3 research outputs found
Transference of fractional Laplacian regularity
In this note we show how to obtain regularity estimates for the fractional
Laplacian on the multidimensional torus from the fractional
Laplacian on . Though at first glance this may seem quite
natural, it must be carefully precised. A reason for that is the simple fact
that functions on the torus can not be identified with functions on
. The transference is achieved through a formula that holds in
the distributional sense. Such an identity allows us to transfer Harnack
inequalities, to relate the extension problems, and to obtain pointwise
formulas and H\"older regularity estimates.Comment: 7 pages. To appear in Special Functions, Partial Differential
Equations and Harmonic Analysis. Proceedings of the conference in honor of
Calixto P. Calder\'on, Roosevelt University at Chicago, November 16-18, 2012.
C. Georgakis, A. Stokolos and W. Urbina (eds
Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications
The analysis of nonlocal discrete equations driven by
fractional powers of the discrete Laplacian on a mesh of size
for , , is performed. The pointwise nonlocal formula for and
the nonlocal discrete mean value property for discrete -harmonic functions are obtained.
We observe that a
characterization of 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)
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 .
As applications, the convergence of our fractional discrete Laplacian to
the (continuous) fractional Laplacian as in H\"older spaces
is analyzed. Indeed, uniform estimates for the error of the approximation
in terms of under minimal regularity assumptions are obtained. We finally prove
that solutions to the Poisson problem for the fractional Laplacian
in , can be approximated by solutions to the Dirichlet problem for our fractional
discrete Laplacian, with explicit uniform error estimates in terms of~