The Hardy inequality and the heat flow in curved wedges

TL;DR

This paper investigates how curvature affects heat decay in wedges, establishing a decay rate formula and an improved Hardy inequality, with implications for understanding heat flow in curved geometries.

Contribution

It introduces a method using self-similar variables to analyze heat decay in curved wedges and proves an improved Hardy inequality for such geometries.

Findings

Decay rate depends on curvature and opening angle

Develops a new method for analyzing heat equations in curved domains

Proposes a conjecture for further decay rate improvements

Abstract

We show that the polynomial decay rate of the heat semigroup of the Dirichlet Laplacian in curved planar wedges equals the sum of the usual dimensional decay rate and a multiple of the reciprocal value of the opening angle. To prove the result, we develop the method of self-similar variables for the associated heat equation and study the asymptotic behaviour of the transformed non-autonomous parabolic problem for large times. We also establish an improved Hardy inequality for the Dirichlet Laplacian in non-trivially curved wedges and state a conjecture about an improved decay rate in this case.