Harmonic maps in connection of phase transitions with higher dimensional potential wells

TL;DR

This paper develops a regularity theory for harmonic maps related to phase transitions with complex boundary conditions and proves the existence of smooth gradient flows under geometric constraints, linking to mean-curvature motion.

Contribution

It establishes a new regularity framework for minimizing maps with non-standard boundary conditions and demonstrates local smooth gradient flows in phase transition models.

Findings

Regularity theory for minimizing maps with complex boundary conditions.

Existence of local smooth gradient flows under geometric constraints.

Connection to mean-curvature motion in phase transition dynamics.

Abstract

This is in the sequel of authors' paper \cite{LPW} in which we had set up a program to verify rigorously some formal statements associated with the multiple component phase transitions with higher dimensional wells. The main goal here is to establish a regularity theory for minimizing maps with a rather non-standard boundary condition at the sharp interface of the transition. We also present a proof, under simplified geometric assumptions, of existence of local smooth gradient flows under such constraints on interfaces which are in the motion by the mean-curvature. In a forthcoming paper, a general theory for such gradient flows and its relation to Keller-Rubinstein-Sternberg's work \cite{KRS1, KRS2} on the fast reaction, slow diffusion and motion by the mean curvature would be addressed.