On the Evolution of Compressible and Incompressible Viscous Fluids with a Sharp Interface

: In this paper, we consider some two phase problems of compressible and incompressible viscous ﬂuids’ ﬂow without surface tension under the assumption that the initial domain is a uniform W 2 − 1/ q q domain in R N ( N ≥ 2). We prove the local in the time unique existence theorem for our problem in the L p in time and L q in space framework with 2 < p < ∞ and N < q < ∞ under our assumption. In our proof, we ﬁrst transform an unknown time-dependent domain into the initial domain by using the Lagrangian transformation. Secondly, we solve the problem by the contraction mapping theorem with the maximal L p - L q regularity of the generalized Stokes operator for the compressible and incompressible viscous ﬂuids’ ﬂow with the free boundary condition. The key step of our proof is to prove the existence of an R -bounded solution operator to resolve the corresponding linearized problem. The Weis operator-valued Fourier multiplier theorem with R -boundedness implies the generation of a continuous analytic semigroup and the maximal L p - L q regularity theorem.


Introduction
It is an important mathematical problem to consider the unsteady motion of a bubble in an incompressible viscous fluid or that of a drop in a compressible viscous one. The problem is, in general, formulated mathematically by the Navier-Stokes equations in a timedependent domain separated by an interface, where one part of the domain is occupied by a compressible viscous fluid and another part by an incompressible viscous fluid. More precisely, we consider two fluids that fill a region Ω ⊂ R N (N ≥ 2). Let Γ ⊂ Ω be a given surface that bounds the region Ω + occupied by a compressible barotropic viscous fluid and the region Ω − occupied by an incompressible viscous one. We assume that the boundary of Ω ± consists of two parts, Γ and Γ ± , where ∂Ω ± = Γ ∪ Γ ± , Γ ± ∩ Γ = ∅, Γ + ∩ Γ − = ∅, and Ω = Ω + ∪ Γ ∪ Ω − . Let Γ t , Γ t− , Ω t+ , and Ω t− with time variable t > 0 be the time evolution of Γ, Γ − , Ω + , and Ω − , respectively. We assume that the two fluids are immiscible, so that Ω t+ ∩ Ω t− = ∅ for any t ≥ 0. Moreover, we assume that no phase transitions occur, and we do not consider the surface tension at the interface Γ t and the boundary Γ t− . Thus, in this paper, we consider that the motion of the fluids is governed by the following system of equations: in Ω t+ , ρ + (∂ t v + + v + · ∇v + ) − Div S + (v + ) + ∇p(ρ + ) = 0 in Ω t+ , on Γ t , boundary conditions: kinematic conditions: for any t > 0, and initial conditions: Here, v ± = (v 1± , . . . , v N± ) are the unknown velocity fields of the fluids, ρ 0± positive numbers describing the mass densities of Ω ± , ρ + the unknown mass density of Ω t+ , π − the unknown pressure, θ 0+ and v 0± the prescribed initial data, p(s) the prescribed pressure, which is a C ∞ function defined on an open interval (ρ 0+ /2, 2ρ 0+ ) satisfying the condition: p (s) ≥ 0 on (ρ 0+ /2, 2ρ 0+ ), n t the unit outward normal to Γ t , pointing from Ω t− to Ω t+ , n t− the unit outward normal to Γ t− , V n the evolution speed of Γ t along n t , and V n− the evolution speed of Γ t− along n t− .
Moreover, for any point and the stress tensors S ± are defined by: with viscosity coefficients µ ± and ν + , which are positive constants in this paper, where D(v) denotes the deformation tensor whose (j, k) components are D jk (v) = ∂ j v k + ∂ k v j with ∂ j = ∂/∂x j and I is the N × N identity matrix. Finally, for an N × N matrix function K = (K ij ), Div K is an N-vector whose ith components are ∑ N j=1 ∂ j K ij , and also, for any vector of functions v = (v 1 , . . . , v N ), we set div v = ∑ N j=1 ∂ j v j and v · ∇v = (∑ N j=1 v j ∂ j v 1 , . . . , ∑ N j=1 v j ∂ j v N ). For any functions f ± defined on Ω ± , f denotes a function defined by f = f ± in Ω ± .
