نبذة مختصرة : For 2a-order strongly elliptic operators P generalizing (−Δ) a , 0 n 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 12a. 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 a C 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 R n ∖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 L 2 -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.
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