Smooth Homogenization of Heat Equations on Tubular Neighborhoods

TL;DR

This paper studies the behavior of heat equations on shrinking tubular neighborhoods around submanifolds, showing that solutions converge to a limit as the neighborhood collapses, with implications for geometric analysis.

Contribution

It introduces a homogenization result for heat equations on tubular neighborhoods, demonstrating convergence of rescaled semigroups in high-order Sobolev spaces.

Findings

Rescaled semigroups converge in Sobolev spaces as tube diameter tends to zero

Limit semigroup describes heat flow on the submanifold

Convergence holds in arbitrarily high Sobolev spaces

Abstract

We consider the heat equation with Dirichlet boundary conditions on the tubular neighborhood of a closed Riemannian submanifold. We show that, as the tube diameter tends to zero, a suitably rescaled and renormalized semigroup converges to a limit semigroup in Sobolev spaces of arbitrarily large Sobolev index.