Denisova [2,3] proved the local well-posedness theorem and the global well-posedness theorem for Equations (1)- (3) and (5) in the L 2 framework. The purpose of this paper is to prove the local well-posedness for Equations (1)- (3) and (5) in the L p in time and L q in space framework with 2 < p < ∞ and N < q < ∞ under the physically reasonable assumption on the viscosity coefficients, that is µ ± > 0 and ν + > 0. The regularity of solutions in our result is optimal in the sense of the maximal regularity, while the L 2 framework used by Denisova [2,3] loses regularity from the point of view of Sobolev's imbedding theorem.
Moreover, we consider the problem with full generality about the domain. Namely, we consider the problem in a uniform W 2−1/q q domain, the conditions of which are satisfied by bounded domains, exterior domains, half-spaces, perturbed half-spaces, and layer domains (cf. Shibata [4]). where ∂ i = ∂/∂x i . For any domain D and 1 ≤ q ≤ ∞, L q (D), W m q (D), and B s q,p (D) denote the standard Lebesgue space, Sobolev space, and Besov space, while · L q (D) , · W m q (D) , and · B s q,p (D) denote their norms. We set W 0 q (D) = L q (D) and W s q (D) = B s q,q (D). In addition, (a, b) D denotes the inner product on D defined by (a, b) D = D a(x)b(x) dx. Let X be any Banach space with norm · X . We set , X) and W m p ((a, b), X) denote the usual Lebesgue space and Sobolev space of X-valued functions defined on an interval (a, b), while · L p ((a,b),X) and · W m p ((a,b),X) denote their norms, respectively. For any N-vector w = (w 1 , . . . , w N ) and z = (z 1 , . . . , z N ), we define < w, z >, T z [w], and N z [w] by: respectively. Here, T z [w] denotes the tangential part of w with respect to z. For 1 < q < ∞, q denotes the dual exponent defined by q = q/(q − 1). We use the letter C to denote generic constants, and C a,b,· denotes that the constant C a,b,· essentially depends on the quantities a, b, · · · . Constants C, C a,b,··· may change from line to line. In this paper, let J q (Ω − ) be a solenoidal space defined by setting: We write div We now introduce a few definitions. Definition 1. Let 1 < r < ∞, and let D be a domain in R N with boundary ∂D. We say that D is a uniform W 2−1/r r domain, if there exist positive constants α, β, and K such that for any x 0 = (x 01 , . . . , x 0N ) ∈ ∂D, there exist a coordinate number j and a W 2−1/r Here, (x 1 , . . . ,x j , . . . , Second, we introduce the assumption of the solvability of the weak Dirichlet problem, which is needed to treat the divergence condition for the incompressible part. Definition 2. Let 1 < q < ∞. We say that the weak Dirichlet problem is uniquely solvable on W 1 q,0 (Ω − ) with exponents q, if for any f ∈ L q (Ω − ) N , there exists a unique solution θ ∈Ŵ 1 q,0 (Ω − ) of the variational problem: (2) When q = 2, the weak Dirichlet problem is uniquely solvable on Ω − without any restriction, but for q ∈ (1, ∞) \ {2}, we do not know the unique solvability in general. For example, we know the unique solvability of the weak Dirichlet problem in bounded domains, exterior domains, half-space, layer, and tube domains. (cf. Galdi [5], as well as Shibata [4,6]).

Remark 2.
Let K be a linear operator defined by K(f) = θ. Then, combining the unique solvability with Banach's closed range theorem implies the estimate: Moreover, for any f ∈ L q (Ω − ) N and g ∈ W 1 which is the space for the pressure term p − in the incompressible part.

