Monotonicity theorems for Laplace Beltrami operator on Riemannian manifolds

TL;DR

This paper extends key monotonicity formulas from Euclidean free boundary problems to Riemannian manifolds, enabling new regularity results for solutions of two-phase free boundary problems in curved spaces.

Contribution

It establishes the first monotonicity theorems for the Laplace-Beltrami operator on Riemannian manifolds, generalizing classical Euclidean results.

Findings

Monotonicity formulas analogous to Euclidean case are proved for Riemannian manifolds.

These formulas are used to show Lipschitz continuity of solutions to two-phase free boundary problems.

The results provide new tools for regularity theory on curved spaces.

Abstract

For free boundary problems on Euclidean spaces, the monotonicity formulas of Alt-Caffarelli-Friedman and Caffarelli-Jerison-Kenig are cornerstones for the regularity theory as well as the existence theory. In this article we establish the analogs of these results for the Laplace-Beltrami operator on Riemannian manifolds. As an application we show that our monotonicity theorems can be employed to prove the Lipschitz continuity for the solutions of a general class of two-phase free boundary problems on Riemannian manifolds.