Abstract
The aim of this paper is to develop a layer potential theory in L2‐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 L2‐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.
Highlights
The layer potential methods play a fundamental role in the analysis of elliptic boundary value problems
Fabes et al.[7] obtained mapping properties of layer potential operators for the constant coefficient Stokes system in Lp spaces by using a technique of harmonic analysis. Further extensions of these results to Lp, Sobolev, Bessel potential, and Besov spaces have been obtained by Mitrea and Wright[5] using layer potential methods to obtain well-posedness results for the main boundary value problems for the standard Stokes system with constant coefficients in arbitrary Lipschitz domains in R3
Kohr et al.[8] obtained mapping properties of the constant-coefficient Stokes and Brinkman layer potential operators in standard and weighted Sobolev spaces in R3
Summary
The layer potential methods play a fundamental role in the analysis of elliptic boundary value problems (see, e.g., previous studies[1,2,3,4,5,6]). We explore equivalent mixed variational formulations and prove the well-posedness of some transmission problems for the anisotropic Stokes system in Lipschitz domains of Rn, with the given data in L2-based weighted Sobolev spaces These results are used to define the volume and layer potentials in terms of solutions of the transmission problems and to obtain the potential properties, without introducing classical explicit integral potential operators. We require the symmetry conditions (1.3) and the ellipticity condition (1.4) only for symmetric zero-trace matrices ξ and develop our results in the symmetric stress setting This approach allows us to obtain properties of layer potentials for the Stokes system with L∞ variable coefficients generalizing well-known results for constant coefficients. Under condition (1.4), the anisotropic Stokes system (1.8) is Agmon–Douglis–Nirenberg elliptic (see Lemma 15)
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