A remark on nonlocal Neumann conditions for the fractional Laplacian

TL;DR

This paper demonstrates how nonlocal Robin boundary conditions can be incorporated into the fractional Laplacian's pointwise expression, revealing a regional operator with a boundary-logarithmic kernel for functions with nonlocal Neumann conditions.

Contribution

It introduces a method to encode nonlocal Robin boundary conditions directly into the fractional Laplacian's pointwise form, connecting boundary conditions with regional operators.

Findings

Fractional Laplacian with nonlocal Neumann conditions expressed as regional operator.

Kernel exhibits logarithmic behavior at the boundary.

Provides a new perspective on boundary conditions for fractional operators.

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.