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Julian Fernandez Bonder

Publications and source records attributed to Julian Fernandez Bonder.

At least 19 recordsLinked to original sources
arXiv2026

An eigenvalue problem for a nonlocal quasilinear anisotropic equation in fractional Orlicz Sobolev spaces without the $Δ_2$--condition

In this paper we analyze an eigenvalue problem associated to fractional operators of the form \[ L_a^s u(x)=2 \text{p.v.}\int_{\mathbb{R}^n}a(x,y,D^su(x,y))\,\frac{dy}{|x-y|^{n+s}},\] which represents a generalization model for nonlocal, nonstandard growth diffusion problems. We study this problem in the context of the fractional Orlicz Sobolev spaces without assuming the so-called $Δ_2$--condition on the Young functions involved. We show existence of a sequence of eigenpairs $(u_k,λ_k)\to (0,+\infty)$.

math.AP↗
arXiv2026

A PINNs approach for the computation of eigenvalues in elliptic problems

In this paper, we propose a method for computing eigenvalues of elliptic problems using Deep Learning techniques. A key feature of our approach is that it is independent of the space dimension and can compute arbitrary eigenvalues without requiring the prior computation of lower ones. Moreover, the method can be easily adapted to handle nonlinear eigenvalue problems.

math.NA↗
arXiv2024

Peridynamics and Anisotropic Fractional Sobolev Spaces with Variable Exponents

In this paper, our primary objective is to develop the peridynamic fractional Sobolev space and establish novel BBM-type results associated with it. We also address the peridynamic fractional anisotropic $p-$Laplacian. A secondary objective is to explore anisotropic fractional Sobolev spaces with variable exponents, where we also derive new BBM-type results. Additionally, we address the eigenvalue problem in the isotropic case.

math.AP↗
arXiv2024

On the first eigenvalue of the generalized laplacian

In this work we investigate the energy of minimizers of Rayleigh-type quotients of the form $$ \frac{\int_ΩA(|\nabla u|)\, dx}{\int_ΩA(|u|)\, dx}. $$ These minimizers are eigenfunctions of the generalized laplacian defined as $Δ_a u = \text{div}\left(a(|\nabla u|)\frac{\nabla u}{|\nabla u|}\right)$ where $a(t)=A'(t)$ and the Rayleigh quotient is comparable to the associated eigenvalue. On the function $A$ we only assume that it is a Young function but no $Δ_2$ condition is imposed. Since the problem is not homogeneous, the energy of minimizers is known to strongly depend on the normalization parameter $α=\int_ΩA(|u|)\, dx$. In this work we precisely analyze this dependence and show differentiability of the energy with respect to $α$ and, moreover, the limits as $α\to 0$ and $α\to \infty$ of the Rayleigh quotient. The nonlocal version of this problem is also analyzed.

math.AP↗
arXiv2024

Ljusternik-Schnirelmann eigenvalues for the fractional $m-$Laplacian without the $Δ_2$ condition

In this work we analyze the eigenvalue problem associated to the fractional $m-$Laplacian, defined as $$ (-Δ_m)^s u(x):=2\text{p.v.}\int_{{\mathbb R}^n} m\left(\frac{|u(x)-u(y)|}{|x-y|^s}\right)\frac{(u(x)-u(y))}{|u(x)-u(y)|}\frac{dy}{|x-y|^{n+s}}, $$ This operator serves as a model for nonlocal, nonstandard growth diffusion problems. In contrast to previous analyses, we explore the eigenvalue problem without presuming the $Δ_2$ condition on $M$ -- the primitive function of $m$. Our results show the existence of a sequence of eigenvalues $λ_k\to\infty$. This research contributes to advancing our understanding of nonlocal diffusion models, specifically those characterized by the fractional $m-$Laplacian, by relaxing the constraints imposed by the $Δ_2$ condition.

math.AP↗
arXiv2023

Effective numerical computation of $p(x)-$Laplace equations in 2D

In this article we implement a method for the computation of a nonlinear elliptic problem with nonstandard growth driven by the $p(x)-$Laplacian operator. Our implementation is based in the {\em decomposition--coordination} method that allows us, via an iterative process, to solve in each step a linear differential equation and a nonlinear algebraic equation. Our code is implemented in {\sc MatLab} in 2 dimensions and turns out to be extremely efficient from the computational point of view.

math.NA↗
arXiv2022

A Bourgain-Brezis-Mironescu formula for anisotropic fractional Sobolev spaces and applications to anisotropic fractional differential equations

In this paper we prove Bourgain-Brezis-Mironescu's type results (cf. \cite{BBM2001}) (BBM for short) for an energy functional which is strongly related to the fractional anisotropic p-Laplacian. We also provide with the analogous of Maz'ya-Shaposhnikova (see \cite{MS}) type results for these energies and finally we apply these results to analyze the stability of solutions to anisotropic fractional $p-$laplacian equations.

math.AP↗
arXiv2022

Homogeneous eigenvalue problems in Orlicz-Sobolev spaces

In this article we consider a homogeneous eigenvalue problem ruled by the fractional $g-$Laplacian operator whose Euler-Lagrange equation is obtained by minimization of a quotient involving Luxemburg norms. We prove existence of an infinite sequence of variational eigenvalues and study its behavior as the fractional parameter $s\uparrow 1$ among other stability results.

math.AP↗
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 $s\uparrow 1$.

math.AP↗
arXiv2017

Optimal design problems for the first $p-$fractional eigenvalue with mixed boundary conditions

In this paper we study an optimal shape design problem for the first eigenvalue of the fractional $p-$laplacian with mixed boundary conditions. The optimization variable is the set where the Dirichlet condition is imposed (that is restricted to have measure equal than a prescribed quantity, $α$). We show existence of an optimal design and analyze the asymptotic behavior when the fractional parameter $s\uparrow 1$ obtaining asymptotic bounds that are independent of $α$.

math.AP↗
arXiv2016

Shape optimization problems for nonlocal operators

In this work we study a general shape optimization problem where the state equation is given in terms of a nonlocal operator. Examples of the problems considered are monotone combinations of fractional eigenvalues. Moreover, we also analyze the transition from nonlocal to local state equations.

math.AP↗
arXiv2014

Quasilinear eigenvalues

In this work, we review and extend some well known results for the eigenvalues of the Dirichlet $p-$Laplace operator to a more general class of monotone quasilinear elliptic operators. As an application we obtain some homogenization results for nonlinear eigenvalues.

math.AP↗
arXiv2013

An extension of a Theorem of V. Šverák to variable exponent spaces

In 1993, V. Šverák proved that if a sequence of uniformly bounded domains $Ω_n\subset {\mathbb R}^2$ such that $Ω_n\to Ω$ in the sense of the Hausdorff complementary topology, verify that the number of connected components of its complements are bounded, then the solutions of the Dirichlet problem for the Laplacian with source $f\in L^2({\mathbb R}^2)$ converges to the solution of the limit domain with same source. In this paper, we extend Šverák result to variable exponent spaces.

math.AP↗