Harmonic map flow with singular targets

In this talk we consider two heat flow problems from Euclidean domains into non-smooth spaces. The first problem, a nonlocal constrained heat flow into a metric tree, was originally motivated by a stationary eigenvalue partition problem. We prove spatial Lipschitz regularity of this flow, free interface regularity, and characterize the limit of the flow as time goes to infinity as a stationary solution of the partition problem. Next, we discuss the extension of this study to non-constrained heat flows into F-connected simplicial complexes, which are natural generalizations of trees to higher dimensions.