Resolvents for fractional-order operators with nonhomogeneous local boundary conditions

Abstract

For $2a$-order strongly elliptic operators $P$ generalizing $(-\Delta )^a$, $0<a<1$, the treatment of the homogeneous Dirichlet problem on a bounded open set $\Omega \subset R^n$ by pseudodifferential methods, has been extended in a recent joint work with Helmut Abels to nonsmooth settings, showing regularity theorems in $L_q$-Sobolev spaces $H_q^s$ for $1<q<\infty $, when $\Omega $ is $C^{\tau +1}$ with a finite $\tau >2a$. Presently, we study the $L_q$-Dirichlet realizations of $P$ and $P^*$, showing invertibility or Fredholmness, finding smoothness results for the kernels and cokernels, and establishing similar results for $P-\lambda I$, $\lambda \in C$. The solution spaces equal $a$-transmission spaces $H_q^{a(s+2a)}(\bar\Omega)$. Similar results are shown for nonhomogeneous Dirichlet problems, prescribing the local Dirichlet trace $(u/d^{a-1})|_{\partial\Omega }$, $d(x)=dist(x,\partial\Omega)$. They are solvable in the larger spaces $H_q^{(a-1)(s+2a)}(\bar\Omega)$. Moreover, the nonhomogeneous problem with a spectral parameter $\lambda \in C$, $$ Pu-\lambda u = f \text { in }\Omega ,\quad u=0 \text { in }R^n\setminus \Omega ,\quad (u/d^{a-1 })|_{\partial\Omega }=\varphi \text{ on }\partial\Omega , $$ is for $q<(1-a)^{-1}$ shown to be uniquely resp. Fredholm solvable when $\lambda $ is in the resolvent set resp. the spectrum of the $L_2$-Dirichlet realization. Finally, we show solvability results for evolution problems $Pu+d_tu= f(x,t)$ in $L_2$ and $L_q$-based spaces over $C^{1+\tau}$-domains, including nonhomogeneous local boundary conditions.