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 $P^{\ast }$, 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 $\lambda \in C$, $$Pu-\lambda u=f\text{ in }\Omega ,u=0\text{ in }{R}^n\setminus \Omega ,(u/d^{a-1}){|}_{\partial \Omega }=\phi \text{ on }\partial \Omega ,$$ 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 $Pu+d_{t}u=f(x,t)$ in $L_2$ and $L_q$-based spaces over $C^{1+\tau }$-domains, including nonhomogeneous local boundary conditions.

47 pages. Minor adjustments. Accepted for publication in Journal of Functional Analysis

References in corpus (3)

• Spectral results for mixed problems and fractional elliptic operators
• Fourier methods for fractional-order operators
• The principal transmission condition

Cited by in corpus (1)

• Discrete approximations to Dirichlet and Neumann Laplacians on a half-space and norm resolvent convergence