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dc.contributor.authorCiaurri Ó.
dc.contributor.authorRoncal, L. 
dc.contributor.authorStinga, P.R.
dc.contributor.authorTorrea, J.L.
dc.contributor.authorVarona, J.L.
dc.date.accessioned2019-02-12T14:44:07Z
dc.date.available2019-02-12T14:44:07Z
dc.date.issued2018
dc.identifier.issn0001-8708
dc.identifier.urihttp://hdl.handle.net/20.500.11824/922
dc.description.abstractThe analysis of nonlocal discrete equations driven by fractional powers of the discrete Laplacian on a mesh of size $h>0$ \[ (-\Delta_h)^su=f, \] for $u,f:\Z_h\to\R$, $0<s<1$, is performed. The pointwise nonlocal formula for $(-\Delta_h)^su$ and the nonlocal discrete mean value property for discrete $s$-harmonic functions are obtained. We observe that a characterization of $(-\Delta_h)^s$ as the Dirichlet-to-Neumann operator for a semidiscrete degenerate elliptic local extension problem is valid. Regularity properties and Schauder estimates in discrete H\"older spaces as well as existence and uniqueness of solutions to the nonlocal Dirichlet problem are shown. For the latter, the fractional discrete Sobolev embedding and the fractional discrete Poincar\'e inequality are proved, which are of independent interest. We introduce the negative power (fundamental solution) \[ u=(-\Delta_h)^{-s}f, \] which can be seen as the Neumann-to-Dirichlet map for the semidiscrete extension problem. We then prove the discrete Hardy--Littlewood--Sobolev inequality for $(-\Delta_h)^{-s}$. As applications, the convergence of our fractional discrete Laplacian to the (continuous) fractional Laplacian as $h\to0$ in H\"older spaces is analyzed. Indeed, uniform estimates for the error of the approximation in terms of $h$ under minimal regularity assumptions are obtained. We finally prove that solutions to the Poisson problem for the fractional Laplacian \[ (-\Delta)^sU=F, \] in $\R$, can be approximated by solutions to the Dirichlet problem for our fractional discrete Laplacian, with explicit uniform error estimates in terms of~$h$.en_US
dc.formatapplication/pdfen_US
dc.language.isoengen_US
dc.rightsReconocimiento-NoComercial-CompartirIgual 3.0 Españaen_US
dc.rights.urihttp://creativecommons.org/licenses/by-nc-sa/3.0/es/en_US
dc.titleNonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applicationsen_US
dc.typeinfo:eu-repo/semantics/articleen_US
dc.identifier.doi10.1016/j.aim.2018.03.023
dc.relation.publisherversionhttps://doi.org/10.1016/j.aim.2018.03.023en_US
dc.rights.accessRightsinfo:eu-repo/semantics/embargoedAccessen_US
dc.type.hasVersioninfo:eu-repo/semantics/publishedVersionen_US
dc.journal.titleAdv. Math.en_US


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Reconocimiento-NoComercial-CompartirIgual 3.0 España
Except where otherwise noted, this item's license is described as Reconocimiento-NoComercial-CompartirIgual 3.0 España