On the linear independence condition for the Bobkov-Tanaka first eigenvalue of the double-phase operator

TL;DR

This paper examines the key linear independence condition necessary for the Bobkov-Tanaka first eigenvalue in double-phase operators, focusing on when the weight function is positive or the domain has symmetry and convexity.

Contribution

It identifies specific geometric and weight conditions that ensure the linear independence condition for the spectrum of double-phase operators.

Findings

The condition holds if the weight is strictly positive throughout the domain.

The condition is satisfied if the domain is convex and symmetric.

The paper clarifies the spectral properties under these geometric and weight assumptions.

Abstract

The paper investigates a pivotal condition for the Bobkov-Tanaka type spectrum for double-phase operators. This condition is satisfied if either the weight $w$ driving the double-phase operator is strictly positive in the whole domain or the domain is convex and fulfils a suitable symmetry condition.