Neumann conditions for the higher order $s$-fractional Laplacian $(-\Delta)^su$ with $s>1$

TL;DR

This paper investigates a variational Neumann problem for the higher order fractional Laplacian with s>1, introducing non-local boundary conditions derived from a Gauss-like formula, expanding understanding of fractional boundary value problems.

Contribution

It introduces a novel non-local Neumann boundary condition for the higher order fractional Laplacian, derived from a Gauss-like integration formula, advancing fractional PDE boundary theory.

Findings

Defined non-local Neumann boundary conditions for s>1

Connected boundary conditions to a Gauss-like integration formula

Provided a variational framework for the problem

Abstract

In this paper we study a variational Neumann problem for the higher order $s$-fractional Laplacian, with $s>1$. In the process, we introduce some non-local Neumann boundary conditions that appear in a natural way from a Gauss-like integration formula.