# Exact Green's formula for the fractional Laplacian and perturbations

## TL;DR

This paper derives an explicit Green's formula for the fractional Laplacian and related pseudodifferential operators, clarifying the boundary terms and their locality or nonlocality depending on the underlying differential operator.

## Contribution

It explicitly determines the boundary operator in Green's formula for fractional powers of elliptic operators, extending previous results to more general operators and clarifying when the boundary term is local or nonlocal.

## Key findings

- B=0 for the standard Laplacian case
- B is local (multiplication by a function) when L includes a first-order term
- B can be nonlocal for more general elliptic operators

## Abstract

Let $\Omega $ be an open, smooth, bounded subset of $ \Bbb R ^n$. In connection with the fractional Laplacian $(-\Delta )^a$ ($a>0$), and more generally for a $2a$-order classical pseudodifferential operator ($\psi $do) $P$ with even symbol, one can define the Dirichlet value $\gamma _0^{a-1}u$ resp. Neumann value $\gamma _1^{a-1}u$ of $u(x)$ as the trace resp. normal derivative of $u/d^{a-1}$ on $\partial\Omega $, where $d(x)$ is the distance from $x\in\Omega $ to $\partial\Omega $; they define well-posed boundary value problems for $P$. A Green's formula was shown in a preceding paper, containing a generally nonlocal term $(B\gamma _0^{a-1}u,\gamma _0^{a-1}v)_{\partial\Omega }$, where $B$ is a first-order $\psi $do on $\partial\Omega $. Presently, we determine $B$ from $L$ in the case $P=L^a$, where $L$ is a strongly elliptic second-order differential operator. A particular result is that $B=0$ when $L=-\Delta $, and that $B$ is multiplication by a function (is local) when $L$ equals $-\Delta $ plus a first-order term. In cases of more general $L$, $B$ can be nonlocal.