Resolvent Estimates and Maximal Regularity in Weighted Lebesgue Spaces of the Stokes Operator in Unbounded Cylinders

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1 Resolvent Estimates and Maximal Regularity in Weighted Lebesgue Spaces of the Stokes Operator in Unbounded Cylinders Myong-Hwan Ri and Reinhard Farwig Abstract We study resolvent estimate and maximal regularity of the Stokes operator in L -spaces with exponential weights in the axial directions of unbounded cylinders of R n, n 3. For straights cylinders we obtain these results in Lebesgue spaces with exponential weights in the axial direction and Muckenhoupt weights in the cross-section. Next, for general cylinders with several exits to infinity we prove that the Stokes operator in L -spaces with exponential weight along the axial directions generates an exponentially decaying analytic semigroup and has maximal regularity. The proofs for straight cylinders use an operator-valued Fourier multiplier theorem and techniues of unconditional Schauder decompositions based on the R-boundedness of the family of solution operators for a system in the cross-section of the cylinder parametrized by the phase variable of the onedimensional partial Fourier transform. For general cylinders we use cut-off techniues based on the result for straight cylinders and the result for the case without exponential weight Mathematical Subject Classification: 35Q30, 76D05, 76D07 Keywords: Maximal regularity; Stokes operator; exponential weights; Stokes semigroup; unbounded cylinder 1 Introduction Let Ω = m Ω i (1.1) i=0 be a cylindrical domain of C 1,1 -class where Ω 0 is a bounded domain and Ω i, i = 1,..., m, are disjoint semi-infinite straight cylinders, that is, in possibly different 1 Ri Myong-Hwan: Institute of Mathematics, Academy of Sciences, DPR Korea, mhri@amss.ac.cn 2 Reinhard Farwig: Department of Mathematics, Darmstadt University of Technology, Darmstadt, Germany, farwig@mathematik.tu-darmstadt.de 1

2 coordinates, Ω i = {x i = (x i 1,..., x i n) R n : x i n > 0, (x i 1,..., x i n 1) Σ i }, where the cross sections Σ i R n 1, i = 1,..., m, are bounded domains and Ω i Ω j = for i j. Given β i 0, i = 1,..., m, introduce the space L b (Ω) = {U L (Ω) : e β ix i n U Ωi L (Ω i )}, U L b (Ω) := ( U L (Ω 0 ) + m i=1 eβ ix i n U L (Ω i ) for 1 < <. Moreover, let W k, b (Ω), k N, be the space of functions whose derivatives up to k-th order belong to L b (Ω), where a norm is endowed in the standard way. Let W 1, 1, 0,b (Ω) = {u Wb (Ω) : u Ω = 0}. Let L σ(ω) and L b,σ (Ω) be the completion of the set C0,σ(Ω) = {u C0 (Ω) n : div u = 0} in the norm of L (Ω) and L b (Ω), respectively. Then we consider the Stokes operator A = A,b = P in L b,σ (Ω) with domain ) 1/ D(A) = W 2, b (Ω)n W 1, 0,b (Ω)n L σ(ω), where P is the Helmholtz projection of L (Ω) onto L σ(ω). The goal of this paper is to study resolvent estimates and maximal L p -regularity of the Stokes operator in Lebesgue spaces with exponential weights in the axial direction. The semigroup approach to instationary Navier-Stokes euations is a very convenient tool to prove existence, uniueness and stability of solutions; to this end, resolvent estimates of the Stokes operator must be obtained. Moreover, maximal