Remark 3.
Here, f ∈ W m p ((0, T), W n q (Ω t± )) denotes that for almost all t ∈ (0, T), ∂ k t f (·, t) ∈ W n q (Ω t± ) and: Theorem 1 is proven by using a standard fixed point argument based on the maximal L p -L q regularity for solutions to the linear problem: Here, γ i = γ i (x) (i = 0, 1, 2) are uniformly continuous functions defined on Ω + such that: for k = 1, 2 and = 0, 1, 2 with some positive constant ρ 2+ and N < r < ∞. We may consider the case where γ 1+ = 0, which corresponds to the Lamé system. Symbols 2. To state our main result for linear Equation (22), we introduce more symbols and functional spaces used throughout this paper. Set: and W 0 p,loc (R, X) = L p,loc (R, X). Moreover, we set: with ∂ 0 t f (t) = f (t) and W 0 p,γ (R, X) = L p,γ (R, X) and W 0 p,γ,0 (R, X) = L p,γ,0 (R, X). Let L and L −1 denote the Laplace transform and the Laplace inverse transform defined by: with λ = γ + iτ ∈ C, respectively. Given s ∈ R and X-valued function f (t), we set: We introduce a Bessel potential space of X-valued functions of order s > 0 as follows: We have the following theorem.
Theorem 3. Let 1 < p, q < ∞, N < r < ∞, 2/p + N/q = 1, and 2/p + N/q = 2. Assume that r ≥ max(q, q ), that Ω ± are uniformly W 2−1/q q domains, and that the weak Dirichlet problem is uniquely solvable on Ω − with exponents q and q . Then, there exists a positive number γ 0 such that the following three assertions are valid.
To prove Theorem 3, Problem (22) is divided into two parts: One is the case where the right side in (22) is considered for all t ∈ R, while the initial conditions are not taken into account. The other case is non-homogeneous initial conditions and a zero right side in (22). In the first case, solutions are represented by the Laplace inverse transform of solution formulas represented by using R-bounded solution operators for the generalized resolvent problem corresponding to (22). Combining the R-boundedness and Weis's operator-valued Fourier multiplier theorem yields the maximal L p -L q estimate of solutions to Equation (22) with zero initial conditions. Moreover, the R-bounded solution operators yield the generation of the continuous analytic semigroup associated with Equation (22), which, combined with some real interpolation technique, yields the L p -L q maximal regularity for the initial problem for Equation (22). Combining these two results gives Theorem 3. To prove the generation of the continuous analytic semigroup, we have to eliminate the pressure term p − in Equation (22), and so, using the assumption of the unique existence of the weak Dirichlet problem, we define the reduced generalized resolvent problem (RGRP) (cf. (41) in Section 2 below) according to Grubb and Solonnikov [8], which is the equivalent system to the generalized resolvent problem (GRP) corresponding to (22).
The paper is organized as follows. In Section 2, first we introduce (GRP) and state main results for (GRP). Secondly, we drive (RGRP) and discuss some equivalence between (GRP) and (RGRP). Thirdly, we state the main results for (RGRP), which implies the results for (GRP) according to the equivalence between (GRP) and (RGRP). In Section 3, we discuss the model problems in R N . In Section 4, we discuss the bent half space problems for (RGRP). In Section 5, we prove the main result for (RGRP) and also Theorem 3. In Section 6, we prove Theorem 1 by the Banach fixed point argument based on Theorem 3.

