Calculus of Variations and Geometric Measure Theory

P. Baroni - T. Kuusi - J. M. Urbano

A quantitative modulus of continuity for the two-phase Stefan problem

created by baroni on 14 Jan 2014
modified on 31 Aug 2017


Published Paper

Inserted: 14 jan 2014
Last Updated: 31 aug 2017

Journal: Arch. Ration. Mech. Anal.
Volume: 214
Number: 2
Year: 2014
Doi: 10.1007/s00205-014-0762-9

ArXiv: 1401.2623 PDF


We derive the quantitative modulus of continuity \[ \omega(r)=\left[ p+\ln \left( \frac{r_0}{r} \right) \right]^{-\alpha (n,p)}, \] 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 (J. reine angew. Math., 1984), in the sense that we discard one logarithm iteration and obtain an explicit value for the exponent $\alpha (n,p)$.