regularity of the Stokes operator helps to deal with the nonlinearity of the Navier-Stokes euations. There are many papers dealing with resolvent estimates ([7], [8], [15], [16], [20]; see Introduction of [10] for more details) or maximal regularity (see e.g. [1], [14], [16]) of Stokes operators for domains with compact as well as noncompact boundaries. General unbounded domains are considered in [6] by replacing the space L by L L 2 or L + L 2. For resolvent estimates and maximal regularity in unbounded cylinders without exponential weights in the axial direction we refer the reader e.g. to [10]- [13] and [31]. For partial results in the Bloch space of uniformly suare integrable functions on a cylinder we refer to [33]. Further results on stationary Stokes and instationary Stokes and Navier-Stokes systems in unbounded cylindrical domains can be found e.g. in [2], [3], [17], [18], [21]-[30], [33]-[35]. Despite of some references showing the existence of stationary flows in L -setting (e.g. [25], [26], [28]) and instationary flows in L 2 -setting (e.g. [29], [30]) that converge at x to some limit states (Poiseuille flow or zero flow) in unbounded cylinders, resolvent estimates and maximal regularity of the Stokes operator in L -spaces with exponential weights on unbounded cylinders do not seem to have been obtained yet. We start our work with consideration of the Stokes operator in straight cylinders; we get resolvent estimate and maximal regularity of the Stokes operator even in 2

3 L β (R; Lr ω(σ)), 1 <, r <, with exponential weight e βxn, β > 0, and arbitrary Muckenhoupt weight ω A r (R n 1 ) with respect to x Σ (see Section 2 for the definition). We note that our resolvent estimate gives, in particular when λ = 0, a new result on the existence of a uniue flow with zero flux for the stationary Stokes system in L β (R, Lr ω(σ)). Next, based on the results for straight cylinders, we get resolvent estimates and maximal L p -regularity of the Stokes operator in L b (Ω), 1 < <, for general cylinders Ω using a cut-off techniue. The proofs for straight cylinders are mainly based on the theory of Fourier analysis. By the application of the partial Fourier transform along the axis of the cylinder Σ R the generalized Stokes resolvent system λu U + P = F in Σ R, (R λ ) div U = G in Σ R, u = 0 on Σ R, is reduced to the parametrized Stokes system in the cross-section Σ: (λ + η 2 )Û + ˆP = ˆF in Σ, (λ + η 2 )Ûn + iη ˆP = ˆF n in Σ, (R λ,η ) div Û + iηûn = Ĝ in Σ, Û = 0, Û n = 0 on Σ, which involves the Fourier phase variable η C as parameter. Now, for fixed β 0 let (û, ˆp, ˆf, ĝ)(ξ) := (Û, ˆP, ˆF, Ĝ)(ξ + iβ). Then (R λ,η ) is reduced to the system (λ + (ξ + iβ) 2 )û (ξ) + ˆp(ξ) = ˆf (ξ) in Σ, (λ + (ξ + iβ) 2 )û n (ξ) + i(ξ + iβ)ˆp(ξ) = ˆf n (ξ) in Σ, (R λ,ξ,β ) div û (ξ) + i(ξ + iβ)û n (ξ) = ĝ(ξ) in Σ, û (ξ) = 0, û n (ξ) = 0 on Σ. We will get estimates of solutions to (R λ,ξ,β ) independent of ξ R := R \ {0} and λ in L r -spaces with Muckenhoupt weights, which yield R-boundedness of the family of solution operators a(ξ) for (R λ,ξ,β ) with g = 0 due to an extrapolation property of operators defined on L r -spaces with Muckenhoupt weights, see Theorem 4.8. Then, an operator-valued Fourier multiplier theorem ([36]) implies the estimate of e βxn U = F 1 (a(ξ)ff) for the solution U to (R λ ) with G = 0 in the straight cylinder Σ R. In order to prove maximal regularity of the Stokes operator in straight cylinders we use that maximal regularity of an operator A in a UMD space X is implied by the R-boundedness of the operator family {λ(λ + A) 1 : λ i R} (1.2) 3

