arXiv subjects

Gabriele Fioravanti

Publications and source records attributed to Gabriele Fioravanti.

9 recordsLinked to original sources

arXiv2026

Optimal partition and segregation problems driven by torsional rigidity

Spectral optimal partition and segregation problems are deeply connected with harmonic maps, eigenfunctions, and the fine structure of nodal sets for linear elliptic equations. In this paper, we show that replacing the spectral energy by torsional rigidity leads to a genuinely different theory. The resulting optimal configurations are governed locally not by harmonic equations, but by torsion-type energies and unstable free boundary problems, thereby creating a natural bridge between optimal partition theory and the analysis of sublinear free boundary phenomena. We prove existence of optimal torsional partitions and segregated torsional configurations, together with optimal Lipschitz regularity of the associated nonlinear eigenfunctions. We establish a strong unique continuation principle, characterize the admissible vanishing orders and the corresponding blow-up profiles, derive sharp Hausdorff dimension estimates for the nodal set and its singular subset, and prove $C^{1,α}$-regularity of the regular part of the free boundary. The proofs combine variational arguments with Almgren-type and Weiss-type monotonicity formulae adapted to the intrinsically sublinear torsional regime, blow-up analysis, and tools from geometric measure theory.

math.AP↗

arXiv2026

A Priori Regularity Estimates for Ratio of Solutions to Elliptic Equations with a Product Structure of Two-Dimensional Nodal Sets

In this paper, we establish optimal a priori $C^{1,α}$ regularity estimates for the ratio $w = v/u$ of two solutions to the same elliptic equation $-\operatorname{div}(A \nabla u )=0$ with Lipschitz coefficients $A$, under the assumption that their nodal sets satisfy $Z(u) \subseteq Z(v)$. We specifically address the case where the zero set $Z(u)$ exhibits a product structure of $2$-dimensional nodal sets, namely $Z(u)=Z(u_1)\times \cdots \times Z(u_{m})$, where the $u_i$ are $2$-dimensional functions. This result extends the regularity estimates previously proved in dimension $2$ by [Logunov and Malinnikova, 2016] and by [Terracini, Tortone, and Vita, 2026].

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arXiv2025

Regularity for elliptic equations with monomial weights

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.

math.AP↗

arXiv2025

Higher order Schauder estimates for degenerate or singular parabolic equations

In this paper, we complete the analysis initiated in [AFV24] establishing some higher order $C^{k+2,α}$ Schauder estimates ($k \in \mathbb{N}$) for a a class of parabolic equations with weights that are degenerate/singular on a characteristic hyperplane. The $C^{2,α}$-estimates are obtained through a blow-up argument and a Liouville theorem, while the higher order estimates are obtained by a fine iteration procedure. As a byproduct, we present two applications. First, we prove similar Schauder estimates when the degeneracy/singularity of the weight occurs on a regular hypersurface of cylindrical type. Second, we provide an alternative proof of the higher order boundary Harnack principles established in [BG16,Kuk22].

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arXiv2025

Schauder estimates for elliptic equations degenerating on lower dimensional manifolds

In this paper we begin exploring a local regularity theory for elliptic equations having coefficients which are degenerate or singular on some lower dimensional manifold $$ -\mathrm{div}(|y|^aA(x,y)\nabla u)=|y|^af+\mathrm{div}(|y|^aF)\qquad\mathrm{in \ } B_1\subset\mathbb R^d, $$ where $z=(x,y)\in\mathbb R^{d-n}\times\mathbb R^n$, $2\leq n\leq d$ are two integers and $a\in\mathbb R$. Such equations are a prototypical example of elliptic equations spoiling their uniform ellipticity on the (possibly very) thin characteristic manifold $Σ_0=\{|y|=0\}$ of dimension $0\leq d-n\leq d-2$, having $$λ|y|^a|ξ|^2\leq |y|^aA(x,y)ξ\cdotξ\leqΛ|y|^a|ξ|^2.$$ Whenever $a+n>0$, the weak solutions with a homogeneous conormal boundary condition at $Σ_0$ are provided to be $C^{0,α}$ or even $C^{1,α}$ regular up to $Σ_0$. Our approach relies on a regularization-approximation scheme which employs domain perforation, very fine blow-up procedures, and a new Liouville theorem in the perforated space. Our theory extends to the case of equations degenerating on suitably smooth curved manifolds.

math.AP↗

arXiv2025

Remarks on elliptic equations degenerating on lower dimensional manifolds

The paper continues the analysis started in [Cora-Fioravanti-Vita-25,Fioravanti-24] on the local regularity theory for elliptic equations having coefficients which are degenerate or singular on some lower dimensional manifold. The model operator is given by $L_au(z)=\mathrm{div}(|y|^a\nabla u)(z)$, where $z=(x,y)\in\mathbb R^{d-n}\times\mathbb R^n$, $2\leq n\leq d$ are two integers and $a\in\mathbb R$. The weight term is degenerate/singular on the (possibly very) thin characteristic manifold $Σ_0=\{|y|=0\}$ of dimension $0\leq d-n\leq d-2$. Whenever $a+n>0$, we prove smoothness of the axially symmetric $L_a$-harmonic functions. In the mid-range $a+n\in(0,2)$, we deal with regularity estimates for solutions with inhomogeneous conormal boundary conditions prescribed at $Σ_0$, and we establish the connection with fractional Laplacians on very thin flat manifolds via Dirichlet-to-Neumann maps, as a higher codimensional analogue of the extension theory developed by Caffarelli and Silvestre. Finally, whenever $a+n<2$ we complement the study in [Fioravanti-24], providing some regularity estimates for solutions having a homogeneous Dirichlet boundary condition prescribed at $Σ_0$ by a boundary Harnack type principle.

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arXiv2025

Extinction rates for nonradial solutions to the Stefan problem

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 $Σ_{n-1}$.

math.AP↗

arXiv2024

The Dirichlet problem on lower dimensional boundaries: Schauder estimates via perforated domains

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.

math.AP↗

arXiv2024

Schauder estimates for parabolic equations with degenerate or singular weights

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.

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