Parameter-elliptic and parabolic pseudodifferential boundary problems in global \(L_ p\) Sobolev spaces (Q1346321)

The results of this paper extend those of the author [Functional calculus of pseudodifferential boundary problems, Birkhäuser, Boston (1986; Zbl 0622.35001)] from the \(L^2\) framework to \(L^p\) Sobolev spaces. More generally, a parameter-dependent pseudodifferential calculus is presented for Besov and Bessel potential spaces, containing as a special case the resolvent construction for an elliptic pseudodifferential boundary value problem. The resolvent considerations are then used to establish solvability results for parabolic initial-boundary value problems of the type \[ \partial_tu+ P_\Omega u+Gu=f\quad\text{for }(x,t)\in\Omega\times I, \] \[ Tu=\varphi\quad\text{for }(x,t)\in\partial\Omega\times I,\quad u|_{t=0}= u_0\quad\text{for }x\in\Omega \] in anisotropic global \(L^p\) spaces.