4 in L(X), see [36]. We show the R-boundedness of (1.2) for the Stokes operator A := A,r;β,ω in L β (R : Lr ω(σ)) by virtue of Schauder decomposition techniues; to be more precise, we use the Schauder decomposition { j } j Z where j = F 1 χ [2 j,2 j+1 )F to get R-boundedness of the family (1.2). The proof for general cylinders, Theorem 2.4 and Theorem 2.5, uses a cut-off techniue based on the result for resolvent estimates and maximal regularity without exponential weights in [13] and the result (Theorem 2.3) for straight cylinders. This paper is organized as follows. In Section 2 the main results of this paper (Theorem 2.1, Corollary 2.2, Theorem 2.3 Theorem 2.5) and preliminaries are given. In Section 3 we obtain the estimate for (R λ,ξ,β ) on bounded domains, see Theorem 3.8. In Section 4 proofs of the main results are given. 2 Main Results and Preliminaries Let Σ R be an infinite cylinder of R n with bounded cross section Σ R n 1 and with generic point x Σ R written in the form x = (x, x n ) Σ R, where x Σ and x n R. Similarly, differential operators in R n are split, in particular, = + 2 n and = (, n ). For (1, ) we use the standard notation L (Σ R) = L (R; L (Σ)) for classical Lebesgue spaces with norm = ;Σ R and W k, (Σ R), k N, for the usual Sobolev spaces with norm k,;σ R. We do not distinguish between spaces of scalar functions and vector-valued functions as long as no confusion arises. In particular, we use the short notation u, v X for u X + v X, even if u and v are tensors of different order. Let 1 < r <. A function 0 ω L 1 loc (Rn 1 ) is called A r -weight (Muckenhoupt weight) on R n 1 iff ( ) ( ) r A r (ω) := sup ω dx ω 1/(r 1) dx < Q Q Q Q Q where the supremum is taken over all cubes of R n 1 and Q denotes the (n 1)- dimensional Lebesgue measure of Q. We call A r (ω) the A r -constant of ω and denote the set of all A r -weights on R n 1 by A r = A r (R n 1 ). Note that ω A r iff ω := ω 1/(r 1) A r, r = r/(r 1), and A r (ω ) = A r (ω) r /r. A constant C = C(ω) is called A r -consistent if for every d > 0 sup {C(ω) : ω A r, A r (ω) < d} <. We write ω(q) for Q ω dx. Typical Muckenhoupt weights are the radial functions ω(x) = x α : it is wellknown that ω A r (R n 1 ) if and only if (n 1) < α < (r 1)(n 1); the same bounds for α hold when ω(x) = (1+ x ) α and ω(x) = x α (log(e+ x ) β for all β R. For further examples we refer to [8]. 4

5 Given ω A r, r (1, ), and an arbitrary domain Σ R n 1 let { ( ) 1/r L r ω(σ) = u L 1 loc( Σ) : u r,ω = u r,ω;σ = u r ω dx < }. For short we will write L r ω for L r ω(σ) provided that the underlying domain Σ is known from the context. It is well-known that L r ω is a separable reflexive Banach space with dense subspace C0 (Σ). In particular (L r ω) = L r k,r ω. As usual, Wω (Σ), k N, denotes the weighted Sobolev space with norm ( 1/r, u k,r,ω = D α u r,ω) r α k where α = α 1 + +α n 1 is the