provides a variational characterization for the first nontrivial eigenvalue $λ_p$, showing existence of a corresponding weak eigenfunction. The results extend the linear ($p=2$) theory to the nonlinear regime $p≠2$ and highlight the influence of cusp geometry on spectral properties, with potential applications in continuum mechanics and related PDE contexts on singular domains.

Abstract

In this article, we consider the nonlinear Steklov eigenvalue problem in outward cuspidal domains. Using the compactness of the weighted trace embedding we obtain the variational characterization of the first non-trivial eigenvalue and prove the existence of a corresponding weak solution.

The nonlinear Steklov problem in outward cuspidal domains

TL;DR

This paper investigates the nonlinear weighted Steklov problem in outward -cuspidal domains, addressing the challenge of noncompact traces in non-Lipschitz geometries. By leveraging the compactness of the weighted trace embedding, it proves a Friedrichs–Poincaré inequality and provides a variational characterization for the first nontrivial eigenvalue , showing existence of a corresponding weak eigenfunction. The results extend the linear () theory to the nonlinear regime and highlight the influence of cusp geometry on spectral properties, with potential applications in continuum mechanics and related PDE contexts on singular domains.

Abstract

In this article, we consider the nonlinear Steklov eigenvalue problem in outward cuspidal domains. Using the compactness of the weighted trace embedding we obtain the variational characterization of the first non-trivial eigenvalue and prove the existence of a corresponding weak solution.
Paper Structure (3 sections, 5 theorems, 39 equations)

This paper contains 3 sections, 5 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Sobolev spaces
  3. Weighted Steklov $p$-eigenvalue problem

Key Result

Lemma 2.1

The space $W^{1,p}(\Omega)$, $1<p<\infty$, is real separable and uniformly convex Banach space.

Theorems & Definitions (8)

  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Definition 3.1
  • Theorem 3.2
  • proof