Resolvents for fractional-order operators with nonhomogeneous local boundary conditions

Abstract

For $2a$-order strongly elliptic operators $P$ generalizing $(-Δ)^a$, $0 2a$. 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 $P-λI$, $λ\in C$. The solution spaces equal $a$-transmission spaces $H_q^{a(s+2a)}(\barΩ)$. Similar results are shown for nonhomogeneous Dirichlet problems, prescribing the local Dirichlet trace $(u/d^{a-1})|_{\partialΩ}$, $d(x)=dist(x,\partialΩ)$. They are solvable in the larger spaces $H_q^{(a-1)(s+2a)}(\barΩ)$. Moreover, the nonhomogeneous problem with a spectral parameter $λ\in C$, $$ Pu-λu = f \text { in }Ω,\quad u=0 \text { in }R^n\setminus Ω,\quad (u/d^{a-1 })|_{\partialΩ}=φ\text{ on }\partialΩ, $$ is for $q<(1-a)^{-1}$ shown to be uniquely resp. Fredholm solvable when $λ$ 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 $C^{1+τ}$-domains, including nonhomogeneous local boundary conditions.