Newtonian and single layer potentials for the Stokes system with $L^{\infty}$ coefficients and the exterior Dirichlet problem
Mirela Kohr Sergey E. Mikhailov Wolfgang L. Wendland


































Newtonian and single layer potentials for the Stokes system]
Newtonian and single layer potentials for the Stokes system with L^∞ coefficients and the exterior Dirichlet problem

M. Kohr]Mirela Kohr

Faculty of Mathematics and Computer Science, Babeş-Bolyai University,

1 M. Kogălniceanu Str.,
400084 Cluj-Napoca, Romania

mkohr@math.ubbcluj.ro

The authors acknowledge the support of the grant EP/M013545/1: "Mathematical Analysis of Boundary-Domain Integral Equations for Nonlinear PDEs" from the EPSRC, UK


S. E. Mikhailov]Sergey E. Mikhailov
Department of Mathematics, Brunel University London,

             Uxbridge, UB8 3PH, United Kingdom

sergey.mikhailov@brunel.ac.uk


W. L. Wendland]Wolfgang L. Wendland
Institut für Angewandte Analysis und Numerische Simulation, Universität Stuttgart,

Pfaffenwaldring, 57,
70569 Stuttgart, Germany

wendland@mathematik.uni-stuttgart.de




Primary 35J25, 35Q35, 42B20, 46E35; Secondary 76D, 76M





Dedicated to Professor H. Begehr on the occasion of his 80th birthday.



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^∞ coefficients on Lipschitz domains in ℝ^3.
Then the solution of the exterior Dirichlet problem for the Stokes system with L^∞ coefficients is presented in terms of these potentials and the inverse of the corresponding single layer operator.




[
    [
    January 1, 2004
===================





§ INTRODUCTION

Let 
u be an unknown vector field, π be an unknown scalar field, and f be a given vector field defined on an exterior Lipschitz domain Ω_-⊂ℝ^3. Let also 𝔼( u) be the symmetric part of the gradient of u, ∇ u. Then the equations

    [ ℒ_μ( u,π ):=div(2μ𝔼( u))-∇π= f, div  u=0 Ω _- ]

determine the Stokes system with a known viscosity coefficient μ∈ L^∞(Ω_-).
This linear PDE system describes the flows of viscous incompressible fluids, when the inertia of such a fluid can be neglected. The coefficient μ is related to the physical properties of the fluid (for further details we refer the reader to the books <cit.> and <cit.> and the references therein).

The methods of layer potential theory have a main role in the analysis of boundary value problems for elliptic partial differential equations (see, e.g., <cit.>). Fabes, Kenig and Verchota <cit.> obtained mapping properties of layer potentials for the constant coefficient Stokes system in L^p spaces.
Mitrea and Wright <cit.> have used various methods of layer potentials in the analysis of the main boundary problems for the Stokes system with constant coefficients in arbitrary Lipschitz domains in ℝ^n. The authors in <cit.> have obtained mapping properties of the constant coefficient Stokes layer potential operators in standard and weighted Sobolev spaces by exploiting results of singular integral operators. Gatica and Wendland <cit.> used the coupling of mixed finite element and boundary integral methods for solving a class of linear and nonlinear elliptic boundary value problems. The authors in <cit.> used the Stokes and Brinkman integral layer potentials and a fixed point theorem to show an existence result for a nonlinear Neumann-transmission problem for the Stokes and Brinkman systems with data in L^p, Sobolev, and Besov spaces (see also <cit.>). All above results are devoted to elliptic boundary value problems with constant coefficients.

Potential theory plays also a main role in the study of elliptic boundary value problems with variable coefficients. Dindos̆ and Mitrea <cit.> have obtained well-posedness results in Sobolev spaces for Poisson problems for the Stokes and Navier-Stokes systems with Dirichlet condition on C^1 and Lipschitz domains in compact Riemannian manifolds by using mapping properties of Stokes layer potentials in Sobolev and Besov spaces. 
Chkadua, Mikhailov and Natroshvili <cit.> obtained direct segregated systems of boundary-domain integral equations for a mixed boundary value problem of Dirichlet-Neumann type for a scalar second-order divergent elliptic partial differential equation with a variable coefficient in an exterior domain of ℝ^3 (see also <cit.>). Hofmann, Mitrea and Morris <cit.> considered layer potentials in L^p spaces for elliptic operators of the form L=-div(A∇ u) acting in the upper half-space ℝ_+^n, n≥ 3, or in more general Lipschitz graph domains, with an L^∞ coefficient matrix A, which is t-independent, and solutions of the equation L u =0 satisfy interior De Giorgi-Nash-Moser estimates. They obtained a Calderón-Zygmund type theory associated to the layer potentials, and well-posedness results of boundary problems for the operator L in L^p and endpoint spaces.

Our variational approach is inspired by that developed by Sayas and Selgas in <cit.> for the constant coefficient Stokes layer potentials on Lipschitz boundaries, and is based on the technique of Nédélec <cit.>. Girault and Sequeira <cit.> obtained a well-posed result in weighted Sobolev spaces for the Dirichlet problem for the standard Stokes system in exterior Lipschitz domains of ℝ^n, n=2,3. Băcuţă, Hassell and Hsiao <cit.> developed a variational approach for the standard Brinkman single layer potential and used it in the analysis of the time dependent exterior Stokes problem with Dirichlet boundary condition in ℝ^n, n=2,3. Barton <cit.> constructed layer potentials for strongly elliptic differential operators in general settings by using the Lax-Milgram theorem, and generalized various properties of layer potentials for harmonic and second order elliptic equations.
Brewster et al. in <cit.> have used a variational approach and a deep analysis to obtain well-posedness results for boundary problems of Dirichlet, Neumann and mixed type for higher order divergence-form elliptic equations with L^∞ coefficients in locally (ϵ,δ )-domains and in Besov and Bessel potential spaces.
Choi and Lee <cit.> have studied the Dirichlet problem for the Stokes system with nonsmooth coefficients, and proved the unique solvability of the problem in Sobolev spaces on a bounded Lipschitz domain Ω⊂ℝ^n (n≥ 3) with a small Lipschitz constant when the coefficients have vanishing mean oscillations with respect to all variables. Choi and Yang <cit.> obtained the existence and pointwise bound of the fundamental solution for the Stokes system with measurable coefficients in ℝ^n, n≥ 3, whenever the weak solutions of the system are locally Hölder continuous. Alliot and Amrouche <cit.> have used a variational approach to obtain weak solutions for the exterior Stokes problem in weighted Sobolev spaces. Also, Amrouche and Nguyen <cit.> proved existence and uniqueness results in weighted Sobolev spaces for the Poisson problem with Dirichlet boundary condition for the Navier-Stokes system in exterior Lipschitz domains in ℝ^3.

The purpose of this work is to show the well-posedness result of the Poisson problem of Dirichlet type for the Stokes system with L^∞ coefficients in L^2-based Sobolev spaces on an exterior Lipschitz domain in ℝ^3. We use a variational approach that reduces this boundary value problem to a mixed variational formulation. A similar variational approach is used to define the Newtonian and layer potentials for the Stokes system with L^∞ coefficients on Lipschitz surfaces in ℝ^3, by using the weak solutions of some transmission problems in L^2-based Sobolev spaces. Finally, the mapping properties of these layer potentials are used to construct explicitly the solution of the exterior Dirichlet problem for the Stokes system with L^∞ coefficients.
The analysis developed in this paper confines to the case n=3, due to its practical interest, but the extension to the case n≥ 3 can be done with similar arguments.



