The Dirichlet problem for nonlocal operators

Abstract

In this note we set up the elliptic and the parabolic Dirichlet problem for linear nonlocal operators. As opposed to the classical case of second order differential operators, here the "boundary data" are prescribed on the complement of a given bounded set. We formulate the problem in the classical framework of Hilbert spaces and prove unique solvability using standard techniques like the Fredholm alternative.

Fractional Diffusion Limit of a Kinetic Equation with Diffusive Boundary Conditions in the Upper-Half Space

“…Outline of the paper. The paper is organised as follows: in Section 2 we recall some useful results about the kinetic equation (8) and the free transport equation. In Section 3 we will focus of the specular reflection boundary condition -case for which the method we are developing in this paper takes its simplest form -and we will prove Theorem 1.1.…”

“…We see that the particular choice of power of ε for the rescaling in time depends on the equilibrium F . This is due to the fact that, for such a linear Boltzmann model as (8), the limit diffusion process will be a 2s-stable Levy process, with s the parameter of the polynomial decay of F , as was proved e.g. in [19,18,2] when Ω = R d .…”

“…Several such problems have been studied in the literature, see e.g. [1,9,8,7]. In [4], the author studied the derivation of fractional diffusion approximation from a fractional Vlasov-Fokker-Planck equation in a bounded domain with specular reflection at the boundary.…”

“…The purpose of the present paper is to consider general Maxwell boundary condition (5) for the Linear Boltzmann model (8) in a half-space Ω = R d + . To that end, we will present an improved version of the technique introduced in [5] for the diffusive case.…”

Regularity of spectral fractional Dirichlet and Neumann problems

“…The properties of the restricted Dirichlet fractional Laplacian (− ) a Dir defined in the introduction were studied e.g. in Blumenthal and Getoor [6], Landkof [34], Hoh and Jacob [29], Kulczycki [33], Chen and Song [14], Jakubowski [31], Silvestre [44], Caffarelli and Silvestre [11], Frank and Geisinger [21], Ros-Oton and Serra [36], [37], Felsinger, Kassmann and Voigt [20], Grubb [25], [26], Bonforte, Sire and Vazquez [8], Servadei and Valdinoci [43], Binlin, Molica Bisci and Servadei [5], and many more papers referred to in these works (see in particular the list in [43]).…”