Spectral bounds for the operator pencil of an elliptic system in an angle

TL;DR

This paper derives spectral bounds for elliptic systems in a plane angle, focusing on Dirichlet and mixed boundary conditions, leading to new regularity results and advancing the understanding of spectral properties in elliptic PDEs.

Contribution

It introduces a novel framework for spectral analysis of elliptic systems in an angle, providing new bounds for mixed boundary conditions and recovering known bounds for Dirichlet cases.

Findings

Optimal bounds on $| ext{Re} \, \lambda|$ for Dirichlet and mixed conditions

New regularity results for elliptic systems with mixed boundary conditions

Framework employing numerical range analysis and accretive operator theory

Abstract

The model problem of a plane angle for a second-order elliptic system subject to Dirichlet, mixed, and Neumann boundary conditions is analyzed. For each boundary condition, the existence of solutions of the form $r^\lambda v$ is reduced to spectral analysis of a particular matrix. Focusing on Dirichlet and mixed boundary conditions, optimal bounds on $|\Re \lambda|$ are derived, employing tools from numerical range analysis and accretive operator theory. The developed framework is novel and recovers known bounds for Dirichlet boundary conditions. The results for mixed boundary conditions are new and represent the central contribution of this work. Immediate applications of these findings are new regularity results for linear second-order elliptic systems subject to mixed boundary conditions.