§ FUNCTIONAL SETTING AND USEFUL RESULTS



Let Ω _+:=Ω⊂ℝ^3 be a bounded Lipschitz domain, i.e., an open connected set whose boundary ∂Ω is locally the graph of a Lipschitz function. Assume that ∂Ω is connected. Let Ω _-:=ℝ^3∖Ω_+ denote the exterior Lipschitz domain. Let E_± denote the operators of extension by zero outside Ω_±.



 §.§ Standard L^2-based Sobolev spaces and related results

Let ℱ and ℱ^-1 denote the Fourier transform and its inverse defined on the
L^1(ℝ^3) functions 
and generalized to the space of tempered distributions 𝒮^*(ℝ^3) (i.e., the topological dual of the space 𝒮(ℝ^3) of all rapidly decreasing infinitely differentiable functions on ℝ^3). The Lebesgue space of (equivalence classes of) measurable, square integrable functions on ℝ^3 is denoted by L^2(ℝ^3), and by L^∞(ℝ^3) we denote the space of (equivalence
classes of) essentially bounded measurable functions on ℝ^3.
Let H^1(ℝ^3) and H^1(ℝ^3)^3 denote the L^2-based Sobolev (Bessel potential) spaces

    H^1(ℝ^3):={f∈𝒮^*(ℝ^3):f_H^1(ℝ^3)=ℱ^-1[(1+|ξ |^2)^1/2ℱf]_L^2(ℝ^3)<∞},
    
       H^1(ℝ^3)^3:={f=(f_1,f_2,f_3):f_j∈ H^1(ℝ^3), j=1,2,3}

Now let Ω ' be Ω_+, Ω_- or ℝ^3. We denote by 
𝒟(Ω '):=C^∞_0(Ω ') the space of infinitely differentiable functions with compact support in Ω ', equipped with the inductive limit topology. 
Let 
𝒟^*(Ω ') denote the corresponding space of distributions on Ω ', i.e., the dual space of 𝒟(Ω ').
Let us consider the space

    H^1(Ω '):={f∈𝒟^*(Ω '):∃ F∈ H^1(ℝ^3)
     F|_Ω '=f} ,

where · |_Ω ' is the restriction operator to Ω '.
The space H^1(Ω ') is the closure of 𝒟(Ω ') in H^1(ℝ^3). This space can be also characterized as

    H^1(Ω '):={f∈ H^1(ℝ^3):supp f⊆Ω '}.

Similar to definition (<ref>), H^1(Ω ')^3 and H^1(Ω ')^3 are the spaces of vector-valued functions whose components belong to the scalar spaces H^1(Ω ') and H^1(Ω '), respectively (see, e.g., <cit.>). 
The Sobolev space H^1(Ω ') can be identified with the closure H^1(Ω ') of 𝒟(Ω ') in the norm of H^1(Ω ') (see, e.g., <cit.>, <cit.>). The space 𝒟(Ω ') is dense in H^1(Ω '), and the following spaces can be isomorphically identified
(cf., e.g., <cit.>)

    (H^1(Ω '))^*=H^-1(Ω '), H^-1(Ω ')=(H^1(Ω '))^*.

For s∈ [0,1], the Sobolev space H^s(∂Ω ) on the boundary ∂Ω can be defined by using the space H^s(ℝ^2), a partition of unity and the pull-backs of the local parametrization of ∂Ω, and H^-s(∂Ω  )=(H^s(∂Ω ))^*. All the above spaces are Hilbert spaces. For further properties of Sobolev spaces on bounded Lipschitz domains and Lipschitz boundaries, we refer to <cit.>.

A useful result for the next arguments is given below (see, e.g., <cit.>, <cit.>). 

 Assume that Ω:=Ω _+⊂ℝ^3 is a bounded Lipschitz domain with connected boundary ∂Ω and denote by Ω _-:=ℝ^3∖Ω the corresponding exterior domain. Then there exist linear and bounded trace operators γ_±:H^1(Ω_±)→ H^1/2(∂Ω) such that γ_±f=f|_∂Ω for any f∈ C^∞(Ω_±).
These operators are surjective and have (non-unique) bounded linear right inverse operators γ^-1_±:H^1/2(∂Ω)→H^1(Ω_±).


The jump of a function u∈ H^1(ℝ^3∖∂Ω ) across ∂Ω is denoted by [γ(u)]:=γ_+(u)-γ_-(u).
For u∈ H^1_ loc(ℝ^3), [γ(u)]=0. The trace operator γ :H^1(ℝ^3)→ H^1/2(∂Ω ) can be also considered and is linear and bounded[The trace operators defined on Sobolev spaces of vector fields on Ω_± or ℝ^3 are also denoted by γ_± and γ, respectively.].

If X is either an open subset or a surface in ℝ^3, then we use the notation ⟨· ,·⟩ _X for the duality pairing of two dual Sobolev spaces defined on X.



 §.§ Some weighted Sobolev spaces and related results


For a point x=(x_1,x_2,x_3)∈ℝ^3, its distance to the origin is denoted by | x|=(x_1^2+x_2^2+x_3^2)^1/2. Let ρ denote the weight function

    ρ ( x)=(1+| x|^2)^1/2 .

For λ∈ℝ, we consider the weighted space L^2(ρ^λ ;ℝ^3) given by

    f∈L^2(ρ^λ ;ℝ^3)⟺ρ^λ f∈ L^2(ℝ^3),

which is a Hilbert space when it is endowed with the inner product and the associated norm,

    (f,g)_L^2(ρ^λ ;ℝ^3) :=∫_ℝ^3fgρ^2λdx, f^2_L^2(ρ^λ ;ℝ^3):= (f,f)_L^2(ρ^λ ;ℝ^3).

We also consider the weighted Sobolev space

    ℋ^1(ℝ^3):   ={f∈𝒟'(ℝ^3):ρ^-1f∈ L^2(ℝ^3), ∇ f∈ L^2(ℝ^3)^3}
       ={f∈ L^2(ρ ^-1;ℝ^3):∇ f∈ L^2(ℝ^3)^3},

which is a Hilbert space with respect to the inner product

    (f,g)_ℋ^1(ℝ^3):=(f,g)_L^2(ρ^-1;ℝ^3)+(∇ f,∇ g)_L^2(ℝ^3)^3

and the associated norm

    f_ℋ^1(ℝ^3)^2:=ρ^-1f_L^2(ℝ^3)^2+∇ f_L^2(ℝ^3)^3^2

(cf. <cit.>; see also <cit.>). The spaces L^2(ρ^λ ;Ω_-) and ℋ^1(Ω _-) can be similarly defined, and 𝒟(Ω_-) is dense in ℋ^1(Ω _-) (see, e.g.,  <cit.>, <cit.>).
The seminorm

    |f|_ℋ^1(Ω _-):=∇ f_L^2(Ω _-)^3

is equivalent to the norm of ℋ^1(Ω _-) defined as in (<ref>), with Ω _- in place of ℝ^3 (see, e.g., <cit.>). The weighted spaces L^2(ρ^-1;Ω_+) and ℋ^1(Ω _+) coincide with the standard spaces L^2(Ω_+) and H^1(Ω _+), respectively (with equivalent norms).

Note that the result in Lemma <ref> extends also to the weighted Sobolev space ℋ^1(Ω_-). Therefore, there exists a linear bounded exterior trace operator

    γ_-:ℋ^1(Ω_-)→ H^1/2(∂Ω),

which is also surjective (see <cit.>). Moreover,
the embedding of the space H^1(Ω _-) into ℋ^1(Ω _-) and Lemma <ref> show the existence of a (non-unique) linear and bounded right inverse γ^-1_-: H^1/2(∂Ω)→ℋ^1(Ω_-) of operator (<ref>)
(see <cit.>, <cit.>).

