Filippo Paiano | Search arXiv

arXiv2026

An elliptic regularization approach to the Stefan problem

In this paper, we develop the theory for the two-phase Stefan problem with finite energy, possibly non-empty mushy region, and space-dependent melting temperature. Specifically, we prove the existence of weak solutions with an elliptic regularization scheme. Our existence theorem provides information about the regularity of the solutions: we prove that the temperature of weak solutions is in $H^1$ for all times, that the enthalpy is well defined and bounded for all times, and that both the enthalpy and the temperature are weakly continuous in time. Finally, we establish a comparison principle for weak solutions on general unbounded domains and use it to show that every weak solution is recovered by the approximation scheme.

math.AP↗

arXiv2025

Regularity for free boundary surfaces minimizing degenerate area functionals

We establish an epsilon-regularity theorem at points in the free boundary of almost-minimizers of the energy $\mathrm{Per}_{w}(E)=\int_{\partial^*E}w\,\mathrm{d} {\mathscr{H}}^{n-1}$, where $w$ is a weight asymptotic to $d(\cdot,\mathbb{R}^n\setminusΩ)^a$ near $\partialΩ$ and $a>0$. This implies that the boundaries of almost-minimizers are $C^{1,γ_0}$-surfaces that touch $\partial Ω$ orthogonally, up to a Singular Set $\mathrm{Sing}(\partial E)$ whose Hausdorff dimension satisfies the bound $d_{\mathscr{H}}(\mathrm{Sing}(\partial E)) \leq n +a -(5+\sqrt{8})$.

math.AP↗