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 L_q-Sobolev spaces H_q^s for 1<q<∞, when Ω is C^τ+1 with a finite τ>2a. We now develop this into existence-and-uniqueness theorems (or Fredholm theorems), by a study of the L_p-Dirichlet realizations of P and P^⁎, showing that there are finite-dimensional kernels and cokernels lying in d^aC^α(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 H_q^a(t)(Ω‾). Moreover, the results are extended to nonhomogeneous Dirichlet problems prescribing the local Dirichlet trace (u/d^a−1)|_∂Ω. They are solvable in the larger spaces H_q^(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/d^a−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)/d^a−1(x))|_x∈∂Ω is prescribed.