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Julián Fernández Bonder

Publications and source records attributed to Julián Fernández Bonder.

At least 19 recordsLinked to original sources
arXiv2026

An eigenvalue problem for a generalized polyharmonic operator in Orlicz-Sobolev spaces without the $Δ_2$-condition

In this paper, we consider a generalized polyharmonic eigenvalue problem of the form $A(u)= λh(u)$ in a bounded smooth domain with Dirichlet boundary conditions in the setting of higher-order Orlicz-Sobolev spaces. Here, $A$ is a very general operator depending on $u$ and arbitrary higher-order derivatives of $u$, whose growth is governed by an Orlicz function, and $h$ is a lower order term. Combining the theories of pseudomonotone operators with complementary systems, we prove that this eigenvalue problem has an infinite number of eigenfunctions and that the corresponding sequence of eigenvalues tends to infinite. We point out that the $Δ_2$-condition is not assumed for the involved Orlicz functions.

math.AP↗
arXiv2025

The size of the sync basin resolved

Sparsely coupled Kuramoto oscillators offer a fertile playground for exploring high-dimensional basins of attraction due to their simple yet multistable dynamics. For $n$ identical Kuramoto oscillators on cycle graphs, it is well known that the only attractors are twisted states, whose phases wind around the circle with a constant gap between neighboring oscillators ($θ_j = 2πq j/n$). It was conjectured in 2006 that basin sizes of these twisted states scale as $e^{-kq^2}$ to the winding number $q$. Here, we provide new numerical and analytical evidence supporting the conjecture and uncover the dynamical mechanism behind the Gaussian scaling. The key idea is that, when starting with a random initial condition, the winding number of the solution stabilizes rapidly at $t \propto \log n$, before long-range correlation can develop among oscillators. This timescale separation allows us to calculate the winding number as a sum of weakly-dependent random variables, leading to a Central Limit Theorem derivation of the basin scaling.

math-ph↗
arXiv2025

$Γ-$convergence of energy functionals in fractional Orlicz spaces beyond the $Δ_2$ condition

Given a Young function $A$, $n\geq 1$ and $s\in(0,1)$ we consider the energy functional $$ \mathcal{J}_s(u)=(1-s)\iint_{\mathbb{R}^n\times \mathbb{R}^n} A\left(\frac{|u(x)-u(y)|}{|x-y|^s}\right)\frac{dxdy}{|x-y|^n}. $$ Without assuming the $Δ_2$ condition on $A$ not its conjugated function $\bar A$, we prove the following liminf inequality: if $u\in E^A(\mathbb{R}^n)$ and $\{u_k\}_{k\in\mathbb{N}}\subset E^A(\mathbb{R}^n)$ is such that $u_k\to u$ in $E^A(\mathbb{R}^n)$, and $s_k\to 1$, then $$ \mathcal{J}(u) \leq \liminf_{k\to\infty } \mathcal{J}_{s_k}(u_k), $$ where $\mathcal{J}$ is a limit functional related with the behavior of the fractional Orlicz-Sobolev spaces as $s\to 1^+$. As a direct consequence, we obtain the $Γ-$convergence of the functional $\mathcal{J}_s$. Finally, we extend our result to the study of the so called \emph{fractional peridynamic} case.

math.AP↗
arXiv2025

Fractional Lane-Emden Hamiltonian systems

In this work, our interest lies in proving the existence of solutions to the following Fractional Lane-Emden Hamiltonian system: $$ \begin{cases} (-Δ)^s u = H_v(x,u,v) & \text{in }Ω,\\ (-Δ)^s v = H_u(x,u,v) & \text{in }Ω,\\ u=v=0 & \text{in } \R^n\setminusΩ. \end{cases} $$ The method, that can be traced back to the work of De Figueiredo and Felmer \cite{DF-F}, is flexible enough to deal with more general nonlocal operators and make use of a combination of fractional order Sobolev spaces together with functional calculus for self-adjoint operators.

math.AP↗
arXiv2024

The energy landscape of the Kuramoto model in random geometric graphs in a circle

We study the energy function of the Kuramoto model in random geometric graphs defined in the unit circle as the number of nodes diverges. We prove the existence of at least one local minimum for each winding number $q \in \mathbb{Z}$ with high probability. Hence providing a large family of graphs that support patterns that are generic. These states are in correspondence with the explicit twisted states found in WSG and other highly symmetric networks, but in our situation there is no explicit formula due to the lack of symmetry. The method of proof is simple and robust. It allows other types of graphs like $k-$nn graphs or the boolean model and holds also for graphs defined in any simple closed curve or even a small neighborhood of the curve and for weighted graphs. It seems plausible that the method can be extended also to higher dimensions, but a more careful analysis is required.

math.PR↗
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.

math.AP↗
arXiv2019

Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems

In this paper we analyze the asymptotic behavior of several fractional eigenvalue problems by means of Gamma-convergence methods. This method allows us to treat different eigenvalue problems under a unified framework. We are able to recover some known results for the behavior of the eigenvalues of the $p-$fractional laplacian when the fractional parameter $s$ goes to 1, and to extend some known results for the behavior of the same eigenvalue problem when $p$ goes to $\infty$. Finally we analyze other eigenvalue problems not previously covered in the literature.

math.AP↗
arXiv2019

Stability of solutions for nonlocal problems

In this paper we deal with the stability of solutions of fractional $p-$Laplace problems with nonlinear sources when the fractional parameter $s$ goes to 1. We prove a general convergence result for general weak solutions which is applied to study the convergence of ground state solutions of $p-$fractional problems in bounded and unbounded domains as $s$ goes to 1. Moreover, our result applies to treat the stability of $p-$fractional eigenvalues as $s$ goes to 1.

math.AP↗
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 Hölder Infinity Laplacian obtained as limit of Orlicz Fractional Laplacians

This paper concerns with the study of the asymptotic behavior of the solutions to a family of fractional type problems on a bounded domain, satisfying homogeneous Dirichlet boundary conditions. The family of differential operators includes the fractional $p_n$-Laplacian when $p_n\to\infty$ as a particular case, tough it could be extended to a function of the Hölder quotient of order $s$, whose primitive is an Orlicz function satisfying appropriated growth conditions. The limit equation involves the Hölder infinity Laplacian.

math.AP↗
arXiv2017

Fractional order Orlicz-Sobolev spaces

In this paper we define the fractional order Orlicz-Sobolev spaces, and prove its convergence to the classical Orlicz-Sobolev spaces when the fractional parameter $s\uparrow 1$ in the spirit of the celebrated result of Bourgain-Brezis-Mironescu. We then deduce some consequences such as $Γ-$convergence of the modulars and convergence of solutions for some fractional versions of the $Δ_g$ operator as the fractional parameter $s\uparrow 1$.

math.AP↗
arXiv2016

Homogenization of Fučík eigencurves

In this work we study the convergence of an homogenization problem for half-eigenvalues and Fučík eigencurves. We provide quantitative bounds on the rate of convergence of the curves for periodic homogenization problems.

math.AP↗