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Ariel Salort

Publications and source records attributed to Ariel Salort.

At least 37 records · Page 2Linked to original sources
arXiv2022

Asymptotic behavior for anisotropic fractional energies

In this paper we investigate the asymptotic behavior of anisotropic fractional energies as the fractional parameter $s\in (0,1)$ approaches both $s\uparrow 1$ and $s\downarrow 0$ in the spirit of the celebrated papers of Bourgain-Brezis-Mironescu \cite{BBM} and Maz'ya-Shaposhnikova \cite{MS}. Then, focusing con the case $s\uparrow 1$ we analyze the behavior of solutions to the corresponding minimization problems and finally, we also study the problem where a homogenization effect is combined with the localization phenomena that occurs when $s\uparrow 1$.

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arXiv2022

Rotating Spirals in segregated reaction-diffusion systems

We give a complete characterization of the boundary traces $φ_i$ ($i=1,\dots,K$) supporting spiraling waves, rotating with a given angular speed $ω$, which appear as singular limits of competition-diffusion systems of the type \[ \frac{\partial}{\partial t} u_i -Δu_i = μu_i -βu_i \sum_{j \neq i} a_{ij} u_j \text{ in } Ω\times\mathbb{R}^+, \qquad u_i = φ_i \text{ on $\partialΩ\times\mathbb{R}^+$}, \qquad u_i(\mathbf{x},0) = u_{i,0}(\mathbf{x}) \text{ for $\mathbf{x} \in Ω$} \] as $β\to +\infty$. Here $Ω$ is a rotationally invariant planar set and $a_{ij}>0$ for every $i$ and $j$. We tackle also the homogeneous Dirichlet and Neumann boundary conditions, as well as entire solutions in the plane. As a byproduct of our analysis we detect explicit families of eternal, entire solutions of the pure heat equation, parameterized by $ω\in\mathbb{R}$, which reduce to homogeneous harmonic polynomials for $ω=0$.

math.AP↗
arXiv2021

On the mixed local-nonlocal Hénon equation

In this paper we consider a Hénon-type equation driven by a nonlinear operator obtained as a combination of a local and nonlocal term. We prove existence and non-existence akin to the classical result by Ni, and a stability result as the fractional parameter $s \to 1$.

math.AP↗
arXiv2021

Maximum principles, Liouville theorem and symmetry results for the fractional $g-$Laplacian

We study different maximum principles for non-local non-linear operators with non-standard growth that arise naturally in the context of fractional Orlicz-Sobolev spaces and whose most notable representative is the fractional $g-$Laplacian: \[ (-Δ_g)^su(x):=\textrm{p.v.}\int_{\mathbb{R}^n}g\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\frac{dy}{|x-y|^{n+s}}, \] being $g$ the derivative of a Young function. We further derive qualitative properties of solutions such as a Liouville type theorem and symmetry results and present several possible extensions and some interesting open questions. These are the first results of this type proved in this setting.

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arXiv2020

Variational Eigenvalues of the fractional $g$-Laplacian

In the present work we study existence of sequences of variational eigenvalues to non-local non-standard growth problems ruled by the fractional $g-$Laplacian operator with different boundary conditions (Dirichlet, Neumann and Robin). Due to the non-homogeneous nature of the operator several drawbacks must be overcome, leading to some results that contrast with the case of power functions.

math.AP↗
arXiv2020

Neumann and Robin type boundary conditions in Fractional Orlicz-Sobolev spaces

In the first part of this article we deal with the existence of at least three non-trivial weak solutions of a nonlocal problem with nonstandard growth involving a nonlocal Robin type boundary condition. The second part of the article is devoted to study eigenvalues and minimizers of several nonlocal problems for the fractional $g-$Laplacian $(-Δ_g)^s$ with different boundary conditions, namely, Dirichlet, Neumann and Robin.