length of the multi-index α = (α 1,..., α n 1 ) N n 1 0 and D α = α α n 1 n 1 ; moreover, W0,ω(Σ) k,r := C0 (Σ) k,r,ω and W k,r 0,ω (Σ) := (W k,r 0,ω (Σ)), where r = r/(r 1). We introduce the weighted homogeneous Sobolev space Ŵω 1,r (Σ) = { u L 1 loc( Σ)/R : u L r ω(σ) } with norm u r,ω and its dual space Ŵ 1,r ω := (Ŵ 1,r ω ) with norm 1,r,ω = 1,r,ω ;Σ. Let, r (1, ). On an infinite cylinder Σ R, where Σ is a bounded C 1,1 - domain of R n 1, we introduce the function space L (L r ω) := L (R; L r ω(σ)) with norm ( ( ) ) /r 1/ u L (L r ω ) = u(x, x n ) r ω(x ) dx dxn. R Σ Furthermore, Wω k;,r (Σ R), k N, denotes the Banach space of all functions in Σ R whose derivatives of order up to k belong to L (L r ω) with norm u W k;,r = ω ( α k Dα u 2 L (L r ω ))1/2, where α N n 0, and let W 1;,r 0,ω (Ω) be the completion of the set C0 (Ω) in Wω 1;,r (Ω). Given β > 0, we denote by with norm e βxn L (L r ω ) and for k N W k;,r β,ω L β (Lr ω) := {u : e βxn u L (L r ω)} (Σ R) := {u : eβxn u Wω k;,r (Σ R)} with norm e βxn W k;,r ω (Σ R). Finally, L (L r ω) σ and L β (Lr ω) σ are completions in the space L (L r ω) and L β (Lr ω) of the set C 0,σ(Σ R) = {u C 0 (Σ R) n ; div u = 0}, respectively. The Fourier transform in the variable x n is denoted by F or and the inverse Fourier transform by F 1 or. For ε (0, π ) we define the complex sector 2 S ε = {λ C; λ 0, argλ < π 2 + ε}. The first main theorem of this paper is as follows. 5 Σ

6 Theorem 2.1 (Weighted Resolvent Estimates) Let Σ be a bounded domain of C 1,1 -class with α 0 > 0 and α 1 > 0 being the least eigenvalue of the Dirichlet and Neumann Laplacian in Σ, and let ᾱ := min{α 0, α 1 }, β (0, ᾱ), α (0, ᾱ β 2 ), 0 < ε < ε := arctan ( ᾱ 1 β β2 α ), 1 <, r < and ω A r. Then for every f L β (R; Lr ω(σ)), and λ α + S ε there exists a uniue solution (u, p) to (R λ ) (with g = 0) such that (λ + α)u, 2 u, p L β (Lr ω) and (λ + α)u, 2 u, p L β (Lr ω ) C f L β (Lr ω ) (2.1) with an A r -consistent constant C = C(, r, α, β, ε, Σ, A r (ω)) independent of λ. In particular we obtain from Theorem 2.1 the following corollary on resolvent estimates of the Stokes operator in the cylinder Ω. Corollary 2.2 (Stokes Semigroup in Straight Cylinders) Let 1 <, r <, ω A r (R n 1 ) and define the Stokes operator A = A,r;β,ω on Σ R by D(A) = W 2;,r 1;,r β,ω (Σ R) W0,β,ω (Σ R) L β (Lr ω) σ L β (Lr ω) σ, Au = P,r;β,ω u, (2.2) where P,r;β,ω is the Helmholtz projection in L β (Lr ω) (see [9]). Then, for every ε (0, ε ) and α (0, ᾱ β 2 ), β (0, ᾱ), α + S ε is contained in the resolvent set of A, and the estimate (λ + A) 1 L(L (L r ω )σ) C λ + α, λ α + S ε, (2.3) holds with an A r -consistent constant C = C(Σ,, r, α, β, ε, A r (ω)). As a conseuence, the Stokes operator generates a bounded analytic semigroup {e ta,r;β,ω ; t 0} on L β (Lr ω) σ satisfying the estimate e ta,r;β,ω L(L β (Lr ω) σ) C e αt α (0, ᾱ β 2 ), t > 0, (2.4) with a constant C = C(, r, α, β, ε, Σ, A r (ω)). The second important result of this paper is the maximal regularity of the Stokes operator in an infinite straight cylinder. Theorem 2.3 (Maximal Regularity in Straight Cylinders) Let 1 < p,, r <, ω A r (R n 1 ) and β (0, ᾱ). Then the Stokes operator A = A,r;β,ω has maximal regularity in L β (Lr ω) σ. To be more precise, for each F L p (R + ; L β (Lr ω) σ ) the instationary problem U t + AU = F, U(0) = 0, (2.5) has a uniue solution U W 1,p (R + ; L β (Lr ω) σ ) L p (R + ; D(A)) such that U, U t, AU L p (R + ;L β (Lr ω )σ) C F L p (R + ;L β (Lr ω )σ). (2.6) 6

