Gradient Continuity for the Parabolic (1, p)-Laplace System with an External Force Term

This talk, part of the Workshop on "Degenerate and Singular PDEs" held at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) in February 2025, explores the gradient continuity for the parabolic (1, p)-Laplace system with external forces. Delve into the challenging open problem of whether a spatial gradient of a weak solution to this singular parabolic system exhibits Hölder continuity. Understand the main difficulty arising from the anisotropic structure of the one-Laplacian, which causes the system to lose uniform parabolicity, particularly on the facet where spatial gradient vanishes. Learn about the qualitative continuity results achieved by considering a spatial gradient that is suitably truncated near the facet, despite the absence of quantitative estimates across this region. The 28-minute presentation also covers several generalizations, including how to handle external force terms in this complex mathematical framework.