Nonlocal problems with Neumann boundary conditions

TL;DR

This paper introduces a novel nonlocal Neumann boundary condition for the fractional Laplacian, exploring its properties, formulations, and solutions for elliptic and parabolic equations with probabilistic interpretations.

Contribution

It formulates a new nonlocal Neumann problem for the fractional Laplacian, analyzes its properties, and establishes existence, boundary behavior, and asymptotic properties of solutions.

Findings

Solutions conserve mass and decrease energy over time

Solutions to the fractional heat equation converge to a constant

Existence of solutions for elliptic problems is established

Abstract

We introduce a new Neumann problem for the fractional Laplacian arising from a simple probabilistic consideration, and we discuss the basic properties of this model. We can consider both elliptic and parabolic equations in any domain. In addition, we formulate problems with nonhomogeneous Neumann conditions, and also with mixed Dirichlet and Neumann conditions, all of them having a clear probabilistic interpretation. We prove that solutions to the fractional heat equation with homogeneous Neumann conditions have the following natural properties: conservation of mass inside $\Omega$, decreasing energy, and convergence to a constant as $t\to \infty$. Moreover, for the elliptic case we give the variational formulation of the problem, and establish existence of solutions. We also study the limit properties and the boundary behavior induced by this nonlocal Neumann condition. For concreteness, one may think that our nonlocal analogue of the classical Neumann condition~$\partial_\nu u=0$ on~$\partial\Omega$ consists in the nonlocal prescription $$ \int_\Omega \frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy=0 \ {\mbox{ for }} x\in\R^n\setminus\overline{\Omega}.$$