Parabolic theory as a high-dimensional limit of elliptic theory

TL;DR

This paper demonstrates how parabolic theorems can be derived from elliptic theorems using high-dimensional limits, providing new proofs for key results in parabolic unique continuation and geometric flows.

Contribution

It introduces a novel approach of deriving parabolic results from elliptic theorems via high-dimensional limiting procedures, simplifying proofs and unifying theories.

Findings

New proofs of Carleman estimates for the heat operator

Monotonicity formulas for frequency functions in parabolic equations

Unified approach to parabolic theorems using elliptic limits

Abstract

The aim of this article is to show how certain parabolic theorems follow from their elliptic counterparts. This technique is demonstrated through new proofs of five important theorems in parabolic unique continuation and the regularity theory of parabolic equations and geometric flows. Specifically, we give new proofs of an $L^2$ Carleman estimate for the heat operator, and the monotonicity formulas for the frequency function associated to the heat operator, the two-phase free boundary problem, the flow of harmonic maps, and the mean curvature flow. The proofs rely only on the underlying elliptic theorems and limiting procedures belonging essentially to probability theory. In particular, each parabolic theorem is proved by taking a high-dimensional limit of the related elliptic result.