7 Analogously, for every F L p (R + ; L β (Lr ω)), the instationary system has a uniue solution U t U + P = F, div U = 0, U(0) = 0, (U, P ) ( W 1,p (R + ; L β (Lr ω) σ ) L p (R + ; D(A)) ) L p (R + ; L β (Lr ω)) satisfying the a priori estimate U t, U, U, 2 U, P L p (R + ;L β (Lr ω)) C F L p (R + ;L β (Lr ω)) (2.7) with C = C(Σ,, r, β, A r (ω)). Moreover, if e αt F L p (R + ; L β (Lr ω)) for some α (0, ᾱ β 2 ), then the solution u satisfies the estimate e αt U, e αt U t, e αt 2 U L p (R + ;L β (Lr ω)) C e αt F L p (R + ;L β (Lr ω)) (2.8) with C = C(Σ,, r, α, β, A r (ω)). As a corollary of Theorem 2.3 we get the maximal regularity result for general cylinder Ω with several exits to infinity given by (1.1). Theorem 2.4 (Stokes Semigroup in General Cylinders) Let a C 1,1 -domain Ω be given by (1.1) and β i > 0 for i = 1,..., m satisfy the same assumptions on β with Σ i in place of Σ. Then, the Stokes operator A,b (Ω) generates an exponentially decaying analytic semigroup {e ta,b }t 0 in L b,σ (Ω). Theorem 2.5 (Maximal Regularity in General Cylinders) Let a C 1,1 -domain Ω be given by (1.1) and β i > 0 for i = 1,..., m satisfy the same assumptions on β with Σ i in place of Σ. Then, the Stokes operator A,b has maximal regularity in L b,σ (Ω); to be more precise, for any F Lp (R + ; L b,σ (Ω)) the Cauchy problem has a uniue solution U such that U t + A,b U = F, U(0) = 0, in L b,σ (Ω), (2.9) U, U t, A,b U L p (R+;L b,σ (Ω)) C F L p (R+;L b,σ (Ω)) (2.10) with some constant C = C(, Ω). Euivalently, if F L p (R + ; L b (Ω)), then the instationary Stokes system has a uniue solution (U, P ) such that U t U + P = F in R + Ω, div U = 0 in R + Ω, U(0) = 0 in Ω, U = 0 on Ω, (U, P ) (L p (R + ; W 2, b (Ω) W 1, 0 (Ω)) L σ(ω)) L p (R + ; L b (Ω)), U t L p (R + ; L b (Ω)), U L p (R + ;W 2, b (Ω) W 1, 0 (Ω)) + U t, P L p (R + ;L b (Ω)) C F L p (R + ;L b (Ω)). (2.11) (2.12) 7

8 Remark 2.6 We note that in (2.5) and in (2.11) we may take nonzero initial values u(0) = u 0 in the interpolation space (L β (Lr ω) σ, D(A,r;β,ω )) 1 1/p,p and U(0) = U 0 (L 2, 1, b (Ω), Wb (Ω) W0,b (Ω)) 1 1/p,p, respectively. For the proofs in Section 3 and Section 4, we need some preliminary results for Muckenhoupt weights. Proposition 2.7 ([9], Lemma 2.4) Let 1 < r < and ω A r (R n 1 ). (1) Let T : R n 1 R n 1 be a bijective, bi-lipschitz vector field. Then, it holds that ω T A r (R n 1 ) and A r (ω T ) c A r (ω) with a constant c = c(t, r) > 0 independent of ω. (2) Define the weight ω(x ) = ω( x 1, x ) for x = (x 1, x ) R n 1. Then ω A r and A r ( ω) 2 r A r (ω). (3) Let Σ R n 1 be a bounded domain. Then there exist s, s (1, ) satisfying L s (Σ) L r ω(σ) L s (Σ). Here s and 1 are A s r-consistent. Moreover, the embedding constants can be chosen uniformly on a set W A r provided that sup A r (ω) <, ω W ω dx = 1 for all ω W, (2.13) for a cube Q R n 1 with Σ Q. Q Proposition 2.8 ([9], Proposition 2.5) Let Σ R n 1 be a bounded Lipschitz domain and let 1 < r <. (1) For every ω A r the continuous embedding