Improved moduli of continuity for degenerate phase transitions

We substantially improve in two scenarios the current state-of-the-art modulus of continuity for weak solutions to the $N$-dimensional, two-phase Stefan problem featuring a $p-$degenerate diffusion: for $p=N\ge 3$, we sharpen it to $$\mathit{\omega }(r)\approx \exp (-c\mathrm{\mid }\ln r{\mathrm{\mid }}^{\frac{1}{N}});$$ for $p>\max \{2,N\}$, we derive an unexpected Hölder modulus.