A Quantitative Modulus of Continuity for the Two-Phase Stefan Problem

Abstract

We derive the quantitative modulus of continuity, which we conjecture to be optimal, for solutions of the p-degenerate two-phase Stefan problem. Even in the classical case p = 2, this represents a twofold improvement with respect to the 1984 state-of-the-art result by DiBenedetto and Friedman [10], in the sense that we discard one logarithm iteration and obtain an explicit value for the exponent α(n, p).

Continuity of the temperature in a multi-phase transition problem

“…For a few years there were basically no further improvements, as far as the continuity issue is concerned. Things changed with [1]: the authors consider (1.3) with p ≥ 2 and (1.2) with a single jump, and they derive an explicit modulus of continuity better than (1.6), namely…”

“…We will consider the regularized version of the Stefan problem (1.3). For a parameter ε ∈ (0, 1 2 d), we introduce the function…”

Existence and Boundary Regularity for Degenerate Phase Transitions

“…Approximation of the problem. Let ρ ε be the standard symmetric, positive, one dimensional mollifier, supported in (−ε, ε), obtained via rescaling of 1) and observe that H a,ε is smooth and…”

“…The outcome of our effort is two-fold: on the one hand, we prove sharp a priori estimates for solutions of (1.1), and obtain the boundary continuity, quantified through a modulus, assuming a mild geometric condition on Ω. On the other hand, we use this "almost uniform" modulus of continuity at the boundary, together with the interior modulus of continuity we deduced in [1], to build a solution to (1.1), which is continuous up to the boundary and enjoys the same modulus of continuity.…”

“…We remark that the solution we build has the interior modulus of continuity described in [1], where we assumed the existence of a solution built in the way described in this paper. Our other main result concerns a precise modulus of continuity up to the boundary for the physical solution obtained in Theorem 1.1, in the case the regularity of the boundary datum does not overcome a threshold we are going to describe.…”