Wω 1,r (Σ) L r ω(σ) is compact. (2) Consider a seuence of weights (ω j ) A r satisfying (2.13) for W = {ω j : j N} and a fixed cube Q R n 1 with Σ Q. Further let (u j ) be a seuence of functions on Σ satisfying sup u j 1,r,ωj < and u j 0 in W 1,s (Σ) j for j where s is given by Proposition 2.7 (3). Then u j r,ωj 0 for j. (3) Under the same assumptions on (ω j ) A r as in (2) consider a seuence of functions (v j ) on Σ satisfying sup v j r,ωj < and v j 0 in L s (Σ) j for j. Then considering v j as functionals on W 1,r ω (Σ) j v j (W 1,r 0 for j. (Σ)) ω j 8

9 Proposition 2.9 Let r (1, ), ω A r and Σ R n 1 be a bounded Lipschitz domain. Then there exists an A r -consistent constant c = c(r, Σ, A r (ω)) > 0 such that u r,ω c u r,ω for all u W 1,r ω (Σ) with vanishing integral mean Σ u dx = 0. Proof: See the proof of [16], Corollary 2.1 and its conclusions; checking the proof, one sees that the constant c = c(r, Σ, A r (ω)) is A r -consistent. Finally we cite the Fourier multiplier theorem in weighted spaces. Theorem 2.10 ([19], Ch. IV, Theorem 3.9) Let m C k (R k \ {0}), k N, admit a constant M R such that η γ D γ m(η) M for all η R k \ {0} and multi-indices γ N k 0 with γ k. Then for all 1 < r < and ω A r (R k ) the multiplier operator T f = F 1 m( )Ff defined for all rapidly decreasing functions f S(R k ) can be uniuely extended to a bounded linear operator from L r ω(r k ) to L r ω(r k ). Moreover, there exists an A r -consistent constant C = C(r, A r (ω)) such that T f r,ω CM f r,ω, f L r ω(r k ). 3 Resolvent estimate of the Stokes operator in weighted spaces on infinite straight cylinders In this section we obtain the resolvent estimate of the Stokes operator in Lebesgue spaces with exponential weight with respect to the axial variable and Muckenhoupt weight for cross-sectional variables in an infinite straight cylinder Σ R, where the cross-section Σ is a C 1,1 -bounded domain. 3.1 Estimate for the problem (R λ,ξ,β ) In this subsection we get estimates for (R λ,ξ,β ) independent of λ and ξ R in L r - spaces with Muckenhoupt weights. To this aim we rely partly on cut-off techniues using the results for (R λ,ξ ) (i.e., the case β = 0) in the whole and bent half spaces in [12] (Theorem 3.1 below). The main existence and uniueness result in weighted L r -spaces for (R λ,ξ,β ) is described in Theorem 3.8. For whole or bent half spaces Σ, g Ŵ ω 1,r (Σ) + L r ω(σ) and η = ξ + iβ, ξ R, β 0, we use notation g; Ŵ 1,r ω + L r ω,1/η = inf{ g 0 1,r,ω + g 1 /η r,ω : g = g 0 + g 1, g 0 Ŵ ω 1,r, g 1 L r ω}. In the following we put R λ,ξ R λ,ξ,0. Theorem 3.1 Let n 3, 1 < r <, ω A r (R n 1 ), 0 < ε < π 2, ξ R,λ S ε, 0 < ε < π/2 and µ = λ + ξ 2 1/2. 9

10 (i) ([12], Theorem 3.1) Let Σ = R n 1. If f L r ω(σ) and g Wω 1,r (Σ), then the problem (R λ,ξ ) has a uniue solution (u, p) Wω 2,r (Σ) Wω 1,r (Σ) satisfying µ 2 u, µ u, 2 u, p, ξp r,ω c ( f, g, ξg r,ω + λg; Ŵ 1,r ω +L r ω,1/ξ ) (3.1) with an A r -consistent constant c = c(ε, r, A r (ω)) independent of λ and ξ. (ii) ([12], Theorem 3.5) Let Σ = H σ = {x = (x 1, x ); x 1 > σ(x ), x R n 2 } for a given function σ C 1,1 (R n 2 ). Then there are A r -consistent constants K 0 = K 0 (r, ε, A r (ω)) > 0 and λ 0 = λ 0 (r, ε, A r (ω)) > 0 independent of λ and ξ such