Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems

Bonder, Julián Fernández; Silva, Analía; Spedaletti, Juan F.

doi:10.3934/dcds.2020355

Cited by 3 publications

(1 citation statement)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Here we show the continuity of the eigenfunctions of L s A with respect to the parameter s, 0 < s ≤ 1. A similar result on s 1 can be found in Theorem 1.2 of [11] for the nonlocal p-Laplacian and Theorem 3.1 of [9] for other nonlocal operators.…”

Section: Therefore On Applying (Lsupporting

confidence: 77%

On an Anisotropic Fractional Stefan-Type Problem with Dirichlet Boundary Conditions

K.¹,

Rodrigues²

2022

Preprint

View full text Add to dashboard Cite

In this work, we consider the fractional Stefan-type problem in a Lipschitz bounded domain Ω ⊂ R d with time-dependent Dirichlet boundary condition for the temperature ϑ = ϑ(x, t), ϑ = g on Ω c ×]0, T [, and initial condition η0 for the enthalpy η = η(x, t), given in Ω×]0, T [ bywhere L s A is an anisotropic fractional operator defined in the distributional sense byβ is a maximal monotone graph, A(x) is a symmetric, strictly elliptic and uniformly bounded matrix, and D s is the distributional Riesz fractional gradient for 0 < s < 1. We show the existence of a unique weak solution with its corresponding weak regularity. We also consider the convergence as s 1 towards the classical local problem, the asymptotic behaviour as t → ∞, and the convergence of the two-phase Stefan-type problem to the one-phase Stefan-type problem by varying the maximal monotone graph β.

show abstract

Section: Therefore On Applying (Lsupporting

confidence: 77%

On an Anisotropic Fractional Stefan-Type Problem with Dirichlet Boundary Conditions

K.¹,

Rodrigues²

2022

Preprint

View full text Add to dashboard Cite

show abstract

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

Dussel¹,

Bonder²

2023

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

On an anisotropic fractional Stefan-type problem with Dirichlet boundary conditions

Catharine¹,

Rodrigues

2023

MINE

View full text Add to dashboard Cite

<abstract>In this work, we consider the fractional Stefan-type problem in a Lipschitz bounded domain $ \Omega\subset\mathbb{R}^d $ with time-dependent Dirichlet boundary condition for the temperature $ \vartheta = \vartheta(x, t) $, $ \vartheta = g $ on $ \Omega^c\times]0, T[$, and initial condition $ \eta_0 $ for the enthalpy $ \eta = \eta(x, t) $, given in $ \Omega\times]0, T[$ by <disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \frac{\partial \eta}{\partial t} +\mathcal{L}_A^s \vartheta = f\quad\text{ with }\eta\in \beta(\vartheta), $\end{document} </tex-math></disp-formula> where $ \mathcal{L}_A^s $ is an anisotropic fractional operator defined in the distributional sense by <disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \langle\mathcal{L}_A^su, v\rangle = \int_{\mathbb{R}^d}AD^su\cdot D^sv\, dx, $\end{document} </tex-math></disp-formula> $ \beta $ is a maximal monotone graph, $ A(x) $ is a symmetric, strictly elliptic and uniformly bounded matrix, and $ D^s $ is the distributional Riesz fractional gradient for $ 0 < s < 1 $. We show the existence of a unique weak solution with its corresponding weak regularity. We also consider the convergence as $ s\nearrow 1 $ towards the classical local problem, the asymptotic behaviour as $ t\to\infty $, and the convergence of the two-phase Stefan-type problem to the one-phase Stefan-type problem by varying the maximal monotone graph $ \beta $.</abstract>

show abstract

Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems

Cited by 3 publications

References 17 publications

On an Anisotropic Fractional Stefan-Type Problem with Dirichlet Boundary Conditions

On an Anisotropic Fractional Stefan-Type Problem with Dirichlet Boundary Conditions

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

On an anisotropic fractional Stefan-type problem with Dirichlet boundary conditions

Contact Info

Product

Resources

About