Layer potential theory for the anisotropic Stokes system with variable <i>L</i><sub><i>∞</i></sub> symmetrically elliptic tensor coefficient
Mirela Kohr, Sergey E. Mikhailov, Wolfgang L. Wendland
Abstract
The aim of this paper is to develop a layer potential theory in L 2 ‐based weighted Sobolev spaces on Lipschitz bounded and exterior domains of , n ≥ 3, for the anisotropic Stokes system with L ∞ viscosity tensor coefficient satisfying an ellipticity condition for symmetric matrices with zero matrix trace. To do this, we explore equivalent mixed variational formulations and prove the well‐posedness of some transmission problems for the anisotropic Stokes system in Lipschitz domains of , with the given data in L 2 ‐based weighted Sobolev spaces. These results are used to define the volume (Newtonian) and layer potentials and to obtain their properties. Then, we analyze the well‐posedness of the exterior Dirichlet and Neumann problems for the anisotropic Stokes system with L ∞ symmetrically elliptic tensor coefficient by representing their solutions in terms of the obtained volume and layer potentials.