arXiv subjects

Stefano Vita

Publications and source records attributed to Stefano Vita.

At least 19 recordsLinked to original sources

arXiv2026

Edge asymptotics for weighted Robin equations on cylinders with applications to spectral fractional Laplacians

For a class of weighted degenerate or singular problems on a cylinder, Almgren-type monotonicity methods are employed to derive asymptotic estimates of solutions near the edge between the base and the lateral surface, where a Robin boundary condition is imposed. As a relevant application, local asymptotics and unique continuation from the boundary are obtained for the Robin and Neumann spectral fractional Laplacians. The local decay rates of the solutions are the same for both the Neumann and the Robin problems and are explicitly quantized by a weighted spherical spectral problem with some symmetry.

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arXiv2026

Local regularity for anisotropic magnetic operators with general codimension singularities

We study local regularity properties of solutions to stationary anisotropic magnetic Schrödinger equations in $\mathbb{R}^d$, $d \ge 2$, arising from singular magnetic potentials concentrated along manifolds of general codimension $2 \le n \le d$. The magnetic interaction is modeled through a covariant gradient of the form \[ \nabla_m u = (iM\nabla + A)u, \] where $M^T M$ is a uniformly elliptic matrix encoding anisotropy and $A$ is a magnetic potential with critical Hardy-type scaling along the $n$-codimensional singular set $Σ_0$; that is, $A\sim \mathrm{dist}(\cdot,Σ_0)^{-1}$. We establish local Hölder $C^{0,α}$ and Schauder $C^{1,α}$ estimates for weak solutions via a blow-up analysis adapted to the magnetic structure. The regularity is deeply influenced by the combined effect of anisotropy and the singular magnetic potential, which determines the spectrum of the limiting spherical Laplace-Beltrami operator arising in the blow-up at the singular set. Our model is motivated by the study of magnetic potentials generated by shrinking solenoids onto an axis $Σ_0$, in the three-dimensional setting $d=3$, $n=2$, leading to Aharonov-Bohm-type (AB) models. In this framework, we show that the geometry of the solenoidal loops plays a crucial role: in particular, any deviation from planar cross-sections orthogonal to $Σ_0$ induces a twofold effect. On the one hand, it breaks the ideal AB configuration, in the sense that the magnetic field outside the solenoid is no longer vanishing. On the other hand, it yields an unexpected regularizing mechanism on the wave functions, through a positive shift in the eigenvalues of the asymptotic spectral problem. This purely three-dimensional effect is consistent with our $C^{1,α}$ regularity.

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arXiv2026

Boundary unique continuation in planar domains by conformal mapping

Let $Ω\subset\mathbb R^2$ be a chord arc domain. We give a simple proof of the the following fact, which is commonly known to be true: a nontrivial harmonic function which vanishes continuously on a relatively open set of the boundary cannot have the norm of the gradient which vanishes on a subset of positive surface measure (arc length). This result is conjectured to be true in higher dimensions by Lin, in Lipschitz domains. Let now $Ω\subset\mathbb R^2$ be a $C^1$ domain with Dini mean oscillations. We prove that a nontrivial harmonic function which vanishes continuously on a relatively open subset of the boundary $\partialΩ\cap B_1$ has a finite number of critical points in $\overlineΩ\cap B_{1/2}$. The latter improves some recent results by Kenig and Zhao. Our technique involves a conformal mapping which moves the boundary where the harmonic function vanishes into an interior nodal line of a new harmonic function, after a further reflection. Then, size estimates of the critical set - up to the boundary - of the original harmonic function can be understood in terms of estimates of the \emph{interior} critical set of the new harmonic function and of the critical set - up to the boundary - of the conformal mapping.

