Resolvents for fractional-order operators with nonhomogeneous local boundary conditions

For 2a -order strongly elliptic operators P generalizing (-Delta)a, 0 < a < 1, the homogeneous Dirichlet problem on a bounded open set 12 subset of Rn has been widely studied. Pseudo differen-tial methods have been applied by the present author when 12 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 < infinity, when 12 is C Tau +1 with a finite Tau > 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 daC alpha(12) with suitable alpha > 0, d(x) = dist(x, partial differential 12). Similar results are established for P - AI, A is an element of C. The solution spaces equal a -transmission spaces Ha(t) q (12).Moreover, the results are extended to nonhomogeneous Dirichlet problems prescribing the local Dirichlet trace (u/da-1)| partial differential Omega. They are solvable in the larger spaces H(a-1)(t) q (12). Furthermore, the nonhomogeneous problem with a spectral parameter A is an element of C,Pu - Au = fin 12, u = 0 in Rn \ 12, (u/da-1)| partial differential Omega =phi on partial differential 12,is for q < (1 - a)-1 shown to be uniquely resp. Fredholm solvable when A 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 non -homogeneous Dirichlet trace (u(x, t)/da-1(x))|x is an element of partial differential Omega is pre-scribed. (c) 2022 The Author(s). Published by Elsevier Inc.