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

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.

Some small mistakes have been corrected and some arguments shortened