13 research outputs found
Fractional powers of second order partial differential operators: extension problem and regularity theory
Tesis doctoral inédita. Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemåticas. Fecha de lectura: 25-06-201
All functions are (locally) -harmonic (up to a small error) - and applications
The classical and the fractional Laplacians exhibit a number of similarities,
but also some rather striking, and sometimes surprising, structural
differences.
A quite important example of these differences is that any function
(regardless of its shape) can be locally approximated by functions with locally
vanishing fractional Laplacian, as it was recently proved by Serena Dipierro,
Ovidiu Savin and myself.
This informal note is an exposition of this result and of some of its
consequences
Fractional integrals on compact Riemannian symmetric spaces of rank one
In this paper we study mixed norm boundedness for fractional integrals related to Laplace-Beltrami operators on compact Riemannian symmetric spaces of rank one. The key point is the analysis of weighted inequalities for fractional integral operators associated to trigonometric Jacobi polynomials expansions. In particular, we find a novel sharp estimate for the Jacobi fractional integral kernel with explicit dependence on the type parameters. © 2012
Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications
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~.Comment: 39 pages. Submitted on 08/31/2016 and accepted on 03/21/2018. To
appear in Advances in Mathematic