Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Heat flow within convex sets

James Dibble

TL;DR

This work extends the classical maximum-principle behavior of the heat flow for maps between Riemannian manifolds to arbitrary compact locally convex target subsets. By marrying Hamilton's geometric estimates with Evans'-style viscosity methods, the author shows that solutions to $\partial_t u=\tau$ cannot leave a compact locally convex set $Y\subset N$ before the boundary image does, provided the initial and boundary data lie in $Y$. The core technical advance is converting the distance to $Y$ into a viscosity subsolution via $\rho=d_Y\circ u$ and applying a parabolic maximum principle, aided by a detailed analysis of projections and the second fundamental form. The paper also discusses a potential Riemannian analogue of Evans' strong maximum principle, which would yield a sharp rigidity statement for the flow when the boundary meets $\partial Y$.

Abstract

A solution to the heat equation between Riemannian manifolds, where the domain is compact and possibly has boundary, will not leave a compact and locally convex set before the image of the boundary does.

Heat flow within convex sets

TL;DR

This work extends the classical maximum-principle behavior of the heat flow for maps between Riemannian manifolds to arbitrary compact locally convex target subsets. By marrying Hamilton's geometric estimates with Evans'-style viscosity methods, the author shows that solutions to cannot leave a compact locally convex set before the boundary image does, provided the initial and boundary data lie in . The core technical advance is converting the distance to into a viscosity subsolution via and applying a parabolic maximum principle, aided by a detailed analysis of projections and the second fundamental form. The paper also discusses a potential Riemannian analogue of Evans' strong maximum principle, which would yield a sharp rigidity statement for the flow when the boundary meets .

Abstract

A solution to the heat equation between Riemannian manifolds, where the domain is compact and possibly has boundary, will not leave a compact and locally convex set before the image of the boundary does.

Paper Structure

This paper contains 5 sections, 10 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Convexity
  3. Viscosity solutions
  4. Proof of Theorem \ref{['maximum principle for heat flow']}
  5. A potential strong maximum principle

Key Result

Theorem 1.1

Let $M$ and $N$ be compact Riemannian manifolds, where $M$ has boundary $\partial M \neq \emptyset$. If $u_0 : M \times \{ 0 \} \rightarrow N$ and $f : \partial M \times [0,T] \rightarrow N$ are smooth maps satisfying $u_0 = f$ on $\partial M \times \{ 0 \}$, then there exists $0 < \varepsilon < T$ has a unique solution $u : M \times [0,\varepsilon) \rightarrow N$, which is continuous on $M \time

Theorems & Definitions (16)

  • Theorem 1.1: Hamilton
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 6 more