In this paper, we define the notion of asymptotic spirallikeness (a generalization of asymptotic starlikeness) in the Euclidean space C n . We consider the connection between this notion and univalent subordination chains. We introduce the notions of A-asymptotic spirallikeness and A-parametric represen-, then a mapping f ∈ S(B n ) is Aasymptotically spirallike if and only if f has A-parametric representation, i.e., if and only if there exists a univalent subordination chain f (z, t) such that Df (0, t) = e At , {e −At f (·, t)} t≥0 is a normal family on B n and f = f (·, 0). In particular, a spirallike mapping with respect to A ∈ L(C n , C n ) with ∞ 0 e (A−2m(A)In )t dt < ∞ has A-parametric representation. We also prove that if f is a spirallike mapping with respect to an operator A such that A + A * = 2In, then f has parametric representation (i.e., with respect to the identity). Finally, we obtain some examples of asymptotically spirallike mappings.

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Abstract. The purpose of this paper is to obtain existence and uniqueness results in weighted Sobolev spaces for transmission problems for the non-linear Darcy-Forchheimer-Brinkman system and the linear Stokes system in two complementary Lipschitz domains in R 3 , one of them is a bounded Lipschitz domain Ω with connected boundary, and the other one is the exterior Lipschitz domain R 3 \Ω. We exploit a layer potential method for the Stokes and Brinkman systems combined with a fixed point theorem in order to show the desired existence and uniqueness results, whenever the given data are suitably small in some weighted Sobolev spaces and boundary Sobolev spaces.Mathematics Subject Classification (2010). Primary 35J25, 35Q35, 42B20, 46E35; Secondary 76D, 76M.Keywords. The Stokes system, the Darcy-Forchheimer-Brinkman system, transmission problems, Lipschitz domains in R 3 , layer potentials, weighted Sobolev spaces, fixed point theorem.

We obtain well-posedness results in Lp-based weighted Sobolev spaces for a transmission problem for anisotropic Stokes and Navier-Stokes systems with L∞ strongly elliptic coefficient tensor, in complementary Lipschitz domains of R n , n ≥ 3. The strong ellipticity allows to explore the associated pseudostress setting. First, we use a variational approach that reduces two linear transmission problems for the anisotropic Stokes system to equivalent mixed variational formulations with data in Lp-based weighted Sobolev and Besov spaces. We show that such a mixed variational formulation is well-posed in the space H 1 p (R n ) n × Lp(R n ), n ≥ 3, for any p in an open interval containing 2. These results are used to define the Newtonian and layer potential operators for the considered anisotropic Stokes system. Various mapping properties of these operators are also obtained. The potentials are employed to show the well-posedness of some linear transmission problems, which then is combined with a fixed point theorem in order to show the well-posedness of the nonlinear transmission problem for the anisotropic Stokes and Navier-Stokes systems in Lp-based weighted Sobolev spaces, whenever the given data are small enough.for an unsteady exterior Stokes problem). The authors in [34] obtained mapping properties of the constant-coefficient Stokes and Brinkman layer potential operators in standard and weighted Sobolev spaces by exploiting results of singular integral operators (see also [35,36]).The methods of layer potential theory play also a significant role in the study of elliptic boundary problems with variable coefficients. Mitrea and Taylor [49, Theorem 7.1] used the technique of layer potentials to prove the well-posedness of the Dirichlet problem for the Stokes system in L p -spaces on arbitrary Lipschitz domains in a compact Riemannian manifold. Dindos and Mitrea [24, Theorems 5.1, 5.6, 7.1,7.3] used a boundary integral approach to show well-posedness results in Sobolev and Besov spaces for Poisson problems of Dirichlet type for the Stokes and Navier-Stokes systems with smooth coefficients in Lipschitz domains on compact Riemannian manifolds. A layer potential analysis of pseudodifferential operators of Agmon-Douglis-Nirenberg type in Lipschitz domains on compact Riemannian manifolds has been developed in [39]. The authors in [37] used a layer potential approach and a fixed point theorem to show well-posedness of transmission problems for the Navier-Stokes and Darcy-Forchheimer-Brinkman systems with smooth coefficients in Lipschitz domains on compact Riemannian manifolds. Choi and Lee [21] proved the well-posedness in Sobolev spaces for the Dirichlet problem for the Stokes system with non-smooth coefficients in a Lipschitz domain Ω ⊂ R n (n ≥ 3) with a small Lipschitz constant when the coefficients have vanishing mean oscillations (VMO) with respect to all variables. Choi and Yang [22] established existence and pointwise bound of the fundamental solution for the Stokes system with measurable coefficients in the spac...

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