R-Bounded Solution Operators
To prove the generation of the continuous analytic semigroup and the maximal L p -L q regularity for the linear problem (22), we show the existence of R-bounded solution operators to the following generalized resolvent problem (GRP) corresponding to: (22): When λ = 0, setting θ + = λ −1 ( f + − γ 2+ div u + ), we transfer the second equation and the fifth equation in (28) to: Γ+0 , being renamed g + and h, respectively, and setting γ 1+ γ 2+ = γ 3+ , from now on, we consider the following problem: Here, δ and λ satisfy one of the following three conditions: . We may include the case where γ 1 = 0, which corresponds to the Lamé system. The former case (C1) is used to prove the existence of R-bounded solution operators to (28), and the latter cases (C2) and (C3) enable the application of a homotopic argument for proving the exponential stability of the analytic semigroup in bounded domains. For the sake of simplicity, we introduce the set Γ ,λ 0 defined by: Note that |δ| ≤ max(|δ 0 |, λ −1 0 ).
Before stating our main results for the linear problem, we introduce a few symbols and the definition of the R-bounded operator family and the operator-valued Fourier multiplier theorem due to Weis [9]. Symbols 3. For any two Banach spaces X and Y, L(X, Y) denotes the set of all bounded linear operators from X to Y, and we write L(X) = L(X, X) for short. Hol (U, X) denotes the set of all X-valued holomorphic functions defined on a complex domain U. Let D(R, X) and S(R, X) be the set of all X-valued C ∞ -functions having compact support and the Schwartz space of rapidly decreasing X-valued functions, respectively, while S (R, X) = L(S(R, C), X). Given M ∈ L 1,loc (R \ {0}, X), we define the operator T M : Here, F x and F −1 x denote the Fourier transform and its inversion defined by:

respectively.
Definition 3. Let X and Y be Banach spaces. A family of operators T ⊂ L(X, Y) is called R-bounded on L(X, Y), if there exist constants C > 0 and p ∈ [1, ∞) such that for any n ∈ N, {T j } n j=1 ⊂ T , {x j } n j=1 ⊂ X, and sequences {r j (u)} n j=1 of independent, symmetric, {−1, 1}valued random variables on [0, 1], there holds the inequality: The smallest such C is called the R-bound of T , which is denoted by R L(X,Y) (T ).
The following theorem was obtained by Weis [9]. Theorem 4. Let X and Y be two UMD spaces and 1 < p < ∞. Let M be a function in C 1 (R \ {0}, L(X, Y)) such that: with some constant κ. Then, the operator T M defined in (32) may uniquely be extended to a bounded linear operator from L p (R, X) to L p (R, Y). Moreover, denoting this extension by T M , we have: for some positive constant C depending on p, X, and Y.

Remark 4.
For the definition of the UMD space, we refer to the monograph by Amann [10]. For 1 < q < ∞ and m ∈ N, Lebesgue spaces L q (Ω) and Sobolev spaces W m q (Ω) are UMD spaces.
For the calculation of the R-norm, we use the following lemmas.

Lemma 1.
(1) Let X and Y be Banach spaces, and let T and S be R-bounded families in L(X, Y). Then, T + S = {T + S | T ∈ T , S ∈ S} is also an R-bounded family in L(X, Y) and: R L(X,Y) (T + S) ≤ R L(X,Y) (T ) + R L(X,Y) (S).
(2) Let X, Y, and Z be Banach spaces, and let T and S be R-bounded families in L(X, Y) and L(Y, Z), respectively. Then, ST = {ST | T ∈ T , S ∈ S} is also an R-bounded family in L(X, Z) and: Lemma 2. Let 1 < p, q < ∞, and let D be a domain in R N .
(1) Let m(λ) be a bounded function defined on a subset Λ ⊂ C, and let M m (λ) be a multiplication operator with m(λ) defined by M m (λ) f = m(λ) f for any f ∈ L q (D). Then, (2) Let n(τ) be a C 1 function defined on R \ {0} that satisfies the conditions: |n(τ)| ≤ γ and |τn (τ)| ≤ γ with some constant γ > 0 for any τ ∈ R \ {0}. Let T n be an operator-valued Fourier multiplier defined by ] for any f with F [φ] ∈ D(R, X). Then, T n is extended to a bounded linear operator from L p (R, L q (D)) into itself. Moreover, denoting this extension also by T n , we have: T n L(L p (R,L q (D))) ≤ C p,q,D γ.

