Gradient continuity for the parabolic (1‚ p)-Laplace system with an external force term

Shuntaro Tsubouchi (U Tokyo)

Feb 25. 2025, 13:30 — 14:00

We consider the parabolic (1‚ p)-Laplace system, the singular parabolic system that consists of the one-Laplacian and the $p$-Laplacian. For this singular problem, it is an open problem whether a spatial gradient of a weak solution is Hölder continuous. The main difficulty is that, thanks to the anisotropic structure of the one-Laplacian, the (1‚ p)-Laplacian becomes not everywhere uniformly parabolic. In particular, this system becomes no longer uniformly parabolic on the facet, the place where a spatial gradient vanishes. This talk aims to prove that a spatial gradient of a weak solution is continuous. Although no quantitative estimate across the facet is known, we can conclude qualitative continuity results by considering a spatial gradient that is suitably truncated near the facet. We also discuss a few generalizations, including the treatment of an external force term.

Further Information
Venue:
ESI Boltzmann Lecture Hall
Recordings:
Recording
Associated Event:
Degenerate and Singular PDEs (Workshop)
Organizer(s):
Verena Bögelein (PLUS, Salzburg)
Ugo Gianazza (U of Pavia)
Juha Kinnunen (Aalto U)
Naian Liao (PLUS, Salzburg)