A remark on nonlocal Neumann conditions for the fractional Laplacian

Nicola Abatangelo

doi:10.1007/s00013-020-01440-9

Abstract

We show how nonlocal boundary conditions of Robin type can be encoded in the pointwise expression of the fractional operator. Notably, the fractional Laplacian of functions satisfying homogeneous nonlocal Neumann conditions can be expressed as a regional operator with a kernel having logarithmic behaviour at the boundary.

Highlights

The purpose of this short note is to put in evidence a special feature of the fractional Laplacian when coupled with nonlocal Robin boundary conditions
By fractional Laplacian we mean the nonlocal operator of order 2s ∈ (0, 2)
A Neumann boundary condition has been proposed by Dipierro, Ros-Oton, and Valdinoci [4], by means of some nonlocal normal derivative

Summary

Introduction

A Neumann boundary condition has been proposed by Dipierro, Ros-Oton, and Valdinoci [4], by means of some nonlocal normal derivative (for which we keep the original notation from the authors) We show how the fractional Laplacian of a function satisfying homogeneous Neumann conditions Nsu = 0 in Rn\Ω can be reformulated as a regional type operator, i.e., of the form Fix Ω ⊂ Rn open, bounded, and with C1,1 boundary.

Results

Conclusion