log-concavity of eigenfunction and Brunn-Minkowski inequality of eigenvalue for weighted p-Laplace operator

TL;DR

This paper studies the log-concavity of the first eigenfunction and establishes a Brunn-Minkowski inequality for the first eigenvalue of the weighted p-Laplace operator in convex domains, advancing understanding of spectral properties in weighted settings.

Contribution

It proves the log-concavity of the first eigenfunction and a Brunn-Minkowski inequality for the first eigenvalue of the weighted p-Laplace operator in convex domains.

Findings

Log-concavity of the first eigenfunction established.

Brunn-Minkowski inequality for the first eigenvalue proved.

Results apply to convex and smooth domains in ^n.

Abstract

In this paper, we investigate the log-concavity property of the first eigenfunction to the weighted $p$-Laplace operator in class of bounded, convex and smooth domain. Moreover, we prove a Brunn-Minkowski-type inequality for the first eigenvalue to the weighted $p$-Laplace operator in the class of $C^2$ convex bodies in $\R^n$