Let ℋ^1(Ω _-)⊂ℋ^1(Ω_-) denote the closure of 𝒟(Ω_-) in ℋ^1(Ω _-). This space can be characterized as

    ℋ^1(Ω _-)={v∈ℋ^1(Ω _-):γ_-v=0 ∂Ω}

(cf., e.g., <cit.>). Also let ℋ^1(Ω _-)⊂ℋ^1(ℝ^3) denote the closure of 𝒟(Ω_-) in ℋ^1(ℝ^3). This space can be also characterized as

    ℋ^1(Ω _-)={u∈ℋ^1(ℝ^3):supp u⊆Ω_-},

and can be isomorphically identified with the space ℋ^1(Ω _-) via the extension by zero operator E_-, i.e., ℋ^1(Ω _-)= E_-ℋ^1(Ω _-) (cf., e.g., <cit.>).
In addition, consider the spaces (see, e.g.,  <cit.>, <cit.>)

    ℋ^-1(ℝ^3):=(ℋ^1(ℝ^3))^*, ℋ^-1(Ω _-):=(ℋ^1(Ω _-))^*,
    ℋ^-1(Ω _-):=(ℋ^1(Ω _-))^*.




§ THE CONORMAL DERIVATIVE OPERATORS FOR THE STOKES SYSTEM WITH L^∞ COEFFICIENTS


In the sequel we assume that the viscosity coefficient μ of the Stokes system (<ref>) belongs to L^∞(ℝ^3) and there exists a constant c_μ>0, such that

    c_μ^-1≤μ≤c_μℝ^3.


Let 𝔼( u):=1/2(∇ u+(∇ u)^⊤) be the strain rate tensor.
If ( u,π)∈C^1(Ω_±)^3×C^0(Ω_±), we can define the classical interior and exterior conormal derivatives (i.e., the boundary traction fields) for the Stokes system (<ref>) with continuously differentiable viscosity coefficient μ by the well-known formula

    t_μ^c±( u,π ):=γ_±(-π𝕀+2μ𝔼( u))ν ,

where ν is the outward unit normal to Ω_ +, defined a.e. on ∂Ω.
Then for any function φ∈𝒟(ℝ^3)^3 we obtain the first Green identity

    ±⟨ t_μ^c±( u,π ),
    φ⟩ __∂Ω
    =    2⟨μ𝔼( u),𝔼(φ )⟩ _Ω_±
    -⟨π,div φ⟩ _Ω_±
    +⟨ℒ_μ( u,π ),φ⟩ _Ω_±.

This formula suggests the following weak definition of the generalized conormal derivative for the Stokes system with L^∞ coefficients in the setting of L^2-weighted Sobolev spaces (cf., e.g., <cit.>, <cit.>, <cit.>, <cit.>, <cit.>).


Let μ∈ L^∞(ℝ^3) satisfy assumption (<ref>). Let

    ℋ^1(Ω_±,ℒ_μ):={( u_±,π_±,f̃_±)∈ℋ^1(Ω_±)^3× L^2(Ω_±)×ℋ^-1(Ω_±)^3:
    ℒ_μ( u_±,π_± )=f̃_±|_Ω_±div  u_±=0 Ω_±}.

Then define the conormal derivative operator t_μ^±:ℋ^1(Ω_±,ℒ_μ)→H^-1/2(∂Ω )^3,

    ℋ^1(Ω_±,ℒ_μ)∋ ( u_±,π_± ,f̃_±)⟼ t_μ^±( u_±,π_± ;f̃_±)∈ H^-1/2(∂Ω )^3,
       ±⟨ t_μ^±( u_±,π_± ;f̃_±),Φ⟩ __∂Ω:=2⟨μ𝔼( u_±),𝔼(γ^-1_±Φ)⟩ _Ω_±
       -⟨π_±,div(γ^-1_±Φ)⟩ _Ω_±+⟨f̃_±,γ^-1_±Φ⟩ _Ω_±, ∀ Φ∈ H^1/2(∂Ω )^3,

where γ^-1_±:H^1/2(∂Ω )^3→ℋ^1(Ω_±)^3 is a (non-unique) bounded right inverse of the trace operator γ_±:
ℋ^1(Ω_±)^3→ H^1/2(∂Ω )^3. 

We use the simplified notation t_μ^±( u_±,π_±) for t_μ^±( u_±,π_±; 0).
The following assertion can be proved similar to <cit.>, <cit.>.


Let μ∈L^∞(ℝ^3) satisfy assumption (<ref>). Then for all w_±∈ℋ^1(Ω_±)^3 and ( u_±,π_±,f̃_±)∈ℋ^1(Ω_±,ℒ_μ)
the following identity holds

    ±⟨ t_μ^±( u_±,π_±;f̃_±),γ_± w_±⟩ __∂Ω=2⟨μ𝔼( u_±),𝔼( w_±)⟩ _Ω_±   -⟨π_±,div  w_±⟩ _Ω_±
       +⟨f̃_±, w_±⟩ _Ω_± .


Let γ denote the trace operator from ℋ^1(ℝ^3)^3 to H^1/2(∂Ω )^3 (cf., e.g., <cit.>, <cit.>).
For
( u_±,π_±,f̃_±)∈ℋ^1(Ω_±,ℒ_μ), let

    u:= E_+ u_+ +  E_- u_-,π:= E_+π_+ +  E_-π_-, f:=f̃_+ + f̃_-
       [ t_μ( u,π; f)]:=
     t_μ^+( u_+,π_+;f̃_+)- t_μ^-( u_-,π_-;f̃_-).

Moreover, if f= 0, we define

    [ t_μ( u,π)]:=[ t_μ( u,π; 0)]
    = t_μ^+( u_+,π_+)- t_μ^-( u_-,π_-).

Then Lemma <ref> leads to the following result.


Let μ∈L^∞(ℝ^3) satisfy assumption (<ref>). Also let
( u_±,π_±,f̃_±)∈ℋ^1(Ω_±,ℒ_μ) and let ( u,π , f) be defined as in (<ref>).
Then for all w∈ℋ^1(ℝ^3)^3, the following formula holds

    ⟨ [ t_μ( u,π; f)],γ w⟩ __∂Ω
    =   2⟨μ𝔼( u_+),𝔼( w)⟩ _Ω_+
    +2⟨μ𝔼( u_-),𝔼( w)⟩ _Ω_-
       -⟨π,div  w⟩_ℝ^3
    +⟨ f, w⟩_ℝ^3 .


We also need the following particular case of Lemma <ref> when f= 0.


Let μ∈L^∞(ℝ^3) satisfy assumption (<ref>). Also let
( u_±,π_±, 0)∈ℋ^1(Ω_±,ℒ_μ). 
Then for all w∈ℋ^1(ℝ^3)^3,

    ⟨ [ t_μ( u,π )],γ w⟩ __∂Ω
    =   2⟨μ𝔼( u_+),𝔼( w)⟩ _Ω_+
    +2⟨μ𝔼( u_-),𝔼( w)⟩ _Ω_-
       -⟨π,div  w⟩_ℝ^3 .






§ NEWTONIAN AND SINGLE LAYER POTENTIALS FOR THE STOKES SYSTEM WITH L^∞ COEFFICIENTS


Recall that the function μ∈L^∞(ℝ^3) satisfies conditions (<ref>). Next, we define the Newtonian and single layer potentials for the L^∞ coefficient Stokes system (<ref>).