math.AP↗
arXiv2020

Reverse Faber-Krahn inequality for a truncated laplacian operator

In this paper we prove a reverse Faber-Krahn inequality for the principal eigenvalue $μ_1(Ω)$ of the fully nonlinear eigenvalue problem \[ \label{eq} \left\{\begin{array}{r c l l} -λ_N(D^2 u) & = & μu & \text{in }Ω, \\ u & = & 0 & \text{on }\partial Ω. \end{array}\right. \] Here $ λ_N(D^2 u)$ stands for the largest eigenvalue of the Hessian matrix of $u$. More precisely, we prove that, for an open, bounded, convex domain $Ω\subset \mathbb{R}^N$, the inequality \[ μ_1(Ω) \leq \frac{π^2}{[\text{diam}(Ω)]^2} = μ_1(B_{\text{diam}(Ω)/2}),\] where $\text{diam}(Ω)$ is the diameter of $Ω$, holds true. The inequality actually implies a stronger result, namely, the maximality of the ball under a diameter constraint. Furthermore, we discuss the minimization of $μ_1(Ω)$ under different kinds of constraints.

math.AP↗
arXiv2020

A Pólya-Szegö principle for general fractional Orlicz-Sobolev spaces

In this article we prove modular and norm Pólya-Szegö inequalities in general fractional Orlicz-Sobolev spaces by using the polarization technique. We introduce a general framework which includes the different definitions of theses spaces in the literature, and we establish some of its basic properties such as the density of smooth functions. As a corollary we prove a Rayleigh-Faber-Krahn type inequality for Dirichlet eigenvalues under nonlocal nonstandard growth operators.

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arXiv2018

Magnetic Fractional order Orlicz-Sobolev spaces

In this paper we define the notion of nonlocal magnetic Sobolev spaces with non-standard growth for Lipschitz magnetic fields. In this context we prove a Bourgain - Brezis - Mironescu type formula for functions in this space as well as for sequences of functions. Finally, we deduce some consequences such as the $Γ-$convergence of modulars and convergence of solutions for some non-local magnetic Laplacian allowing non-standard growth laws to its local counterpart.

math.AP↗
arXiv2018

A note on homogeneous Sobolev spaces of fractional order

We consider a homogeneous fractional Sobolev space obtained by completion of the space of smooth test functions, with respect to a Sobolev--Slobodecki\uı norm. We compare it to the fractional Sobolev space obtained by the $K-$method in real interpolation theory. We show that the two spaces do not always coincide and give some sufficient conditions on the open sets for this to happen. We also highlight some unnatural behaviors of the interpolation space. The treatment is as self-contained as possible.

math.FA↗
arXiv2018

Compactness and dichotomy in nonlocal shape optimization

We prove a general result about the behaviour of minimizing sequences for nonlocal shape functionals satisfying suitable structural assumptions. Typical examples include functions of the eigenvalues of the fractional Laplacian under homogeneous Dirichlet boundary conditions. Exploiting a nonlocal version of Lions' concentration-compactness principle, we prove that either an optimal shape exists, or there exists a minimizing sequence consisting of two "pieces" whose mutual distance tends to infinity. Our work is inspired by similar results obtained by Bucur in the local case.

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arXiv2018

A constrained shape optimization problem in Orlicz-Sobolev spaces

In this manuscript we study the following optimization problem: given a bounded and regular domain $Ω\subset \mathbb{R}^N$ we look for an optimal shape for the "$\mathrm{W}-$vanishing window" on the boundary with prescribed measure over all admissible profiles in the framework of the Orlicz-Sobolev spaces associated to constant for the "Sobolev trace embedding". In this direction, we establish existence of minimizer profiles and optimal sets, as well as we obtain further properties for such extremals. Finally, we also place special emphasis on analyzing the corresponding optimization problem involving an "$\mathrm{A}-$vanishing hole" (inside the domain) with volume constraint.

math.AP↗
arXiv2018

Weighted Inequalities for the Fractional Laplacian and the Existence of Extremals

In this article we obtain improved versions of Stein-Weiss and Caffarelli-Kohn-Nirenberg inequalities, involving Besov norms of negative smoothness. As an application of the former, we derive the existence of extremals of the Stein-Weiss inequality in certain cases, some of which are not contained in the celebrated theorem of E. Lieb.

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