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 $p=N\geq 3$, we sharpen it to $$\boldsymbol{\omega}(r) \approx \exp (-c| \ln r|^{\frac1N});$$ for $p>\max\{2,N\}$, we derive an unexpected H\"older modulus.