Improved moduli of continuity for degenerate phase transitions

TL;DR

This paper advances the understanding of the regularity of solutions to the degenerate phase transition problem by providing sharper continuity estimates and revealing new H"older regularity results.

Contribution

It introduces improved modulus of continuity estimates for weak solutions to the degenerate two-phase Stefan problem, including a sharp exponential modulus and an unexpected H"older regularity.

Findings

Sharpened exponential modulus of continuity for p=N≥3

Derived H"older continuity for p>max{2,N}

Enhanced regularity results for degenerate phase transition solutions

Abstract

We substantially improve in two scenarios the current state-of-the-art modulus of continuity for weak solutions to the $N$-dimensional, two-phase Stefan problem featuring a $p-$degenerate diffusion: for $p=N\geq 3$, we sharpen it to $$ \boldsymbol{\omega}(r) \approx \exp (-c| \ln r|^{\frac1N}); $$ for $p>\max\{2,N\}$, we derive an unexpected H\"older modulus.