Newtonian and single layer potentials for the Stokes system with $L^{\infty}$ coefficients and the exterior Dirichlet problem

TL;DR

This paper develops Newtonian and layer potentials for the Stokes system with $L^{ olinebreak}^ ext{infinity}$ coefficients on Lipschitz domains, providing a new approach to solving the exterior Dirichlet problem using these potentials.

Contribution

It introduces a mixed variational formulation to define potentials for the Stokes system with $L^{ olinebreak}^ ext{infinity}$ coefficients, enabling solution of the exterior Dirichlet problem.

Findings

Defined Newtonian and layer potentials for the Stokes system with $L^{ olinebreak}^ ext{infinity}$ coefficients.

Expressed the exterior Dirichlet problem solution in terms of these potentials.

Demonstrated the invertibility of the single layer operator on Lipschitz domains.

Abstract

A mixed variational formulation of some problems in $L^2$-based Sobolev spaces is used to define the Newtonian and layer potentials for the Stokes system with $L^{\infty}$ coefficients on Lipschitz domains in ${\mathbb R}^3$. Then the solution of the exterior Dirichlet problem for the Stokes system with $L^{\infty}$ coefficients is presented in terms of these potentials and the inverse of the corresponding single layer operator.