that, if σ K 0, for every f L r ω(σ) and g Wω 1,r (Σ) the problem (R λ,ξ ) has a uniue solution (u, p) (Wω 2,r (Σ) W0,ω(Σ)) 1,r Wω 1,r (Σ). This solution satisfies the estimate µ 2 u, µ u, 2 u, p, ξp r,ω c ( f, g, ξg r,ω + λg; Ŵ (3.2) ω 1,r (Σ) + L r ω,1/ξ (Σ) ) with an A r -consistent constant c = c(r, ε, A r (ω)). On the bounded domain Σ R n 1 of C 1,1 -class let α 0 and α 1 denote the smallest eigenvalue of the Dirichlet and Neumann Laplacian, respectively, i.e., α 0 := inf{ u 2 2 : u W 1,2 0 (Σ), u 2 = 1} > 0, α 1 := inf{ u 2 2 : u W 1,2 (Σ), u n Σ = 0, u 2 = 1} > 0, ᾱ := min{α 0, α 1 }. (3.3) For fixed λ C\(, α 0 ], η = ξ +iβ, ξ R, β 0, and ω A r we introduce the parametrized Stokes operator S = Sr,λ,η ω by (λ + η 2 )u + p S(u, p) = (λ + η 2 )u n + iηp div η u defined on D(S) = D( r,ω) W 1,r ω (Σ), where D( r,ω) = W 2,r ω (Σ) W 1,r 0,ω(Σ) and div η u = div u + iηu n. For ω 1 the operator Sr,λ,η ω will be denoted by S r,λ,η. Note that the image of D(S) by div η is included in Wω 1,r (Σ) and Wω 1,r (Σ) L r 0,ω(Σ) + L r ω(σ), where L r 0,ω(Σ) := { u L r ω(σ) : u dx = 0 }. Using Poincaré s ineuality in weighted spaces, see Proposition 2.9, one can easily check the continuous embedding L r 0,ω(Σ) Ŵ ω 1,r (Σ); more precisely, u 1,r,ω c u r,ω, 10 Σ u L r 0,ω(Σ),

11 with an A r -consistent constant c > 0. For bounded domain Σ we use the notation g; L r 0,ω + L r ω,1/η 0 := inf{ g 0 1,r,ω + g 1 /η r,ω : g = g 0 + g 1, g 0 L r 0,ω, g 1 L r ω}; note that this norm is euivalent to the norm (W 1,r ω,η ) weighted Sobolev space on Σ with norm u, ηu r,ω. where W 1,r ω,η is the usual First we consider Hilbert space setting of (R λ,ξ,β ). For η = ξ + iβ, ξ R, β 0, let us introduce a closed subspace of W 1,r 0 (Σ) as V η := {u W 1,r 0 (Σ) : div η u = 0}. Lemma 3.2 Let φ = (φ, φ n ) W 1,2 (Σ) be such that φ, v W 1,2 (Σ),W 1,2 0 (Σ) = 0 for all v W 1,2 0 (Σ). Then, there is some p L 2 (Σ) with φ = ( p, iηp). Proof: This lemma can be proved just by copy of the proof of [10], Lemma 3.1 with ξ R replaced by η = ξ + iβ. Lemma 3.3 (i) For any g W 1,2 (Σ), η = ξ + iβ, x R, β 0, the euation div η u = g has at least one solution u W 2,2 (Σ) W 1,2 0 (Σ) and u 2,2 c( g 1,2 + 1 g dx ), η where c is independent of g. (ii) Let ε (0, π/2), β (0, α 0 ) and λ { α 0 + β 2 (Im λ)2 + S ε } {λ C : Re λ > α 4β β 2 }. (3.4) Then, for any f L 2 (Σ), g W 1,2 (Σ) the system (R λ,ξ,β ) has a uniue solution (u, p) (W 2,2 (Σ) W 1,2 (Σ)) W 1,2 (Σ). Proof: Proof of (i): Let a scalar function w C0 (Σ) be such that w Σ dx = 0. Given g W 1,2 (Σ), let ḡ = g dx and consider a divergence problem in Σ, that is, Σ div u = g ḡw, u Σ = 0, which has a solution u W 2,2 (Σ) W 1,2 0 (Σ) with u 2,2 c (g ḡw) 2 c g 1,2, by [7], Theorem 1.2. Then, u := (u, ḡw ) satisfies div iη ηu = g and reuired estimate. Proof of (ii): By the assertion (i) of the lemma, we may assume w.l.o.g. that g 0. Now, for fixed λ α 0 + β 2 + S ε define the bilinear form b : V η V η R by ( b(u, v) := (λ + η 2 )u v + u v ) dx. Σ Obviously, b is continuous in V η V η. Moreover, b is coercive, that is, Σ b(u, u) l(λ, ξ, β) u 2 1,2 (3.5) 11

