On the logarithmic type boundary modulus of continuity for the Stefan problem

TL;DR

This paper establishes a logarithmic modulus of continuity for weak solutions to the two-phase Stefan problem, providing sharp estimates and insights into boundary regularity under various boundary conditions.

Contribution

It introduces a measure-theoretic approach to prove logarithmic boundary regularity for the Stefan problem, refining existing methods and suggesting optimality of the results.

Findings

Logarithmic boundary modulus of continuity is proven for weak solutions.

The approach combines De Giorgi's iteration with DiBenedetto's techniques.

Results are applicable to both Dirichlet and Neumann boundary conditions.

Abstract

A logarithmic type modulus of continuity is established for weak solutions to a two-phase Stefan problem, up to the parabolic boundary of a cylindrical space-time domain. For the Dirichlet problem, we merely assume that the spatial domain satisfies a measure density property, and the boundary datum has a logarithmic type modulus of continuity. For the Neumann problem, we assume that the lateral boundary is smooth, and the boundary datum is bounded. The proofs are measure theoretical in nature, exploiting De Giorgi's iteration and refining DiBenedetto's approach. Based on the sharp quantitative estimates, construction of continuous weak (physical) solutions is also indicated. The logarithmic type modulus of continuity has been conjectured to be optimal as a structural property for weak solutions to such partial differential equations.