Resolvents for fractional-order operators with nonhomogeneous local boundary conditions

Abstract

For 2a-order strongly elliptic operators P generalizing (−Δ)a, 0<a<1, the homogeneous Dirichlet problem on a bounded open set Ω⊂Rn has been widely studied. Pseudodifferential methods have been applied by the present author when Ω is smooth; this is extended in a recent joint work with Helmut Abels showing exact regularity theorems in the scale of Lq-Sobolev spaces Hqs for 1<q<∞, when Ω is Cτ+1 with a finite τ>2a. We now develop this into existence-and-uniqueness theorems (or Fredholm theorems), by a study of the Lp-Dirichlet realizations of P and P⁎, showing that there are finite-dimensional kernels and cokernels lying in daCα(Ω‾) with suitable α>0, d(x)=dist(x,∂Ω). Similar results are established for P−λI, λ∈C. The solution spaces equal a-transmission spaces Hqa(t)(Ω‾).Moreover, the results are extended to nonhomogeneous Dirichlet problems prescribing the local Dirichlet trace (u/da−1)|∂Ω. They are solvable in the larger spaces Hq(a−1)(t)(Ω‾). Furthermore, the nonhomogeneous problem with a spectral parameter λ∈C,Pu−λu=f in Ω,u=0 in Rn∖Ω,(u/da−1)|∂Ω=φ on ∂Ω, 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 L2-Dirichlet realization.The results open up for applications of functional analysis methods. Here we establish solvability results for evolution problems with a time-parameter t, both in the case of the homogeneous Dirichlet condition, and the case where a nonhomogeneous Dirichlet trace (u(x,t)/da−1(x))|x∈∂Ω is prescribed.