 §.§ Variational solution of the variable-coefficient Stokes system in ℝ^3.

First we show the following useful well-posedness result.


Let a_μ(· ,· ):ℋ^1(ℝ^3)^3×ℋ^1(ℝ^3)^3→ℝ and b(· ,·):ℋ^1(ℝ^3)^3× L^2(ℝ^3)→ℝ be the bilinear forms given by

    a_μ( u, v)   :=2⟨μ𝔼( u),𝔼( v)⟩ _ℝ^3, ∀  u, v∈ℋ^1(ℝ^3)^3,
    
    b( v,q)   :=-⟨div  v,q⟩ _ℝ^3, ∀  v∈ℋ^1(ℝ^3)^3, ∀  q∈ L^2(ℝ^3).

Also let ℓ :ℋ^1(ℝ^3)^3→ℝ be a linear and bounded map. Then the mixed variational formulation

    {[ a_μ( u, v)+b( v,p)=ℓ ( v), ∀  v∈ℋ^1(ℝ^3)^3,;                   b( u,q)=0, ∀  q∈ L^2(ℝ^3) ].

is well-posed. Hence, (<ref>) has a unique solution ( u,p)∈ℋ^1(ℝ^3)^3× L^2(ℝ^3) and there exists a constant C=C(μ )>0 such that

    u_ℋ^1(ℝ^3)^3+p_L^2(ℝ^3)≤
    Cℓ_ℋ^-1(ℝ^3)^3.



By using conditions (<ref>) and definition (<ref>) of the norm of the weighted Sobolev space ℋ^1(ℝ^3) we obtain that

    |a_μ( u, v)|   ≤ 2c_μ𝔼( u)_L^2(ℝ^3)^3× 3𝔼( v)_L^2(ℝ^3)^3× 3
       ≤ 2c_μ u_ℋ^1(ℝ^3)^3 v_ℋ^1(ℝ^3)^3, ∀  u, v∈ℋ^1(ℝ^3)^3.

Moreover, by using the Korn type inequality for functions in ℋ^1(ℝ^3)^3,

    grad  v_L^2(ℝ^3)^3× 3≤ 2^1/2𝔼 ( v)_L^2(ℝ^3)^3× 3

(cf., e.g., <cit.>) and since the seminorm

    |g|_ℋ^1(ℝ^3):=∇ g_L^2(ℝ^3)^3

is a norm in ℋ^1(ℝ^3)^3 equivalent to the norm defined by (<ref>) (see, e.g., <cit.>), there exists a constant c_1 such that

    a_μ( u, u)≥ 2c_μ^-1𝔼( u)_L^2(ℝ^3)^3× 3^2   ≥ c_μ^-1∇ u_L^2(ℝ^3)^3× 3^2
       ≥ c_μ^-1c_1 u_ℋ^1(ℝ^3)^3^2, ∀  u∈ℋ^1(ℝ^3)^3.

Inequalities (<ref>) and (<ref>) show that a_μ(· ,· ):ℋ^1(ℝ^3)^3×ℋ^1(ℝ^3)^3→ℝ is a bounded and coercive bilinear form. Moreover, since the divergence operator

    div:ℋ^1(ℝ^3)^3→ L^2(ℝ^3)

is bounded, then the bilinear form b(· ,·):ℋ^1(ℝ^3)^3× L^2(ℝ^3)→ℝ is bounded as well. In addition, the operator in (<ref>) is surjective (cf. <cit.>, <cit.>) and also 

    ℋ^1_ div(ℝ^3)^3:=   { w∈ℋ^1(ℝ^3)^3: div  w=0}
    
    =   { w∈ℋ^1(ℝ^3)^3: b( w,q)=0, ∀  q∈ L^2(ℝ^3) }.

In addition, the operator in (<ref>) is surjective (cf. <cit.>, <cit.>), and hence the operator 
    -div:ℋ^1(ℝ^3)^3/ℋ^1_ div(ℝ^3)^3→ L^2(ℝ^3)
 is an isomorphism. Then by Lemma <ref>(ii) the bounded bilinear form b(· ,·):ℋ^1(ℝ^3)^3× L^2(ℝ^3)→ℝ satisfies the inf-sup condition (<ref>). Hence, there exists β _0∈ (0,∞ ) such that

    inf _q∈ L^2(ℝ^3)∖{0}sup _ w∈ℋ^1(ℝ^3)^3∖{ 0}b( w,q)/ w_ℋ^1(ℝ^3)^3q_L^2(ℝ^3)≥β _0.

By applying Theorem <ref>, with X=ℋ^1(ℝ^3)^3, M=L^2(ℝ^3), V=ℋ^1_div(ℝ^3)^3, we conclude that the mixed variational formulation (<ref>) has a unique solution ( u,p)∈ℋ^1(ℝ^3)^3× L^2(ℝ^3) and there exists a constant C=C(μ )>0 such that ( u,p) satisfies inequality (<ref>).

Next we use the result of Lemma <ref> in order to show the well-posedness of the L^∞ coefficient Stokes system in the space ℋ^1(ℝ^3)^3× L^2(ℝ^3) (see also <cit.> for the constant-coefficient case).


Let μ∈L^∞(ℝ^3) satisfy conditions (<ref>).
Then the L^∞ coefficient Stokes system

    {[ div(2μ𝔼( u))-∇π =ℓ ,       ℓ∈ℋ^-1(ℝ^3)^3,;            div  u=0,                 ℝ^3, ].

has a unique solution ( u,p)∈ℋ^1(ℝ^3)^3× L^2(ℝ^3), and there exists a constant C_0=C_0(μ )>0 such that

    u_ℋ^1(ℝ^3)^3+ p_L^2(ℝ^3)≤
    C_0ℓ_ℋ^-1(ℝ^3)^3.



Note that the Stokes system (<ref>) is equivalent to the variational problem (<ref>) as follows from the density of 𝒟(ℝ^3)^3 in the space ℋ^1(ℝ^3)^3 (cf., e.g., <cit.>, <cit.>). Then the well-posedness result of the Stokes system with L^∞ coefficients (<ref>) follows from Lemma <ref>.




 §.§ Newtonian potential for the Stokes system with L^∞ coefficients


The well-posedness of problem (<ref>) allows us to define the Newtonian potential for the Stokes system with L^∞ coefficients as follows.


For any ℓ∈ℋ^-1(ℝ^3)^3, we define the Newtonian velocity and pressure potentials for the Stokes system with L^∞ coefficients
as

    𝒩_μ ;ℝ^3ℓ:= u, 𝒬_μ ;ℝ^3ℓ:=π,

where ( u,π )∈ℋ^1(ℝ^3)^3× L^2(ℝ^3) is the unique solution of problem (<ref>) with the given datum ℓ.

Moreover, the well-posedness of problem (<ref>) 
yields the continuity of the above operators as stated in the following assertion (cf. also <cit.> for μ =1).


The Newtonian velocity and pressure potential operators

    𝒩_μ ;ℝ^3:ℋ^-1(ℝ^3)^3→ℋ^1(ℝ^3)^3,𝒬_μ ;ℝ^3:ℋ^-1(ℝ^3)^3→ L^2(ℝ^3)

are linear and continuous.




