On the First Eigenvalue of the Degenerate $p$-Laplace Operator in Non-Convex Domains

TL;DR

This paper provides lower bounds for the first non-trivial eigenvalues of the degenerate p-Laplace operator in various non-convex and fractal-boundary domains using geometric composition operator theory.

Contribution

It introduces a novel approach applying geometric composition operators to estimate eigenvalues in complex non-convex domains.

Findings

Lower bounds for eigenvalues in non-convex domains derived

Application to quasidiscs and snowflake-type fractal domains

Establishment of Poincaré-Sobolev inequality constants

Abstract

In this paper we obtain lower estimates of the first non-trivial eigenvalues of the degenerate $p$-Laplace operator, $p>2$, in a large class of non-convex domains. This study is based on applications of the geometric theory of composition operators on Sobolev spaces that permits us to estimates constants of Poincar\'e-Sobolev inequalities and as an application to derive lower estimates of the first non-trivial eigenvalues for the Alhfors domains (i.e. to quasidiscs). This class of domains includes some snowflakes type domains with fractal boundaries.