*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

**Abstract:**

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 $\alpha (n,p)$.

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