arXiv subjects
Publications and source records attributed to Ariel M. Salort.
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.
In this article we consider the following weighted nonlinear eigenvalue problem for the $g-$Laplacian $$ -\mathop{\text{ div}}\left( g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\right) = λw(x) h(|u|)\frac{u}{|u|} \quad \text{ in }Ω\subset \mathbb{R}^n, n\geq 1 $$ with Dirichlet boundary conditions. Here $w$ is a suitable weight and $g=G'$ and $h=H'$ are appropriated Young functions satisfying the so called $Δ'$ condition, which includes for instance logarithmic perturbation of powers and different power behaviors near zero and infinity. We prove several properties on its spectrum, being our main goal to obtain lower bounds of eigenvalues in terms of $G$, $H$, $w$ and the normalization $μ$ of the corresponding eigenfunctions. We introduce some new strategies to obtain results that generalize several inequalities from the literature of $p-$Laplacian type eigenvalues.
In this article we study the homogenization rates of eigenvalues of a Steklov problem with rapidly oscillating periodic weight functions. The results are obtained via a careful study of oscillating functions on the boundary and a precise estimate of the $L^\infty$ bound of eigenfunctions. As an application we provide some estimates on the first nontrivial curve of the Dancer-{F}u{č}{\'ı}k spectrum.
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.
In this article we study eigenvalues and minimizers of a fractional non-standard growth problem. We prove several properties on this quantities and their corresponding eigenfunctions.
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.
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 $s\uparrow 1$.
In this note we analyze how perturbations of a ball $\mathfrak{B}_r \subset \mathbb{R}^n$ behaves in terms of their first (non-trivial) Neumann and Dirichlet $\infty-$eigenvalues when a volume constraint $\\mathscr{L}^n(Ω) = \mathscr{L}^n(\mathfrak{B}_r)$ is imposed. Our main result states that $Ω$ is uniformly close to a ball when it has first Neumann and Dirichlet eigenvalues close to the ones for the ball of the same volume $\mathfrak{B}_r$. In fact, we show that, if $$ |λ_{1,\infty}^D(Ω) - λ_{1,\infty}^D(\mathfrak{B}_r)| = δ_1 \quad \text{and} \quad |λ_{1,\infty}^N(Ω) - λ_{1,\infty}^N(\mathfrak{B}_r)| = δ_2, $$ then there are two balls such that $$\mathfrak{B}_{\frac{r}{δ_1 r+1}} \subset Ω\subset \mathfrak{B}_{\frac{r+δ_2 r}{1-δ_2 r}}.$$ In addition, we also obtain a result concerning stability of the Dirichlet $\infty-$eigen-functions.
In this article we prove that the first eigenvalue of the $\infty-$Laplacian $$ \left\{ \begin{array}{rclcl} \min\{ -Δ_\infty v,\, |\nabla v|-λ_{1, \infty}(Ω) v \} & = & 0 & \text{in} & Ωv & = & 0 & \text{on} & \partial Ω, \end{array} \right. $$ has a unique (up to scalar multiplication) maximal solution. This maximal solution can be obtained as the limit as $\ell \nearrow 1$ of concave problems of the form $$ \left\{ \begin{array}{rclcl} \min\{ -Δ_\infty v_{\ell},\, |\nabla v_{\ell}|-λ_{1, \infty}(Ω) v_{\ell}^{\ell} \} & = & 0 & \text{in} & Ωv_{\ell} & = & 0 & \text{on} & \partial Ω. \end{array} \right. $$ In this way we obtain that the maximal eigenfunction is the unique one that is the limit of the concave problems as happens for the usual eigenvalue problem for the $p-$Laplacian for a fixed $1<p<\infty$.
In this article we study the behavior as $p \nearrow+\infty$ of the Fucik spectrum for $p$-Laplace operator with zero Dirichlet boundary conditions in a bounded domain $Ω\subset \mathbb{R}^n$. We characterize the limit equation, and we provide a description of the limit spectrum. Furthermore, we show some explicit computations of the spectrum for certain configurations of the domain.
In this paper we analyze possible extensions of the classical Steklov eigenvalue problem to the fractional setting. In particular, we find a nonlocal eigenvalue problem of fractional type that approximate, when taking a suitable limit, the classical Steklov eigenvalue problem.
Given a bounded domain $Ω$ in $\mathbb{R}^N$, $N\geq 1$ we study the asymptotic behavior as $\varepsilon \to 0$ of the eigencurves of $$ -Δ_p u_\varepsilon=α_\varepsilon m(\tfrac{x}{\varepsilon})(u_\varepsilon^+ )^{p-1} - β_\varepsilon n(\tfrac{x}{\varepsilon})(u_\varepsilon^- )^{p-1} \quad \textrm{ in } Ω$$ with Dirichlet boundary conditions, where $m$ and $n$ are bounded periodic weights. In this work we obtain accurate bounds of the convergence rates of these curves to some limit curves as $\varepsilon \to 0$.
In this work we study the homogenization for eigenvalues of the fractional $p-$Laplace in a bounded domain both with Dirichlet and Neumann conditions. We obtain the convergence of eigenvalues and the explicit order of the convergence rates.
In this work we study the asymptotic behavior of the first non-zero Neumann $p-$fractional eigenvalue $λ_1(s,p)$ as $s\to 1^-$ and as $p\to\infty.$ We show that there exists a constant $\mathcal{K}$ such that $\mathcal{K}(1-s)λ_1(s,p)$ goes to the first non-zero Neumann eigenvalue of the $p-$Laplacian. While in the limit case $p\to \infty,$ we prove that $λ_1(1,s)^{1/p}$ goes to an eigenvalue of the Hölder $\infty-$Laplacian.
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.
In this work we consider the Fuucik problem for a family of weights depending on $\ve$ with Dirichlet and Neumann boundary conditions. We study the homogenization of the spectrum. We also deal with the special case of periodic homogenization and we obtain the rate of convergence of the first non-trivial curve of the spectrum.
In this work we study the homogenization problem for (nonlinear) eigenvalues of quasilinear elliptic operators. We prove convergence of the first and second eigenvalues and, in the case where the operator is independent of $\varepsilon$, convergence of the full (variational) spectrum together whit an explicit in $k$ and in $\varepsilon$ order of convergence.
In this paper we study the rate of convergence of the eigenvalues of 1-dimensional rapidly oscillating $p-$laplacian type problems and find explicit order of convergence both in $k$ and in $\ve$. Moreover, explicit bounds on the constant entering in the estimate are also obtained.