Layer potentials and Poisson problems for the nonsmooth coefficient Brinkman system in Sobolev and Besov spaces (Q1756639)

The authors consider the Poisson problem with Dirichlet, Neumann and mixed boundary conditions for the Brinkman system with measurable coefficients in \(L^p\)-based Sobolev and Besov spaces in a Lipschitz subset of a compact Riemaniann manifold with no boundary of dimension \(m\geq 2\). The authors first treat the case with homogeneous boundary conditions and show that the problem is well posed for \(p=2\) and deduce that the problem is well posed for \(p\) close to \(p=2\) by exploiting a stability result for isomorphisms in function space interpolation scales. Then the authors consider the case with nonhomogeneous boundary conditions. In the specific case \(p=2\) the authors can also prove a layer potential representation for the solutions by exploiting layer potentials that can be introduced as solutions of certain boundary value problems.