Optimizing the first eigenvalue of some quasilinear operators with respect to the boundary conditions

TL;DR

This paper investigates how to optimize the first eigenvalue of certain quasilinear operators on bounded domains by adjusting Robin-type boundary conditions, providing bounds and characterizations of optimal conditions.

Contribution

It characterizes the boundary conditions that optimize the first eigenvalue for a class of quasilinear operators and establishes bounds on these eigenvalues.

Findings

Identifies boundary conditions that maximize or minimize the first eigenvalue.

Provides explicit upper and lower bounds for the eigenvalues.

Describes the structure of optimal boundary conditions.

Abstract

We consider a class of quasilinear operators on a bounded domain $\Omega\subset \mathbb R^n$ and address the question of optimizing the first eigenvalue with respect to the boundary conditions, which are of the Robin-type. We describe the optimizing boundary conditions and establish upper and lower bounds on the respective maximal and minimal eigenvalue.