 §.§ Single layer potential for the Stokes system with L^∞ coefficients


For a given φ∈ H^-1/2(∂Ω )^3, we now consider the following transmission problem for the Stokes system with L^∞ coefficients

    {[ div(2μ𝔼( u_φ))-∇π _φ= 0                ℝ^3∖∂Ω ,;              div  u_φ=0                ℝ^3∖∂Ω ,;    [ t_μ( u_φ,π _φ)]
    =φ                    ∂Ω , ].

and show that this problem has a unique solution ( u_φ,π _φ)∈ℋ^1(ℝ^3)^3×L^2(ℝ^3)
(cf. also <cit.> for μ=1). Note that the membership of u_φ in ℋ^1(ℝ^3)^3 implies the transmission condition

    [γ( u_φ)]= 0∂Ω ,

and the first equation in (<ref>) implies also that the jump [ t_μ( u_φ,π _φ)] is well defined.


Let φ∈ H^-1/2(∂Ω )^3 be given. Then the transmission problem (<ref>) has the following equivalent mixed variational formulation: Find  ( u_φ,π _φ)∈ℋ^1(ℝ^3)^3× L^2(ℝ^3) such that

    {[ 2⟨μ𝔼( u_φ),𝔼( v)⟩ _ℝ^3-⟨π _φ,div  v⟩ _ℝ^3
    =⟨φ ,γ v⟩ _∂Ω, ∀  v∈ℋ^1(ℝ^3)^3,;                                      ⟨div  u_φ,q⟩ _ℝ^3=0, ∀  q∈ L^2(ℝ^3). ].

Moreover, problem (<ref>) is well-posed. Hence (<ref>) has a unique solution ( u_φ,π _φ)∈ℋ^1(ℝ^3)^3× L^2(ℝ^3), and there exists a constant C=C(μ ) such that

    u_φ_ℋ^1(ℝ^3)^3+π _φ_L^2(ℝ^3)≤ Cφ_H^-1/2(∂Ω )^3.



The equivalence between the transmission problem (<ref>) and the variational problem (<ref>) follows from the density of the space 𝒟(ℝ^3)^3 in ℋ^1(ℝ^3)^3 and formula (<ref>), while the well-posedness of the variational problem (<ref>) is an immediate consequence of Lemma <ref> with the linear form ℓ :ℋ^1(ℝ^3)^3→ℝ given by

    ℓ ( v):=⟨φ ,γ v⟩ _∂Ω=
    ⟨γ ^*φ , v⟩ _ℝ^3, ∀  v∈ℋ^1(ℝ^3)^3,

and hence ℓ =γ ^*φ, where γ ^*:H^-1/2(∂Ω )^3→ℋ^-1(ℝ^3)^3 is the adjoint of the trace operator γ :ℋ^1(ℝ^3)^3→ H^1/2(∂Ω )^3.

Theorem <ref> leads to the following definition (cf. <cit.> for μ=1).


For any φ∈ H^-1/2(∂Ω )^3, we define the single layer velocity and pressure potentials for the Stokes system with L^∞ coefficients (<ref>) as

    V_μ ;∂Ωφ:= u_φ, 𝒬^s_μ ;∂Ωφ:=π _φ,

and the potential operators 𝒱_μ ;∂Ω:𝒱_μ ;∂Ω:H^-1/2(∂Ω )^3→ H^1/2(∂Ω )^3 and K_μ ;∂Ω^*:𝒱_μ ;∂Ω:H^-1/2(∂Ω )^3→ H^-1/2(∂Ω )^3 as

    𝒱_μ ;∂Ωφ:=γ u_φ, K_μ ;∂Ω^*φ:=
    1/2(
     t_μ^+( u_φ,π_φ)
    + t_μ^-( u_φ,π_φ)),

where ( u_φ,π _φ) is the unique solution of problem (<ref>) in ℋ^1(ℝ^3)^3× L^2(ℝ^3).

The next result shows the continuity of single layer velocity and pressure operators for the variable coefficient Stokes system (cf. <cit.>, <cit.> and <cit.> in the case μ =1).


The following operators are linear and continuous

    V_μ ;∂Ω:H^-1/2(∂Ω )^3→ℋ^1(ℝ^3)^3,𝒬_μ ;∂Ω^s:H^-1/2(∂Ω )^3→ L^2(ℝ^3),
    
       𝒱_μ ;∂Ω:H^-1/2(∂Ω )^3→ H^1/2(∂Ω )^3,  K_μ ;∂Ω^*:H^-1/2(∂Ω )^3→ H^-1/2(∂Ω )^3.



The continuity of operators (<ref>) and (<ref>) follows from the well-posedness of the transmission problem (<ref>) and Definition <ref>. 

The next result yields the jump relations of the single layer potential and its conormal derivative across ∂Ω (see also <cit.> for μ =1).


Let φ∈H^-1/2(∂Ω )^3. Then almost everywhere on ∂Ω,

    [γ V_μ ;∂Ωφ]= 0,
       [ t_μ( V_μ ;∂Ωφ,𝒬_μ ;∂Ω^sφ)]=φ ,  t_μ^±( V_μ ;∂Ωφ,𝒬_μ ;∂Ω^sφ)=±1/2φ+ K_μ ;∂Ω^*φ .



Formulas (<ref>) and (<ref>) follow from Definition <ref> and the transmission condition in (<ref>), as well as the transmission condition in the third line of (<ref>).

Let ℝν ={cν :c∈ℝ}. Let Ker{T:X→ Y}:={x∈ X:T(x)=0} denote the null space of the map T:X→ Y.
We next obtain the main properties of the single layer potential operator (cf., e.g., <cit.>, and <cit.> and <cit.> for μ =1 and α∈ [0,∞ )).

The following properties hold

    V_μ ;∂Ων= 0ℝ^3, 𝒬^s_μ ;∂Ων=-χ _Ω _+
       Ker{𝒱_μ ;∂Ω:H^-1/2(∂Ω )^3→ H^1/2(∂Ω )^3}=ℝν ,
       𝒱_μ ;∂Ωφ∈ H_ν^1/2(∂Ω )^3, ∀ φ∈ H^-1/2(∂Ω )^3,

where 
χ _Ω _+=1 in Ω _+, χ _Ω _+=0 in Ω _-, and

    H_ν^1/2(∂Ω )^3:={ϕ∈ H^1/2(∂Ω )^3: ⟨ν ,ϕ⟩ _∂Ω=0}.



First, we consider the transmission problem (<ref>) with the datum φ=ν∈H^-1/2(∂Ω )^3. Then the solution of this problem is given by

    ( u_ν,π _ν)=( 0,-χ _Ω _+)∈ℋ^1(ℝ^3)^3× L^2(ℝ^3).

Indeed, the pair ( u_ν,π _ν) satisfies the equations and the transmission condition in (<ref>), as well as the transmission condition (<ref>), and, in view of formula (<ref>) and the divergence theorem,

    ⟨ [ t_μ( u_ν,π_ν)],γ v⟩ _∂Ω=-⟨π _ν,div  v⟩ _ℝ^3=⟨ν,γ v⟩ _∂Ω, ∀  v∈𝒟(ℝ^3)^3.

Then by formula (<ref>), Lemma <ref>, the dense embedding of the space 𝒟(ℝ^3)^3 in H^1(ℝ^3)^3, and the above equality, we obtain that ⟨ [ t_μ( u_ν,π_ν)],Φ⟩ _∂Ω=⟨ν,Φ⟩ _∂Ω for any Φ∈ H^1/2(∂Ω )^3. Hence, [ t_μ( u_ν,π_ν)]=ν, as asserted.
Then Definition <ref> implies relations (<ref>). Moreover, 𝒱_μ ;∂Ων= 0, i.e.,
ℝν⊆Ker{𝒱_μ ;∂Ω:H^-1/2(∂Ω )^3→ H^1/2(∂Ω )^3}.