Existence of R-Bounded Solution Operators for Problems
Theorem 5. Let 1 < q < ∞, 0 < < π/2 and N < r < ∞. Assume that r ≥ max(q, q ), that Ω ± are uniform W 2−1/r r domains, and that the weak Dirichlet problem is uniquely solvable on Ω − with exponents q and q . Let X 0 q (Ω) and X 0 q (Ω) be the spaces defined by: Then, there exist a constant λ 0 > 0 and operator families A 0 ± (λ) and B 0 − (λ) with:
, that Ω ± are uniform W 2−1/r r domains, and that the weak Dirichlet problem is uniquely solvable on Ω − with exponents q and q . Let X 1 q (Ω) and X 1 q (Ω) be the sets defined by: Then, there exist a constant λ 0 > 0 and operator families A 1 ± (λ) and B 1 ± (λ) with: q (Ω), and: Remark 7. The variable F 10 corresponds to f + , and we set:

Reduced Generalized Resolvent Problem
Since the pressure term p − has no time evolution in (22), we eliminate p − from (29) and derive a reduced problem. Before this discussion, we consider the resolvent problem for the Laplace operator with non-homogeneous Dirichlet condition of the form: subject to w| Γ = g 1 and w| Γ − = g 2 . Here and in the following, we write (·, ·) = (·, ·) Ω − for short. Note that: We can show the following theorem by using the method in Shibata [13].
(ii) Since R-boundedness implies the usual boundedness, by (34) and (35) we have: We start our main discussion in this subsection. Given Since we can choose some uniform covering of Ω ± (cf. Proposition 4 in Section 5 below), Ext − [w + ] is defined by the even extension of w + in each local chart. For and K is the operator defined in Remark 2. Note that K(u + , u − ) ∈ W 1 q (Ω − ) +Ŵ 1 q,0 (Ω − ) and satisfies the variational equation: subject to: and the estimate: The reduced generalized resolvent problem (RGRP) is the following: (14), we can write the interface condition and free boundary condition in (41) as follows: We (Ω) and u ± and p − satisfy Equation (29). In this subsection, we show the equivalence of the solutions between (29) and (41).
From the second equation of (29), it follows that for any Thus, the uniqueness yields that p − = K(u + , u − ), and so, (u + , u − ) is a solution to (41) with (g + , g − , h, h − ).

Assertion 2. If (41) is solvable, then so is (29).
In fact, given Setting for any ϕ ∈Ŵ 1 q ,0 (Ω − ), which yields that: (44), using the divergence theorem of Gauss, and noticing that div f − = f − give that: Moreover, from the third equation in (42) and (39), it follows that: Thus, the uniqueness yields that div u − = f − in Ω − . Inserting this fact into (44) and using the fact

Existence of R-Bounded Solution Operators for Problem (41)
The following theorem is concerned with the existence of R-bounded solution operators to Problem (41). Theorem 8. Let 1 < q < ∞, 0 < < π/2 and N < r < ∞. Assume that r ≥ max(q, q ), that Ω ± are uniform W 2−1/r r domains, and the weak Dirichlet problem is uniquely solvable in Ω − with exponents q and q . Let X q (Ω) and X q (Ω) be the sets defined by: Then, there exist a constant λ 0 > 0 and operator families S ± (λ) ∈ Hol(Γ ,λ 0 , L(X q (Ω), Remark 9. For any subdomain G ⊂ Ω, we set: Obviously, according to Assertion 2 in Section 2.2, by Theorem 8, Lemma 1, and Lemma 2 we have Theorem 6. Thus, we shall prove Theorem 8 only.

The Uniqueness of Solutions to Problem (41)
Assuming the existence of solutions to Problem (41) with exponent q , we prove the uniqueness of solutions to (41). Namely, we prove the following lemma. Lemma 3. Let 1 < q < ∞ and N < r < ∞. Assume that r ≥ max(q, q ), that Ω ± are uniform W 2−1/r r domains, and that the weak Dirichlet problem is uniquely solvable on Ω − with exponents q and q . If there exists a λ 0 > 0 such that Problem (41) is solvable with exponent q for any λ ∈ Γ ,λ 0 , then the uniqueness for (41) with exponent q is valid for any λ ∈ Γ ,λ 0 .

