Existence of weak solutions for two-phase matrix-valued harmonic map flows

Abstract

We study the existence of weak solutions to a two-phase matrix-valued harmonic map flow with lifespan determined by that of the prescribed interface. This system arises as the limiting equation for the matrix-valued Keller--Rubinstein--Sternberg problem studied in (Invent. Math., 233(1):1--80, 2023). Our construction is based on a modified minimizing movement scheme. Precisely, we discretize time and build approximate solutions by interpolating minimizers of the associated variational problem on each time step.

35 pages. Final version accepted for publication. Several unclear points in the previous version have been clarified