Now let φ _0∈Ker{𝒱_μ ;∂Ω
:H^-1/2(∂Ω )^3→ H^1/2(∂Ω )^3}. Let
( u_φ _0,π_φ _0)=( V_μ ;∂Ωφ _0,𝒬^s_μ ;∂Ωφ _0)∈ℋ^1(ℝ^3)^3×L^2(ℝ^3) be the unique solution of problem (<ref>) with datum φ _0. Since γ u_φ _0= 0 on ∂Ω, formula (<ref>) yields that

    0=⟨ [ t_μ( u_φ _0,π_φ _0)],γ u_φ _0⟩ _∂Ω=a_μ( u_φ _0, u_φ _0),

and hence u_φ _0= 0, π _φ _0=cχ _Ω _+ in ℝ^3, where c∈ℝ. In view of formula (<ref>),

    ⟨ [ t_μ( u_φ _0,π_φ _0)],γ w⟩ _∂Ω=-⟨π _φ _0,div  w⟩ _ℝ^3=-c⟨ν ,γ w⟩ _∂Ω, ∀  w∈𝒟(ℝ^3)^3,

and, thus, φ _0=[ t_μ( u_φ,π_φ _0)]=-cν. Hence, formula (<ref>) follows.

Now let φ∈ H^-1/2(∂Ω )^3. By using the first formula in (<ref>),
we obtain for any φ∈ H^-1/2(∂Ω )^3 that
⟨𝒱_μ ;∂Ωφ ,ν⟩ _∂Ω=
⟨γ𝐮_φ,ν⟩ _∂Ω
=⟨div 𝐮_φ, 1 ⟩_Ω=0,
where u_φ= V_μ ;∂Ωφ.
Thus, we get relation (<ref>).


Next we use the notation [[·]] for the equivalence classes of the space H^-1/2(∂Ω ,Λ ^1TM)/ℝν. Thus, any [[φ]]∈H^-1/2(∂Ω )^3/ℝν can be written as [[φ]]=φ+ℝν, where φ∈ H^-1/2(∂Ω )^3.

Exploiting properties (<ref>) and (<ref>), we now show the following invertibility result (cf. <cit.>, <cit.>, <cit.> for μ =1 and α≥ 0 constant).


The following operator is an isomorphism

    𝒱_μ ;∂Ω:H^-1/2(∂Ω )^3/ℝν→ H_ν^1/2(∂Ω )^3.



We use arguments similar to those for Proposition 5.5 in <cit.>. First, Lemma <ref> and the membership relation (<ref>) imply that the linear operator in (<ref>) is continuous.
We show that this operator is also H^-1/2(∂Ω )^3/ℝν-elliptic 
i.e., that there exists a constant c=c(∂Ω ) such that

    ⟨𝒱_μ ;∂Ω[[φ]],[[φ]]⟩ _∂Ω≥ c[[φ]]_H^-1/2(∂Ω )^3/ℝν^2, ∀ [[φ]]∈H^-1/2(∂Ω )^3/ℝν.

Let [[φ]]∈H^-1/2(∂Ω )^3/ℝν. Thus, [[φ]]=φ+ℝν, where φ∈ H^-1/2(∂Ω )^3. In view of formula (<ref>), Definition <ref>, relations (<ref>), (<ref>), and inequality (<ref>),

    ⟨𝒱_μ ;∂Ω([[φ]]),[[φ]]⟩ _∂Ω   =⟨𝒱_μ ;∂Ω(φ ),φ⟩ _∂Ω=⟨γ u_φ,[ t_μ( u_φ,π_φ)]⟩ _∂Ω
       =a_μ( u_φ, u_φ)≥c_μ^-1 u_φ_H^1(ℝ^3)^3^2,

where u_φ= V_μ ;∂Ωφ and π_φ=𝒬^s_μ ;∂Ωφ.
Now we use the property that the trace operator

    γ :ℋ_ div^1(ℝ^3)^3→ H_ν^1/2(∂Ω )^3

is surjective having a bounded right inverse γ ^-1:H_ν^1/2(∂Ω )^3→ℋ_ div^1(ℝ^3)^3 (cf., e.g., <cit.>). Hence, for any Φ∈ H_ν^1/2(∂Ω )^3, we have that w=γ ^-1Φ∈ℋ_ div^1(ℝ^3)^3. Then there exists c'≡ c'(∂Ω )∈ (0,∞ ) such that

    |⟨[[φ]],Φ⟩ _∂Ω|   =|⟨φ ,Φ⟩ _∂Ω|=|⟨ [ t_μ( u_φ,π_φ)],γ w⟩ _∂Ω|=|a_μ( u_φ, w)|
       ≤2c_μ u_φ_ℋ^1(ℝ^3)^3γ ^-1Φ_ℋ^1(ℝ^3)^3≤2c_μ c' u_φ_ℋ^1(ℝ^3)^3Φ_H^1/2(∂Ω )^3,

where the first equality in (<ref>) follows from the relation [[φ]]=φ +ℝν and the membership of Φ in H_ν^1/2(∂Ω )^3, the second equality follows from Definition <ref>, 
and the third equality is a consequence of formula (<ref>). 
Since the space H_ν^1/2(∂Ω )^3 is the dual of the space H^-1/2(∂Ω )^3/ℝν, formula (<ref>) yields that

    [[φ]]_H^-1/2(∂Ω )^3/ℝν≤ 2c_μ c' u_φ_ℋ^1(ℝ^3)^3.

Then by (<ref>) and (<ref>) we obtain inequality (<ref>), and
the Lax-Milgram lemma yields that operator (<ref>) is an isomorphism.



The fundamental solution of the constant-coefficient Stokes system in ℝ^3 is well known and leads to the construction of Newtonian and boundary layer potentials via the integral approach (see, e.g., <cit.>).
In view of Theorems <ref> and <ref>, the Newtonian and single layer potentials provided by the variational approach (in the case μ=1) coincide with classical ones expressed in terms of the fundamental solution, since they satisfy the same boundary value problems (<ref>) and (<ref>), respectively (see also
<cit.> for μ=1). The assumption μ =1 is a particular case of a more general case of L^∞ coefficients analyzed in this paper.
We also note that an alternative approach, reducing various boundary value problems for variable-coefficient elliptic partial differential equations to boundary-domain integral equations, by employing the explicit parametrix-based integral potentials, was explored in, e.g., <cit.>.





§ EXTERIOR DIRICHLET PROBLEM FOR THE STOKES SYSTEM WITH L^∞ COEFFICIENTS


In this section we analyze the exterior Dirichlet problem for the Stokes system with L^∞ coefficients

    {[ div(2μ𝔼( u))-∇π = f               Ω _-,;            div  u=0               Ω _-,;             γ_- u=ϕ                ∂Ω , ].

with given data ( f,ϕ)∈ℋ^-1(Ω _-)^3× H^1/2(∂Ω )^3.



 §.§ Variational approach

First, we use a variational approach and show that problem (<ref>) has a unique solution ( u,π )∈ℋ^1(Ω _-)^3× L^2(Ω _-) (see also <cit.> and <cit.> for the constant-coefficient Stokes system).



Assume that μ∈ L^∞(Ω _-) satisfies conditions (<ref>).
Then for all given data ( f,ϕ)∈ℋ^-1(Ω _-)^3× H^1/2(∂Ω )^3 the exterior Dirichlet problem for the L^∞ coefficient Stokes system (<ref>)
is well posed. Hence problem (<ref>) has a unique solution ( u,π )∈ℋ^1(Ω _-)^3× L^2(Ω _-) and there exists 
a constant C≡ C(∂Ω ;μ )>0 such that

    u_ℋ^1(Ω _-)^3+π_L^2(Ω _-)≤
    C( f_ℋ^-1(Ω _-)^3+ϕ_H^1/2(∂Ω )^3).



