Boundary value problems in weighted Sobolev spaces on Lipschitz manifolds (Q2927721)

The authors find out the extent to which well-posedness results for the Poisson problem with a Dirichlet boundary condition hold in the setting of weighted Sobolev spaces. They study the extent to which it is possible to depart from basic case and consider \(L^p\)-based Sobolev spaces with \(p\) not necessarily equal to 2.NEWLINENEWLINE On geometric side, the main novelty is the fact that they succeed in formulating the main well-posedness results in the rather general setting of Lipschitz manifolds.NEWLINENEWLINE The results are sharp, by means of counterexamples, for a multitude of perspectives.NEWLINENEWLINE The authors continue the study made in the paper [\textit{L. R. Duduchava} et al., Math. Nachr. 279, No. 9--10, 996--1023 (2006; Zbl 1112.58020)].NEWLINENEWLINEThe atuhors consider weighted Sobolev spaces of arbitrary smoothness in Euclidean Lipschitz domains and prove that Stein's extension operator continues to work in this setting.NEWLINENEWLINE Moreover, this is used to establish a very useful interpolation result.NEWLINENEWLINE Later the authors study the trace theorem for such weighted Sobolev spaces and construct a boundary extension operator, which serves as an inverse from the right for the trace mapping.NEWLINENEWLINE Also boundary value problems for elliptic systems with bounded measurable coefficients in Euclidean Lipschitz domains are treated. Further, these results are generalized to the setting of compact Lipschitz manifolds with boundary.