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arXiv2026

Notes on Schauder estimates by scaling for elliptic PDEs in divergence form

These are the notes of a part of the PhD course Regularity for free boundary problems and for elliptic PDEs, held in Pavia in the spring of 2025. The aim is to provide a comprehensive and self-contained treatment of classical interior and local Schauder estimates for second-order linear elliptic PDEs in divergence form via scaling in the spirit of Simon's work. The main techniques presented here are geometric in nature and were primarily developed in the study of geometric problems such as minimal surfaces. The adopted approach relies on compactness and blow-up arguments, combined with rigidity results (Liouville theorems), and shares many features with the one used in the study of free boundary problems, which was the main topic of the other part of the PhD course.

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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.

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arXiv2025

A priori Hölder estimates for equations degenerating on nodal sets

We prove a priori Hölder bounds for continuous solutions to degenerate equations with variable coefficients of type $$ \mathrm{div}\left(u^2 A\nabla w\right)=0\quad\mathrm{in \ }Ω\subset\mathbb R^n,\qquad \mbox{with}\qquad \mathrm{div}\left(A\nabla u\right)=0, $$ where $A$ is a Lipschitz continuous, uniformly elliptic matrix (possibly $u$ has non-trivial singular nodal set). Such estimates are uniform with respect to $u$ in a class of normalized solutions that have a bounded Almgren frequency. As a consequence, a boundary Harnack principle holds for the quotient of two solutions vanishing on a common set. This analysis relies on a detailed study of the associated weighted Sobolev spaces, including integrability of the weight, capacitary properties of the nodal set, and uniform Sobolev inequalities yielding local boundedness of solutions.

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arXiv2025

A priori regularity estimates for equations degenerating on nodal sets

We prove a priori and a posteriori Hölder bounds and Schauder $C^{1,α}$ estimates for continuous solutions of degenerate elliptic equations with variable coefficients of the form $$ \mathrm{div}\left(|u|^a A\nabla w\right)=0\qquad\mathrm{in \ }Ω\subset\mathbb R^2,\quad a\in\mathbb R, $$ where the weight $u$ is itself a solution to an elliptic equation of the type $\mathrm{div}(A \nabla u) = 0$, with $A$ a Lipschitz-continuous, uniformly elliptic matrix. The function $u$ is allowed to have a nontrivial, possibly singular nodal set. The estimates are uniform with respect to $u$ within a class of normalized solutions having bounded Almgren frequency. In the special case $a = 2$, our results apply to the ratio of two solutions to the same elliptic equation sharing a common zero set. Precisely, we prove higher-order boundary Harnack principles on nodal domains, via the derived Schauder estimates for the associated degenerate equations. The results are based upon a fine blow-up argument, a Liouville theorem, and quasiconformal maps.

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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.

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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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arXiv2024

Schauder type estimates for degenerate or singular elliptic equations with DMO coefficients

In this paper, we study degenerate or singular elliptic equations in divergence form $$-\text{div}(x_n^αA\nabla u)=\text{div}(x_n^α\mathbf{g})\quad\text{in }B_1\cap\{x_n>0\}.$$ When $α>-1$, we establish boundary Schauder type estimates under the conormal boundary condition on the flat boundary, provided that the coefficients satisfy Dini mean oscillation (DMO) type conditions. Additionally, as an application, we derive higher-order boundary Harnack principles for uniformly elliptic equations in divergence form with DMO coefficients.

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arXiv2024

Unique continuation from conical boundary points for fractional equations

We provide fine asymptotics of solutions of fractional elliptic equations at boundary points where the domain is locally conical; that is, corner type singularities appear. Our method relies on a suitable smoothing of the corner singularity and an approximation scheme, which allow us to provide a Pohozaev type inequality. Then, the asymptotics of solutions at the conical point follow by an Almgren type monotonicity formula, blow-up analysis and Fourier decomposition on eigenspaces of a spherical eigenvalue problem. A strong unique continuation principle follows as a corollary.