First, we note that the density of the space 𝒟(Ω _-)^3
in ℋ^1(Ω _-)^3
implies that the exterior Dirichlet problem (<ref>) has the following equivalent variational formulation: Find ( u,π )∈ℋ^1(Ω _-)^3× L^2(Ω _-) such that

    {[ 2⟨μ𝔼( u),𝔼(ṽ)⟩ _Ω _--⟨π ,div ṽ⟩ _Ω _-
     =-⟨ f,ṽ⟩ _Ω _-, ∀ ṽ∈ℋ^1(Ω _-)^3,;                                     ⟨div  u,q⟩ _Ω _-=0, ∀  q∈L^2(Ω _-),;                                                           γ_-( u)=ϕ∂Ω . ].

Next, we consider u_0∈ℋ^1(Ω _-)^3 such that

    {[ div  u_0=0      Ω _-,; γ _- u_0=ϕ       ∂Ω . ].

Particularly, we can choose u_0 as  the solution of the Dirichlet problem for
a constant-coefficient Brinkman system

    {[ ( -α𝕀) u_0-∇π _0=0,div  u_0=0                         Ω _-,;                    γ _- u_0=ϕ                          ∂Ω , ].

where α >0 is an arbitrary constant. The solution is given by the corresponding double layer potential

    u_0= W_α ;∂Ω(1/2𝕀+ K_α ;∂Ω)^-1ϕ,

where K_α ;∂Ω:H^1/2(∂Ω )^3→ H^1/2(∂Ω )^3 is the corresponding Brinkman double-layer boundary potential operator. Note that

    ( W_αh)_j( x):=∫_∂ΩS^α_ijℓ( x, y)ν _ℓ( y) h_i( y)dσ _ y.

The explicit form of the kernel S^α_ijℓ( x, y)
can be found in <cit.> and <cit.>.

In addition, the operator 1/2𝕀+ K_α ;∂Ω:H^1/2(∂Ω )^3→ H^1/2(∂Ω )^3 is an isomorphism, and u_0 belongs to the space H^1(Ω _-)^3 and satisfies (<ref>), and hence (<ref>).
Moreover, the embedding H^1(Ω _-)^3⊂ℋ^1(Ω _-)^3 shows that u_0 belongs also to the space ℋ^1(Ω _-)^3 (see also <cit.>).

Then with the new variable u:= u- u_0∈ℋ^1(Ω _-)^3, the variational problem (<ref>) reduces to the following mixed variational formulation (c.f. Problem (Q) in p. 324 of <cit.> for the
constant-coefficient Stokes system): Find ( u,π )∈ℋ^1(Ω _-)^3× L^2(Ω _-) such that

    {[ a_μ ;Ω _-( u, v)+b_Ω _-( v,π )=
    𝔉_μ ; u_0( v), ∀  v∈ℋ^1(Ω _-)^3,;                                b__Ω _-( u,q)=0, ∀  q∈ L^2(Ω _-), ].

where a_μ ;Ω _-:ℋ^1(Ω _-)^3×ℋ^1(Ω _-)^3→ℝ and b__Ω _-:ℋ^1(Ω _-)^3× L^2(Ω _-)→ℝ are the bilinear forms given by

    a_μ ;Ω _-( w, v):=2⟨μ𝔼( w),𝔼( v)⟩ _Ω _-, ∀  v, w∈ℋ^1(Ω _-)^3,
       b__Ω _-( v,q):=-⟨div  v,q⟩ _Ω _-, ∀  v∈ℋ^1(Ω _-)^3 , q∈ L^2(Ω _-),

and 𝔉_μ ; u_0:ℋ^1(Ω _-)^3→ℝ is the linear form given by

    𝔉_μ ; u_0( v):=-(⟨ f, E v⟩ _Ω _-+2⟨μ𝔼( u_0),𝔼( v)⟩ _Ω _-), ∀  v∈ℋ^1(Ω _-)^3.

Here we took into account that the spaces ℋ^1(Ω _-)^3 and ℋ^1(Ω _-)^3 can be identified through the isomorphism
E_-:ℋ^1(Ω _-)^3→ℋ^1(Ω _-)^3.
Note that

    ℋ_div^1(Ω_-)^3:
       ={ v∈ℋ^1(Ω_-)^3: div  v=0 Ω _-}
       ={ v∈ℋ^1(Ω_-)^3: b__Ω _-( v,q)=0, ∀  q∈ L^2(Ω _-)}.

Now, formula (<ref>), inequality (<ref>) and the Hölder inequality yield that

    |a_μ ;Ω _-( v_1, v_2)|   ≤ 2c_μ𝔼( v_1)_L^2(Ω _-)^3× 3𝔼( v_2)_L^2(Ω _-)^3× 3
       ≤ 2c_μ v_1_ℋ^1(Ω _-)^3 v_ℋ^1(Ω _-)^3, ∀  v_1,  v_2∈ℋ^1(Ω _-)^3.

Moreover, the formula

    2𝔼( v)_L^2(Ω _-)^3× 3^2= grad  v_L^2(Ω_-)^3× 3^2+ div  v_L^2(Ω_-)^2, ∀  v∈𝒟(Ω _-)^3

(cf., e.g., the proof of Corollary 2.2 in <cit.>), and the density of the space 𝒟(Ω _-)^3 in ℋ^1(Ω_-)^3 show that the same formula holds also for any function in ℋ^1(Ω_-)^3. Therefore, we obtain the following Korn type inequality

    grad  v_L^2(Ω_-)^3× 3≤ 2^1/2𝔼 ( v)_L^2(Ω_-)^3× 3 ,  ∀  v∈ℋ^1(Ω_-)^3.


Then by using inequality (<ref>), the equivalence of seminorm (<ref>) to the norm (<ref>) in the space ℋ^1(Ω_-)^3, and assumption (<ref>) we deduce that there exists a constant C=C(Ω _-)>0 such that

    u_ℋ^1(Ω _-)^3^2   ≤ C grad  u_L^2(Ω_-)^3× 3^2≤ 2C𝔼( u)_L^2(Ω _-)^3× 3^2
       ≤ 2Cc_μμ𝔼( u)_L^2(Ω _-)^3× 3^2
    =2Cc_μa_μ ;Ω _-( u, u), ∀  u∈ℋ^1(Ω _-)^3,

and accordingly that 

    a_μ ;Ω _-( u, u)≥1/2Cc_μ u_ℋ^1(Ω _-)^3^2, ∀  u∈ℋ^1(Ω _-)^3.

