Line segment energy and applications

TL;DR

This paper introduces derivatives of line segment energy for symmetric tensor fields and applies them to derive generalized log-concavity estimates for heat equation solutions and eigenfunctions on convex domains.

Contribution

It presents new derivative formulas for line segment energy and extends log-concavity estimates to broader classes of solutions on convex domains.

Findings

Derived derivatives of line segment energy for symmetric tensor fields.

Extended log-concavity estimates for heat solutions and eigenfunctions.

Applicable to bounded strictly convex domains.

Abstract

In this paper, we compute the derivatives of the line segment energy for a symmetric tensor field and apply them to obtain slightly more general log-concavity estimates for positive solutions of heat equations and first eigenfunctions on bounded strictly convex domains.