Showing papers by "Ariel Martin Salort published in 2021"

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

Abstract: In this article, we prove modular and norm Polya–Szego inequalities in general fractional Orlicz–Sobolev spaces by using the polarization technique. We introduce a general framework which includes ...

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

Abstract: 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 (-Δg )s with different boundary conditions, namely, Dirichlet, Neumann and Robin.

A Hölder Infinity Laplacian obtained as limit of Orlicz Fractional Laplacians

Abstract: This paper concerns 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 $$p_n\rightarrow \infty $$ as a particular case, tough it could be extended to a function of the Holder quotient of order s, whose primitive is an Orlicz function satisfying appropriated growth conditions. The limit equation involves the Holder infinity Laplacian.

Asymptotic Behaviours in Fractional Orlicz–Sobolev Spaces on Carnot Groups

Abstract: In this article, we define a class of fractional Orlicz–Sobolev spaces on Carnot groups, and in the spirit of the celebrated results of Bourgain–Brezis–Mironescu and of Maz’ya–Shaposhnikova, we study the asymptotic behaviour of the Orlicz functionals when the fractional parameter goes to 1 and 0.

Magnetic Fractional order Orlicz-Sobolev spaces

Abstract: 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 $\Gamma-$convergence of modulars and convergence of solutions for some non-local magnetic Laplacian allowing non-standard growth laws to its local counterpart.

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

Abstract: 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 ) s u ( x ) ≔ p.v ∫ R n g u ( x ) − u ( y ) | x − y | s d y | 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.

Lower bounds for Orlicz eigenvalues

Abstract: In this article we consider the following weighted nonlinear eigenvalue problem for the \begin{document}$ g- $\end{document} Laplacian \begin{document}$ -{\text{ div}}\left( g(| abla u|)\frac{ abla u}{| abla u|}\right) = \lambda w(x) h(|u|)\frac{u}{|u|} \quad \text{ in }\Omega\subset \mathbb R^n, n\geq 1 $\end{document} with Dirichlet boundary conditions. Here \begin{document}$ w $\end{document} is a suitable weight and \begin{document}$ g = G' $\end{document} and \begin{document}$ h = H' $\end{document} are appropriated Young functions satisfying the so called \begin{document}$ \Delta' $\end{document} condition, which includes for instance logarithmic perturbation of powers and different power behaviors near zero and infinity. We prove several properties on its spectrum, being our main goal to obtain lower bounds of eigenvalues in terms of \begin{document}$ G $\end{document} , \begin{document}$ H $\end{document} , \begin{document}$ w $\end{document} and the normalization \begin{document}$ \mu $\end{document} of the corresponding eigenfunctions. We introduce some new strategies to obtain results that generalize several inequalities from the literature of \begin{document}$ p- $\end{document} Laplacian type eigenvalues.

Maz’ya–Shaposhnikova formula in magnetic fractional Orlicz–Sobolev spaces

Abstract: In this note we prove the validity of the Maz'ya-Shaposhnikova formula in magnetic fractional Orlicz-Sobolev spaces. This complements a previous study of the limit as $s \uparrow 1$ performed by the second author in [21].

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

Abstract: 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: \[ (-\Delta_g)^su(x):=\textrm{p.v.}\int_{\mathbb{R}^n}g\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\frac{dy}{|x-y|^{n+s}}, \] 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.

On the mixed local-nonlocal H\'enon equation

Abstract: In this paper we consider a Henon-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 $s \to 1$.

Global H\"older regularity for eigenfunctions of the fractional $g$-Laplacian

Abstract: We establish global H\"older regularity for eigenfunctions of the fractional $g-$Laplacian with Dirichlet boundary conditions where $g=G'$ and $G$ is a Young functions satisfying the so called $\Delta_2$ condition. Our results apply to more general semilinear equations of the form $(-\Delta_g)^s u = f(u)$.

A computer assisted proof of the symmetries of least energy nodal solutions on squares

Abstract: In this article, we prove that the least energy nodal solutions to Lane-Emden equation $-{\Delta}u = |u|^{p-2}u$ with zero Dirichlet boundary conditions on a square are odd with respect to one diagonal and even with respect to the other one when $p$ is close to 2. We also show that this symmetry breaks on rectangles close to squares.

A nonstandard growth Steklov optimization problem with volume constraint

Abstract: In this article we study an optimal design problem for a nonstandard growth Steklov eigenvalues ruled by the $g-$Laplacian operator. More precisely, given $\Omega\subset \mathbb{R}^n$ and $\alpha,c>0$ we analyze existence and symmetry properties of solution of the optimization problem $\inf \{ \lambda(\alpha,E)\colon E\subset \Omega, |E|=c \}$, where, for a suitable function $u(\alpha,E)$, $\lambda(\alpha,E)$ solves \begin{equation*}\begin{cases} -÷(g( | abla u |)\frac{ abla u}{| abla u|}) + (1+\alpha \chi_E)g( | abla u |)\frac{ abla u}{| abla u|} =0& \text{ in } \Omega,\\ g(| abla u|)\frac{ abla u}{| abla u|} \cdot \eta = \lambda g(|u|)\frac{u}{|u|} &\text{ on } \partial\Omega \end{cases} \end{equation*} being $g$ the derivative of a Young function, and $\eta$ the unit outward normal derivative. We analyze the behavior of the optimization problem as $\alpha$ approaches infinity and its connection of the trace embedding for Orlicz-Sobolev functions.

Continuity of solutions for the Δϕ-Laplacian operator

Abstract: In this paper we give sufficient conditions to obtain continuity results of solutions for the so called ϕ-Laplacian Δϕ with respect to domain perturbations. We point out that this kind of results can be extended to a more general class of operators including, for instance, nonlocal nonstandard growth type operators.

On the asymptotic behavior of $p$-fractional eigenvalues

Abstract: In this note we obtain an asymptotic estimate for growth behavior of variational eigenvalues of the $p-$fractional eigenvalue problem on a smooth bounded domain with Dirichlet boundary condition.

Lower bounds for Orlicz eigenvalues

Abstract: In this article we consider the following weighted nonlinear eigenvalue problem for the $g-$Laplacian $$ -\mathop{\text{ div}}\left( g(| abla u|)\frac{ abla u}{| abla u|}\right) = \lambda w(x) h(|u|)\frac{u}{|u|} \quad \text{ in }\Omega\subset \mathbb{R}^n, n\geq 1 $$ with Dirichlet boundary conditions. Here $w$ is a suitable weight and $g=G'$ and $h=H'$ are appropriated Young functions satisfying the so called $\Delta'$ 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 $\mu$ of the corresponding eigenfunctions. We introduce some new strategies to obtain results that generalize several inequalities from the literature of $p-$Laplacian type eigenvalues.