Finsler-Rellich inequalities involving the distance to the boundary

TL;DR

This paper investigates Rellich inequalities linked to higher-order elliptic operators using Finsler metrics, providing sharp constants for half-spaces and improved estimates for convex domains.

Contribution

It introduces Finsler metric-based Rellich inequalities and derives sharp constants for half-spaces along with enhanced estimates for convex domains.

Findings

Sharp constant for half-space case

Improved estimates for convex domains

Rellich inequalities expressed via Finsler metric

Abstract

We study Rellich inequalities associated to higher-order elliptic operators in the Euclidean space. The inequalities are expressed in terms of an associated Finsler metric. In the case of half-spaces we obtain the sharp constant while for a general convex domain we obtain estimates that are better than those obtained by comparison with the polyharmonic operator.