Exact Green's formula for the fractional Laplacian and perturbations

Abstract

Let Ω be an open, smooth, bounded subset of $ \mathbb{R}^n $. In connection with the fractional Laplacian $(-\Delta )^a$ ($a>0$), and more generally for a $2a$-order classical pseudodifferential operator (ψdo) $P$ 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 ψdo 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.