In view of inequalities (<ref>) and (<ref>) it follows that the bilinear form a_μ ;Ω _-(· ,· ):ℋ^1(Ω _-)^3×ℋ^1(Ω _-)^3→ℝ is bounded and coercive. Moreover, arguments similar to those for inequality (<ref>) imply that the bilinear form b__Ω _-(· ,·):ℋ^1(Ω _-)^3× L^2(Ω _-)→ℝ and the linear form 𝔉_μ ; u_0:ℋ^1(ℝ^3)^3→ℝ given by (<ref>) and (<ref>), are also bounded.
Since the operator

    div:ℋ^1(Ω_-)^3→ L^2(Ω _-)

is surjective (cf., e.g., <cit.>), then by Lemma <ref>, the bounded bilinear form b__Ω _-(· ,·):ℋ^1(Ω _-)^3× L^2(Ω _-)→ℝ satisfies the inf-sup condition 

    inf _q∈ L^2(Ω _-)∖{0}sup _ v∈ℋ^1(Ω _-)^3∖{ 0}b__Ω _-( v,q)/ v_ℋ^1(Ω _-)^3q_L^2(Ω _-)≥β _D

with some constant β _D>0 (cf. <cit.>).
Then Theorem <ref> (with X=ℋ^1(Ω _-)^3 and M=L^2(Ω _-)) implies that the variational problem (<ref>) has a unique solution ( u,π )∈ℋ^1(Ω _-)^3× L^2(Ω _-). Moreover, the pair ( u,π )=( u+ u_0,π )∈ℋ^1(Ω _-)^3× L^2(Ω _-), where u_0∈ℋ^1(Ω _-)^3 satisfies relations (<ref>), is the unique solution of the mixed variational formulation (<ref>) and depends continuously on the given data ( f,ϕ)∈ℋ^-1(Ω _-)^3× H^1/2(∂Ω )^3. The equivalence between the variational problem (<ref>) and the exterior Dirichlet problem (<ref>) shows that problem (<ref>) is also well-posed, as asserted.




 §.§ Potential approach

Theorem <ref> asserts the well-posedness of the exterior Dirichlet problem for the Stokes system with L^∞ coefficients. However, if the given data ( f,ϕ) belong to the space ℋ^-1(Ω _-)^3× H_ν^1/2(∂Ω )^3, then the solution can be
expressed in terms of the Newtonian and single layer potential and of the inverse of the single layer operator as follows (cf. <cit.> for μ>0 constant, <cit.> and <cit.> for the Laplace operator).


If f∈ℋ^-1(Ω _-)^3 and ϕ∈ H_ν^1/2(∂Ω )^3 then the exterior Dirichlet problem (<ref>)
has a unique solution ( u,π )∈ℋ^1(Ω _-)^3× L^2(Ω _-), given by

    u=𝒩_μ ;ℝ^3(f̃)|_Ω _-+ V_μ ;∂Ω(𝒱_μ ;∂Ω^-1(ϕ-γ _-(𝒩_μ ;ℝ^3(f̃)))),
       π =𝒬_μ ;ℝ^3(f̃)|_Ω _-+𝒬_μ ;∂Ω^s(𝒱_μ ;∂Ω^-1(ϕ-γ _-(𝒩_μ ;ℝ^3(f̃)))) Ω _-,

where f̃ is an extension of f to an element of ℋ^1(ℝ^3)^3.


The result follows from Definition <ref> and Lemmas <ref>, <ref>, <ref>.






§ MIXED VARIATIONAL FORMULATIONS AND THEIR WELL-POSEDNESS PROPERTY



Here we make a brief review of well-posedness results due to Babus̆ka <cit.> and Brezzi <cit.> for mixed variational formulations related to bounded bilinear forms in reflexive Banach spaces. We follow <cit.>, <cit.>, <cit.>.

Let X and ℳ be reflexive Banach spaces, and let X^* and ℳ^* be their dual spaces. Let a(· ,·):X×X→ℝ, b(· ,·):X×ℳ→ℝ be bounded bilinear forms. 
Then we consider the following abstract mixed variational formulation.

For f∈ X^*, g∈ℳ^* given, find a pair (u,p)∈ X×ℳ such that

    {[    a(u,v)+b(v,p) =f(v),   ∀ v∈ X,;           b(u,q)  =g(q),   ∀ q∈ℳ. ].

Let A:X→ X^* be the bounded linear operator defined by

    ⟨ Av,w⟩ =a(v,w), ∀  v,w∈ X,

where ⟨· ,·⟩ :=_X^*⟨· ,·⟩ _X is the duality pairing of the dual spaces X^* and X. We also use the notation ⟨· ,·⟩ for the duality pairing _ℳ^*⟨· ,·⟩ _ℳ. Let B:X→ℳ^* and B^*:ℳ→ X^* be the bounded linear and transpose operators given by

    ⟨ Bv,q⟩ =b(v,q), ⟨ v,B^*q⟩ =⟨ Bv,q⟩ , ∀  v∈ X, ∀  q∈ℳ.

In addition, we consider the spaces

    V:=Ker  B={v∈ X: b(v,q)=0, ∀  q∈ℳ},
       V^⊥:={T∈ X^*: ⟨ T,v⟩ =0, ∀  v∈ V}.

Then the following well-posedness result holds (cf., e.g., <cit.>).


Let X and ℳ be reflexive Banach spaces, f∈ X^* and g∈ℳ^*, and a(· ,·):X× X→ℝ and b(· ,·):X×ℳ→ℝ be bounded bilinear forms.
Let V be the subspace of X defined by (<ref>).
Then the variational problem (<ref>) is well-posed if and only if
a(· ,· ) satisfies the conditions

    {[ ∃ λ >0  inf _u∈ V∖{0}sup_v∈ V∖{0}a(u,v)/u_Xv_X≥λ ,;                      {v∈ V: a(u,v)=0, ∀ u∈ V}={0}, ].

and b(· ,·) satisfies the inf-sup (Ladyzhenskaya-Babus̆ka-Brezzi) condition,

    ∃ β >0  inf _q∈ℳ∖{0}sup_v∈ X∖{0}b(v,q)/v_Xq_ℳ≥β .

Moreover, there exists a constant C depending on β, λ and the norm of a(· ,· ), such that the unique solution (u,p)∈X×ℳ of (<ref>) satisfies the inequality

    u_X+p_ℳ≤ C(f_X^*+g_ℳ^*).


In addition, we have (see <cit.>, <cit.>).


Let X,ℳ be reflexive Banach spaces. Let b(· ,·):X×ℳ→ℝ be a bounded bilinear form. Let B:X→ℳ^* and B^*:ℳ→X^* be the operators defined by (<ref>), 
and let V=Ker  B. Then the following results are equivalent:


  (i)
There exists a constant β >0 such that b(· ,·) satisfies 
condition (<ref>).

  (ii)
B:X/V→ℳ^* is an isomorphism and
Bw_ℳ^*≥βw_X/V for any w∈X/V.

  (iii)
B^*:ℳ→V^⊥ is an isomorphism and B^*q_X^*≥βq_ℳ for any q∈ℳ.





Let X be a reflexive Banach space and V be a closed subspace of X. If a bounded bilinear form a(· ,· ):V× V→ℝ is coercive on V, i.e., there exists a constant c_a>0 such that

    a(w,w)≥ c_aw_X^2, ∀  w∈ V,

then the conditions (<ref>) are satisfied as well (see, e.g., <cit.>).

The next result known as the Babus̆ka-Brezzi theorem is the version of Theorem <ref> for Hilbert spaces (see <cit.>, <cit.>).


Let X and ℳ be two real Hilbert spaces. Let a(· ,·):X× X→ℝ and b(· ,·):X×ℳ→ℝ be bounded bilinear forms. Let f∈ X^* and g∈ℳ^*. Let V be the subspace of X defined by (<ref>). Assume that a(· ,· ):V× V→ℝ is coercive and that b(· ,·):X×ℳ→ℝ satisfies
the inf-sup condition (<ref>).
Then the variational problem (<ref>) is well-posed.




§ ACKNOWLEDGEMENTS

The research has been supported by the grant EP/M013545/1: "Mathematical Analysis of Boundary-Domain Integral Equations for Nonlinear PDEs" from the EPSRC, UK. 
Part of this work was done in April/May 2018, when M. Kohr visited the Department of Mathematics of the University of Toronto. She is grateful to the members of this department for their hospitality.

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