L2-Homogenization of Heat Equations on Tubular Neighborhoods

TL;DR

This paper studies the behavior of heat equations on tubular neighborhoods of submanifolds, demonstrating that as the neighborhood shrinks, the heat dynamics converge to a Hamiltonian on the submanifold influenced by geometric properties.

Contribution

It introduces a method to approximate heat equations on shrinking tubular neighborhoods by a Hamiltonian on the submanifold, incorporating geometric effects.

Findings

Effective description of heat semigroup via Hamiltonian

As tube radius decreases, convergence to submanifold Hamiltonian

Geometry influences the limiting potential

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 radius decreases, the semigroup of a suitably rescaled and renormalized generator can be effectively described by a Hamiltonian on the submanifold with a potential that depends on the geometry of the submanifold and of the embedding.