Remark 10. (i)
The reason why we assume that r ≥ max(q, q ) is that we use the existence of solutions to the dual problem to prove the uniqueness.
Secondly, for any with: , by the divergence theorem of Gauss, we have: (47) and the fact that (u − , ∇ψ) Ω − = 0, we have: Since f ± are chosen arbitrarily, we have u ± = 0, which completes the proof of Lemma 3.

Model Problems
In this section, we consider a model problem for the incompressible-compressible viscous fluid in R N . In what follows, we set: and n 0 = (0, . . . , 0, 1). Before stating the main results of this section, we notice that the following two variational problems are uniquely solvable: . We define an operator P acting on f by setting v = P f.
Moreover, for any This assertion is also known (cf. [13]). In particular, we have . In this section, assuming that γ 0+ and γ 3+ are positive constants such that: we consider the following interface problem in R N : Here, g ± ∈ L q (R N ± ) and h ∈ W 1 q (R N ) are prescribed functions, and for notational simplicity, we set: Moreover, v = K 0 I (u + , u − ) is a unique solution to the variational problem: We prove the following theorem.
Remark 11. We set: According to Assertion 1 in Section 2.2, we consider the following system of equations: Then, Theorem 9 follows from the following theorem, because (49) is uniquely solvable.
Summing up, we proved the following proposition.
Concerning the compressible part, we consider the equations: We know the following theorem, which was proven by Götz and Shibata [14].
First, we consider the following problem: Moreover, v = K I (u + , u − ) is a solution to the weak Dirichlet problem: We have the following theorem.
Proof. The idea of the proof here follows Shibata [16] and von Below, Enomoto, and Shibata [17]. Using the change of variable: x = Φ −1 (y) with y ∈ D and x ∈ R N and the change of unknown functions: ) and γ 3+ =γ 3+ + (γ 3+ −γ 3+ ), and setting p = K I (u + , u − ), we see that Problem (71) is transferred to the following equivalent problem: subject to the interface condition: v + | x N =0+ = v − | x N =0− and: p satisfies the following variational equation: subject to: Here, we write (·, ·) = (·, ·) R N − for short, and G ± = A −1 g ± • Φ, H = A −1 h • Φ, and F i ± (v ± ) are the vector of functions of the forms: for i = 2, 3 and j = 2, 3, 4. In view of (67)-(70), we can assume that R i ± , S ± , and P i possesses the following estimate: for i = 1, 2, 3, j = 1, 2, 3, 4, and k = 1, 2. Following Shibata ([16] Section 4), we treat the R N − side as follows: (52), which satisfies the estimate: Setting p = K 0 I (v + , v − ) + p 1 , we see that p 1 satisfies the variational equation: subject to: Since P 2 L ∞ (R N ) is small enough, we can show the following lemma by the small perturbation from the weak Dirichlet problem in R N − .
To solve (81) for any right members (81), where E ± (λ) are operators given in Theorem 9, and then, we have: subject to the interface conditions v + | x N =0+ = v − | x N =0− and: Here, we set: and Ext ± [ f ∓ ] denote the even extension of functions f ∓ defined on R N ∓ to R N . Note that: with ∇ 0 f ∓ = f ∓ . Let us define the corresponding Rbounded operators R 1 ± (λ) and R 2 (λ) by: To obtain: we use the following lemma (cf. Shibata ([4] Lemma 2.4)).
Next, for the compressible part, we consider the following two problems.

Some Preparations for the Proof of Theorem 8
We first give several properties of the uniform W 2−1/r r domain in the following proposition. Proposition 4. Let N < r < ∞, and let Ω ± be uniform W 2−1/r r domains in R N . Let M 1 be the number given in (67). Then, there exist constants M 2 > 0, 0 < d i < 1 (i = 1, . . . , 5), at most countably many N-vectors of functions Φ i j ∈ W 2 r (R N ) N (i = 1, . . . , 5, j ∈ N), and points x 1 j ∈ Γ, x 2 j ∈ Γ + , x 3 j ∈ Γ − , x 4 j ∈ Ω + and x 5 j ∈ Ω − such that the following assertions hold: Here and in the following, we set Γ 1 = Γ, There exists a natural number L ≥ 2 such that any L + 1 distinct sets of {B d i (x i j ) | i = 1, . . . , 5, j ∈ N} have an empty intersection.

