Local continuity of weak solutions to the Stefan problem involving the singular $p$-Laplacian

TL;DR

This paper proves local continuity and quantifies a logarithmic modulus of continuity for weak solutions to a doubly singular parabolic equation modeling phase transitions, involving the p-Laplacian with singularities.

Contribution

It establishes the local continuity of solutions to a singular p-Laplacian parabolic equation and provides an optimal logarithmic modulus of continuity estimate.

Findings

Proved local continuity of weak solutions.

Quantified an optimal logarithmic modulus of continuity.

Addressed the phase transition model with singular p-Laplacian.

Abstract

We establish the local continuity of locally bounded weak solutions (temperatures) to the doubly singular parabolic equation modeling the phase transition of a material: \[ \partial_t \beta(u)-\Delta_p u\ni 0\quad\text{ for }\tfrac{2N}{N+1}<p<2, \] where $\beta$ is a maximal monotone graph with a jump at zero and $\Delta_p$ is the $p$-Laplacian. Moreover, a logarithmic type modulus of continuity is quantified, which has been conjectured to be optimal.