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

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.

Continuity of the temperature in a multi-phase transition problem

“…There is yet another notion of solution, which requires a solution to possess time derivative in the Sobolev sense, cf. [4,5,16,17]. Theorem 1.1 continues to hold for that kind of notion and the proof calls for minor modifications from the one given here.…”

“…Further progress has been recently made in [16,17]. Indeed, interior moduli sharper than (1.7) are provided in [17] for p = 2 and N = 1, 2.…”

“…Indeed, interior moduli sharper than (1.7) are provided in [17] for p = 2 and N = 1, 2. On the other hand, under the same general conditions as in [2], the boundary modulus of continuity has been improved to (1.7) in [16]: for Dirichlet boundary conditions, any p ≥ 2 can do; whereas for Neumann boundary conditions only p = 2 could be dealt with, while the case p > 2 remains an open problem.…”

“…The other novelty is given by the explicit modulus of continuity in Theorem 1.1: to our knowledge, it is the first time that a modulus is explicitly stated for a β that is more general than the one considered in [1,2,16,17]. Due to the wide generality assumed on β, i.e.…”

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

“…The local continuity of weak solutions to (1.1) is previously known only for p ≥ 2; see for instance [1,2,4,10,11,12]. In particular, a modulus of type (1.3) has been achieved in [1] for all p ≥ 2, which has been conjectured to be optimal as a structural property of weak solutions; see also [10]. On the other hand, the case p < 2 presents considerable difficulty.…”