Green's formula and a Dirichlet-to-Neumann operator for fractional-order pseudodifferential operators

Abstract

The paper treats boundary value problems for the fractional Laplacian $(-\Delta )^a$, $a>0$, and more generally for classical pseudodifferential operators ($\psi $do's) $P$ of order $2a$ with even symbol, applied to functions on a smooth subset $\Omega $ of ${\Bbb R}^n$. There are several meaningful local boundary conditions, such as the Dirichlet and Neumann conditions $\gamma_k^{a-1}u=\varphi $, $k=0,1$, where $\gamma_k^{a-1}u=c_k\partial_n^k(u/d^{a-1})|_{\partial\Omega }$, $d(x)=\operatorname{dist}(x,\partial\Omega )$. We show a new Green's formula $$(Pu,v)_\Omega -(u,P^*v)_\Omega =(s_0\gamma_1^{a-1}u+B\gamma_0^{a-1}u,\gamma_0^{a-1}v)_{\partial\Omega }-(s_0\gamma_0^{a-1}u,\gamma_1^{a-1}v)_{\partial\Omega },$$ where $B$ is a first-order $\psi $do on $\partial\Omega $ depending on the first two terms in the symbol of $P$. Moreover, we show in the elliptic case how the Poisson-like solution operator $K_D$ for the nonhomogeneous Dirichlet problem is constructed from $P^+$ in the factorization $P\sim P^-P^+$ obtained in earlier work. The Dirichlet-to-Neumann operator $S_{DN}=\gamma_1^{a-1}K_D$ is derived from this as a first-order $\psi $do on $\partial\Omega $, with an explicit formula for the symbol. This leads to a characterization of those operators $P$ for which the Neumann problem is Fredholm solvable.