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Gabriele Cora

Publications and source records attributed to Gabriele Cora.

14 recordsLinked to original sources
arXiv2026

Entire spacelike radial graphs with prescribed mean curvature in the Lorentz--Minkowski space

In this paper we address the existence and uniqueness of entire spacelike hypersurfaces in the Lorentz--Minkowski space $\mathbb{L}^{m+1}$ with prescribed mean curvature that are star-shaped with respect to a point and asymptotic to a light cone. We also establish a Willmore-type inequality and prove a non-existence result for spacelike radial graphs asymptotic to the light cone whose mean curvature belongs to $L^p$ for $1 \leq p\leq m$, in particular in the case of compactly supported mean curvature.

math.AP↗
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 $\omega(y)=\prod_{i=1}^ny_i^{a_i}$ with $a_i>-1$, $y_i=\mathrm{dist}(z,\Sigma_i)$ and $\Sigma_i=\{y_i=0\}$. We prove $C^{0,\alpha}$ and $C^{1,\alpha}$ 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

Hardy-Sobolev inequalities involving mixed radially and cylindrically symmetric weights

We deal with weighted Hardy-Sobolev type inequalities for functions on $\mathbb{R}^d$, $d\geq 2$. The weights involved are anisotropic, given by products of powers of the distance to the origin and to a nontrivial subspace. We establish necessary and sufficient conditions for validity of these inequalities, and investigate the existence/nonexistence of extremal functions.

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 $\Sigma_0=\{|y|=0\}$ 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 $\Sigma_0$, 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 $\Sigma_0$ by a boundary Harnack type principle.

math.AP↗
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 $\Sigma_0=\{|y|=0\}$ of dimension $0\leq d-n\leq d-2$, having $$\lambda|y|^a|\xi|^2\leq |y|^aA(x,y)\xi\cdot\xi\leq\Lambda|y|^a|\xi|^2.$$ Whenever $a+n>0$, the weak solutions with a homogeneous conormal boundary condition at $\Sigma_0$ are provided to be $C^{0,\alpha}$ or even $C^{1,\alpha}$ regular up to $\Sigma_0$. 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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arXiv2023

Hardy type inequalities with mixed weights in cones

We study Hardy type inequalities involving mixed cylindrical and spherical weights, for functions supported in cones. These inequalities are related to some singular or degenerate differential operators.

math.AP↗
arXiv2019

Sign-changing bubble-tower solutions to fractional semilinear elliptic problems

We study the asymptotic and qualitative properties of least energy radial sign-changing solutions to fractional semilinear elliptic problems of the form \[ \begin{cases} (-Δ)^s u = |u|^{2^*_s-2-\varepsilon}u &\text{in } B_R, \\ u = 0 &\text{in }\mathbb{R}^n \setminus B_R, \end{cases} \] where $s \in (0,1)$, $(-Δ)^s$ is the s-Laplacian, $B_R$ is a ball of $\mathbb{R}^n$, $2^*_s := \frac{2n}{n-2s}$ is the critical Sobolev exponent and $\varepsilon>0$ is a small parameter. We prove that such solutions have the limit profile of a "tower of bubbles", as $ \varepsilon \to 0^+$, i.e. the positive and negative parts concentrate at the same point with different concentration speeds. Moreover, we provide information about the nodal set of these solutions.

math.AP↗
arXiv2019

On the dynamics of a charged particle in magnetic fields with cylindrical symmetry

We study the motion of a charged particle under the action of a magnetic field with cylindrical symmetry. In particular we consider magnetic fields with constant direction and with magnitude depending on the distance $r$ from the symmetry axis of the form $1 + Ar^{-γ}$ as $r\to\infty$, with $A\ne 0$ and $γ> 1$. With perturbative-variational techniques, we can prove the existence of infinitely many trajectories whose projection on a plane orthogonal to the direction of the field describe bounded curves given by the superposition of two motions: a rotation with constant angular speed at a unit distance about a point which moves along a circumference of large radius $ρ$ with a slow angular speed $\varepsilon$. The values $ρ$ and $\varepsilon$ are suitably related to each other. This problem has some interest also in the context of planar curves with prescribed curvature.

math.DS↗
arXiv2019

Nonlinear variational problems with lack of compactness

In this thesis we deal with two different classes of variational problems: 1) the problem of closed curves with prescribed curvature, or $H$-loop problem; 2) the study of the nodal solutions of the fractional Brezis-Nirenberg problem. In both cases we deal with nonlinear equations (an ODE system for problem 1, and an elliptic equation for problem 2) which admit a variational structure. Nevertheless, in both cases, a lack of compactness occurs and this constitutes a strong obstruction for the application of the direct methods of the calculus of variations or even standard variational methods. Therefore, we are lead to use more refined techniques or also to follow different approaches, like the Lyapunov-Schmidt method and blow-up analysis. This will allow us not only to obtain existence and multiplicity results, but also qualitative properties of the solutions.

math.AP↗
arXiv2018

On the structure of the nodal set and asymptotics of least energy sign-changing radial solutions of the fractional Brezis-Nirenberg problem

In this paper we study the asymptotic and qualitative properties of least energy radial sign-changing solutions of the fractional Brezis--Nirenberg problem ruled by the s-laplacian, in a ball of $\mathbb{R}^n$, when $s \in (0,1)$ and $n > 6s$. As usual, $λ$ is the (positive) parameter in the linear part in $u$, and we consider $λ$ close to zero. We prove that if such solutions vanish at the center of the ball then they vanish everywhere, we establish a bound on the number of sign-changes and, when $s$ is close to $1$, for a suitable value of the parameter $λ$ such solutions change sign exactly once. Moreover, for any $s \in (0,1)$ and $λ$ sufficiently small we prove that the number of connected components of the complement of the nodal set corresponds to the number of sign-changes plus one. In addition, for any $s \in (\frac{1}{2},1)$, we prove that least energy nodal solutions which change sign exactly once have the limit profile of a "tower of bubbles", as $λ\to 0^+$, i.e. the positive and negative parts concentrate at the same point (which is the center of the ball) with different concentration speeds.

math.AP↗
arXiv2014

Entire solutions for a class of fourth order semilinear elliptic equations with weights

We investigate the problem of entire solutions for a class of fourth order, dilation invariant, semilinear elliptic equations with power-type weights and with subcritical or critical growth in the nonlinear term. These equations define non compact variational problems and are characterized by the presence of a term containing lower order derivatives, whose strength is ruled by a parameter λ. We can prove existence of entire solutions found as extremal functions for some Rellich-Sobolev type inequalities. Moreover, when the nonlinearity is suitably close to the critical one and the parameter λ is large, symmetry breaking phenomena occur and in some cases the asymptotic behavior of radial and non radial ground states can be somehow described.

math.AP↗