The Dirichlet problem for nonlocal operators

By Matthieu Felsinger, M. Kassmann, P.-G. Voigt. 2013

233 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/e7ebdf942c6fbf035acc365202c1cae863c6226e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

In Mechanics of Materials

Citations in the corpus

Interior regularity of solutions of non-local equations in Sobolev and Nikol’skii spaces (2016, methodology): “In this regard, we refer the interested reader to [FKV13], where a class of integro-differential equations with non-symmetric kernels are studied.”
A Simple Solver for the Fractional Laplacian in Multiple Dimensions (2018, background): “, Ros-Oton [43], D’Elia and Gunzburger [12], Felsinger, Kassmann, and Voigt [14], and Lischke et al.”
Fractional Adams–Moser–Trudinger type inequalities (2015, unknown): “1 in [15].”
Fractional Piola identity and polyconvexity in fractional spaces (2018, background): “We will study complement value problems (as in, for example, [16]), so we are interested in the case in which functions are prescribed in the complement of a given set.”
Nonlocal problems with Neumann boundary conditions (2014, background): “In this case, the probability distribution u of the process solves the heat equation with homogeneous Dirichlet conditions u = 0 in Rn\Ω, and solutions to this problem are well understood; see for example [18, 14, 13, 2].”
Local and nonlocal boundary conditions for μ-transmission and fractional elliptic pseudodifferential operators (2014, methodology): “ported in Ω, hence lies in Ha(s) p (Ω)+H˙ s p(Ω) = H a(s) p (Ω) and satisﬁes (2.1) with f= 0, hence it must be 0. This reduction allows a study of higher regularity of the solutions. The treatment in [FKV14] seems primarily directed towards the regularity involved in variational formulations (p= 2, ”