Gabriele Fioravanti

We study regularity properties for solutions to elliptic equations that are degenerate or singular along orthogonal hyperplanes. The degenerate ellipticity is carried out by a weight term which is the monomial product of different powers of the distance functions to each hyperplane; that is, given the space dimension $d\geq2$, the number of orthogonally crossing hyperplanes $1\leq n\leq d$ and the generic variable point $z=(x,y)\in\mathbb R^{d-n}\times\mathbb R^n$, then the weight is given by $ω(y)=\prod_{i=1}^ny_i^{a_i}$ with $a_i>-1$, $y_i=\mathrm{dist}(z,Σ_i)$ and $Σ_i=\{y_i=0\}$. We prove $C^{0,α}$ and $C^{1,α}$ estimates up to the corners formed by the intersections of two or more hyperplanes, for solutions of the conormal problem with variable coefficients. This is done by a regularization-approximation procedure, a blow-up argument and Liouville theorems. Finally, we provide smoothness of solutions when the equation is isotropic and homogeneous, and we show an application to Caffarelli-Kohn-Nirenberg inequalities with monomial weights.

We consider the one-phase Stefan problem describing the evolution of melting ice. On the one hand, we focus on understanding the evolution of the free boundary near isolated singular points, and we establish for the first time upper and (more surprisingly) lower estimates for its evolution. In 2D, these bounds almost match the best known ones for radial solutions, but hold for all solutions to the Stefan problem, with no extra assumption on the initial or boundary data. On the other hand, as a consequence of our results, we also characterize the global regularity of the free boundary, as follows: it can be written as a graph $t = Γ(x)$, where $Γ$ is $C^1$ (and not $C^2$) near any singular points in the lower strata $Σ_m$, $m \leq n - 2$. Moreover, $Γ$ is not $C^1$ at singular points in $Σ_{n-1}$.

In this paper, we investigate the Dirichlet problem on lower dimensional manifolds for a class of weighted elliptic equations with coefficients that are singular on such sets. Specifically, we study the problem \[\begin{cases} -{\rm div}(|y|^a A(x,y) \nabla u) = |y|^a f + {\rm div}(|y|^a F), \\ u = ψ, \quad \text{ on } Σ_0, \end{cases} \] where $(x,y) \in \mathbb{R}^{d-n} \times \mathbb{R}^n$, $2 \leq n \leq d$, $a + n \in (0,2)$, and $Σ_0 = \{|y| = 0\}$ is the lower dimensional manifold where the equation loses uniform ellipticity. Our primary objective is to establish $C^{0,α}$ and $C^{1,α}$ regularity estimates up to $Σ_0$, under suitable assumptions on the coefficients and the data. Our approach combines perforated domain approximations, Liouville-type theorems and a fine blow-up argument.

We establish some $$C^{0,\alpha }$$C 0 , α and $$C^{1,\alpha }$$C 1 , α regularity estimates for a class of weighted parabolic problems in divergence form. The main novelty is that the weights may vanish or explode on a characteristic hyperplane $$\Sigma $$Σ as a power $$a > -1$$a > - 1 of the distance to $$\Sigma $$Σ . The estimates we obtain are sharp with respect to the assumptions on coefficients and data. Our methods rely on a regularization of the equation and some uniform regularity estimates combined with a Liouville theorem and an approximation argument. As a corollary of our main result, we obtain similar $$C^{1,\alpha }$$C 1 , α estimates when the degeneracy/singularity of the weight occurs on a regular hypersurface of cylindrical type.

We establish some $C^{0,α}$ and $C^{1,α}$ regularity estimates for a class of weighted parabolic problems in divergence form. The main novelty is that the weights may vanish or explode on a characteristic hyperplane $Σ$ as a power $a > -1$ of the distance to $Σ$. The estimates we obtain are sharp with respect to the assumptions on coefficients and data. Our methods rely on a regularization of the equation and some uniform regularity estimates combined with a Liouville theorem and an approximation argument. As a corollary of our main result, we obtain similar $C^{1,α}$ estimates when the degeneracy/singularity of the weight occurs on a regular hypersurface of cylindrical type.