Singular sets of homogeneous Dirichlet heat flows at large time

Abstract

We study homogeneous Dirichlet heat flows for real symmetric, uniformly elliptic divergence-form operators. We establish quantitative stability of singular sets near homogeneous harmonic polynomials and eigenfunctions. Based on these, in the smooth coefficient case, we prove the spatial singular set at sufficiently large time is codimension-two and has a uniform measure upper bound independent of time. However, this codimension-two property fails in the Lipschitz coefficient case. On a three-dimensional ball, we construct a fixed self-adjoint operator and a heat flow consisting of two eigenmodes whose singular sets contain two-dimensional disks at a sequence of times tending to infinity.