Resolvents for fractional-order operators with nonhomogeneous local boundary conditions

TL;DR

This paper extends the analysis of fractional-order elliptic operators with nonhomogeneous boundary conditions, establishing invertibility, regularity, and spectral properties in various function spaces, and applies these results to evolution problems.

Contribution

It provides new results on the invertibility, regularity, and spectral analysis of fractional elliptic operators with nonhomogeneous boundary conditions, including evolution equations, in nonsmooth domains.

Findings

Invertibility and Fredholm properties of $L_q$-Dirichlet realizations.

Regularity results for kernels and cokernels.

Solvability of evolution problems with nonhomogeneous 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.