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arXiv2024

Higher order boundary Harnack principle via degenerate equations

As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type: \[ -\mathrm{div}\left(ρ^aA\nabla w\right)=ρ^af+\mathrm{div}\left(ρ^aF\right) \quad\textrm{in}\; Ω\] for exponents $a>-1$, where the weight $ρ$ vanishes in a non degenerate manner on a regular hypersurface $Γ$ which can be either a part of the boundary of $Ω$ or mostly contained in its interior. As an application, we extend such estimates to the ratio $v/u$ of two solutions to a second order elliptic equation in divergence form when the zero set of $v$ includes the zero set of $u$ which is not singular in the domain (in this case $ρ=u$, $a=2$ and $w=v/u$). We prove first $C^{k,α}$-regularity of the ratio from one side of the regular part of the nodal set of $u$ in the spirit of the higher order boundary Harnack principle established by De Silva and Savin. Then, by a gluing Lemma, the estimates extend across the regular part of the nodal set. Finally, using conformal mapping in dimension $n=2$, we provide local gradient estimates for the ratio which hold also across the singular set.

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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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arXiv2023

Higher order boundary Harnack principles in Dini type domains

Aim of this paper is to provide higher order boundary Harnack principles [De Silva-Savin 15] for elliptic equations in divergence form under Dini type regularity assumptions on boundaries, coefficients and forcing terms. As it was proven in [Terracini-Tortone-Vita 22], the ratio $v/u$ of two solutions vanishing on a common portion $Γ$ of a regular boundary solves a degenerate elliptic equation whose coefficients behave as $u^2$ at $Γ$. Hence, for any $k\geq 1$ we provide $C^k$ estimates for solutions to the auxiliary degenerate equation under double Dini conditions, actually for general powers of the weight $a>-1$, and we imply $C^k$ estimates for the ratio $v/u$ under triple Dini conditions, as a corollary in the case $a=2$.

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arXiv2022

Boundary regularity estimates in Hölder spaces with variable exponent

We present a general blow-up technique to obtain local regularity estimates for solutions, and their derivatives, of second order elliptic equations in divergence form in Hölder spaces with variable exponent. The procedure allows to extend the estimates up to a portion of the boundary where Dirichlet or Neumann boundary conditions are prescribed and produces a Schauder theory for partial derivatives of solutions of any order $k\in\mathbb{N}$. The strategy relies on the construction of a class of suitable regularizing problems and an approximation argument. The estimates we obtain are sharp with respect to the regularity or integrability conditions on variable coefficients, boundaries, boundary data and right hand sides respectively in Hölder and Lebesgue spaces, both with variable exponent

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arXiv2022

Strong unique continuation and local asymptotics at the boundary for fractional elliptic equations

We study local asymptotics of solutions to fractional elliptic equations at boundary points, under some outer homogeneous Dirichlet boundary condition. Our analysis is based on a blow-up procedure which involves some Almgren type monotonicity formulae and provides a classification of all possible homogeneity degrees of limiting entire profiles. As a consequence, we establish a strong unique continuation principle from boundary points.

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arXiv2020

Liouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions

We consider a class of equations in divergence form with a singular/degenerate weight $$-\mathrm{div}(|y|^a A(x,y)\nabla u)=|y|^a f(x,y)\; \quad\textrm{or} \ \textrm{div}(|y|^aF(x,y))\;.$$ Under suitable regularity assumptions for the matrix $A$ and $f$ (resp. $F$) we prove Hölder continuity of solutions which are even in $y\in\mathbb{R}$, and possibly of their derivatives up to order two or more (Schauder estimates). In addition, we show stability of the $C^{0,α}$ and $C^{1,α}$ a priori bounds for approximating problems in the form $$-\mathrm{div}((\varepsilon^2+y^2)^{a/2} A(x,y)\nabla u)=(\varepsilon^2+y^2)^{a/2} f(x,y)\; \quad\textrm{or} \ \textrm{div}((\varepsilon^2+y^2)^{a/2}F(x,y))$$ as $\varepsilon\to 0$. Finally, we derive $C^{0,α}$ and $C^{1,α}$ bounds for inhomogenous Neumann boundary problems as well. Our method is based upon blow-up and appropriate Liouville type theorems.

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