Proof. For a detailed proof, we refer to Enomoto and Shibata ([18] Appendix).
In the following, choosing M 2 larger if necessary, we may assume that , which is a weaker assumption than the last condition in (23). Since functions in W 1 r are Hölder continuous of order α with 0 < α < 1 − N/r, as follows from Sobolev's imbedding theorem, we have |γ with some constant C independent of j, and so choosing d i > 0 smaller and more points x i j suitably, we may assume that |γ k+ (x) − γ k+ (x i j )| ≤ M 1 for x ∈ B d i (x i j ) (k = 0, 3, i = 1, 3, 5, j ∈ N). Here and in the following, constants denoted by C are independent of j ∈ N. In addition, in view of (68), we may assume that each unit outward normal n i j to Φ i j (R N 0 ) (i = 1, 3, 4, j ∈ N) is defined on R N and satisfies the conditions: n i j L ∞ (R N ) = 1 and ∇n i j L r (R N ) ≤ CM 2 . Note that n = n 1 j on B d 1 (x 1 j ) ∩ Γ and n − = n 4 j on B d 4 (x 4 j ) ∩ Γ − . Summing up, from now on, we may assume that: and that both n and n − are defined on R N with n L ∞ (Ω) = 1 and n − L ∞ (Ω − ) = 1, respectively. Next, we prepare two lemmas used to construct a parametrix.
Lemma 6. Let X be a Banach space and X * its dual space, while · X , · X * and < ·, · > are the norm of X, the norm of X * , and the duality of X and X * , respectively. Let n ∈ N, and for i = 1, . . . , n, let a i ∈ C, let { f j } ∞ j=1 and {h j } ∞ j=1 be sequences of positive numbers. Assume that there exist maps N j : X → [0, ∞) such that: for any ϕ ∈ L q (D) with some constant M 3 independent of j ∈ N. If: j exists in the strong topology of X * and:

Lemma 7.
Let D be a domain in R N , and assume that there exists at most countably many covering {B j } ∞ j=1 such that D ⊂ ∞ j=1 B j and {B j } ∞ j=1 has a finite intersection property of order L, that is any L + 1 distinct sets of {B j } ∞ j=1 have an empty intersection. Let 1 < q < ∞. Then, the following assertions hold.
(i) There exists a constant C q,L such that: . . , m) be sequences of positive numbers. Assume that: for any ϕ ∈ L q (D) and = 0, 1, . . . , m with some constant M 3 independent of j ∈ N. Then, f = ∑ ∞ j=1 f j exists in the strong topology of W m q (D) and:

Remark 15.
To prove Lemma 6, we consider the difference of finite sum ∑ N j=1 f (i) j and use the Hölder inequality for the sequence. The assertion (i) of Lemma 7 follows immediately from the property of the Lebesgue measure and suitable decomposition of covering sets {B j } ∞ j=1 , and the assertion (ii) of Lemma 7 follows from Lemma 6 and Lemma 7 (i).

Local Solutions
In the following, we write for short. n 1 j denote the unit outward normals to Γ 1 j pointing from H 1 −j to H 1 +j , and n 3 j denote the unit outward normals to Γ 3 j for j ∈ N. In view of (90), we define the functions γ i jk by: for k = 0, 3, i = 1, 2, 4, and j ∈ N. Noting that 0 ≤ζ i j ≤ 1 and ∇ζ i j L ∞ (R N ) ≤ c 0 , by (90) and (23): for k = 0, 3, i = 1, 2, 4, and j ∈ N. In addition, we have: becauseζ i j = 1 on supp ζ i j . For G = (g + , g − , h, h − ) ∈ X q (Ω), we consider the equations: Here is a unique solution to the variational problem: Here and in the following, X q (Ω) and X q (Ω) denote the spaces defined in Theorem 8 in Section 2.3.
for i = 1, 3 and j ∈ N, where C q is a constant independent of j ∈ N.