12 with some l(λ, ξ, β) > 0. In fact, b(u, u) = ((Re λ + ξ 2 β 2 ) u 2 + u 2 ) dx + i Σ Note that, due to Poincaré s ineuality, (ξ 2 α 0 ) u u 2 2 > 0, ξ R. Σ (Im λ + 2ξβ) u 2 dx. Hence, if Re λ + α 0 β 2 0, then b(u, u) ((Re λ + ξ 2 β 2 ) u 2 + u 2 ) dx (ξ 2 α 0 ) u u 2 2, where Σ if ξ 2 α 0 0 and (ξ 2 α 0 ) u u 2 2 u 2 2 (ξ 2 α 0 ) u u 2 2 (ξ 2 /α 0 1) u u 2 2 ξ2 α 0 u 2 2 if ξ 2 α 0 < 0. Therefore, it remained to prove (3.5) for the case Re λ + α 0 β 2 < 0. Note that if Im λ + 2ξβ 0 then (R λ,ξ,β ) coincides with (R λ1,ξ) where λ 1 = λ β 2 + 2iξβ α 0 + S ε1 with ε 1 = max{ε, arctan Re λ+α 0 β 2 } (0, π/2). Hence, Im λ+2ξβ (3.5) can be proved in the same way as the proof of [10], Lemma 3.2 (ii). Now, suppose that Im λ + 2ξβ = 0, i.e., ξ = Im λ 2β. Since (3.5) is trivial for the case Re λ + ξ 2 β 2 0, we assume that Re λ + ξ 2 β 2 < 0. In this case, note that due to the condition Re λ + c(λ, β) > 0 such that (Im λ)2 4β 2 β 2 > α 0 there is some Then, 0 > Re λ + Finally, (3.5) is proved. (Im λ)2 4β 2 β 2 > c(λ, β) α 0, c(λ, β) α 0 < 0. b(u, u) (Im λ)2 ((Re λ + β 2 ) u 2 + u 2 ) dx Σ 4β 2 (c(λ, β) α Σ 0) u 2 + u 2 ) dx c(λ,β) α 0 u

13 By Lax-Milgram s lemma in view of (3.5), the variational problem b(u, v) = f v dx, v V η, Σ has a uniue solution u in V η. Then, by Lemma 3.2, there is some p L 2 (Σ) such that (λ + η 2 )u + p = f, (λ + η 2 )u n + iηp = f n. Now, applying the well-known regularity theory for Stokes system and Poisson s euation in Σ to and u + p = f (λ + η 2 )u, div u = iηu n, u Σ = 0 u n = f n (λ + η 2 )u n iηp, u n Σ = 0, respectively, we have (u, p) (W 2,2 (Σ) W 1,2 0 (Σ)) W 1,2 (Σ). Thus, the assertion (ii) of the lemma is proved. Remark 3.4 It is seen by elementary calculation that the assumption (3.4) on λ of Lemma 3.3 is satisfied for all λ α + S ε if either α (0, α 0 β 2 ) and ε α (0, arctan 0 β 2 α ) or if α (0, ᾱ β 2 ᾱ β ) and ε (0, arctan 2 α ). Note that β β ᾱ < α 0, see (3.3). Now, we turn in considering (R λ,ξ,β ) in weighted spaces with weights w.r.t. crosssection as well. α 0 β 2 α Lemma 3.5 Let ξ R, β (0, α 0 ), α (0, α 0 β 2 ), ε (0, arctan ), β λ α + S ε, and ω A r, 1 < r <. Then the operator S = Sr,λ,η ω is injective and the range R(S) of S is dense in L r ω(σ) Wω 1,r (Σ). Proof: Since, by Proposition 2.7 (3), there is an s (1, r) such that L r ω(σ) L s (Σ), one sees immediately that D(Sr,λ,η ω ) D(S s,λ,η). Therefore, Sr,λ,η ω (u, p) = 0 for some (u, p) D(Sr,λ,η ω ) yields (u, p) D(S s,λ,η) and S s,λ,η (u, p) = 0. Here note that S s,λ,η (u, p) = 0 implies S s,λ,η (u, p) = ((β 2 2iξβ)u, (β 2 2iξβ)u n + βp, βu n ) T. Hence, by applying [10], Theorem 3.4 finite number of times and the Sobolev embedding theorem, we get that (u, p) (W 2,2 (Σ) W 1,2 0 (Σ)) W 1,2 (Σ). Therefore, by Lemma 3.3 we get that (u, p) = 0, i.e., Sr,λ,η ω is injective. On the other hand, by Proposition 2.7 (3), there is an s (r, ) such that S s,λ,η Sr,λ,η ω. Moreover, by Lemma 3.3, for every (f, g) C 0 (Σ) C ( Σ), there is some (u, p) D(S 2,λ,η ) with S 2,λ,η (u, p) = (f, g). Applying the regularity result of [7], Theorem 1,2 for the Stokes resolvent system in Σ finite number of times using the Sobolev embedding theorem, it can be seen that (u, p) D(S,λ,η ) for all (1, ), in particular, for = s. Therefore, C0 (Σ) C ( Σ) R(S s,λ,η ) R(Sr,λ,η) ω L r ω(σ) Wω 1,r (Σ), 13

14 which proves the assertion on the denseness of R(S). The following lemma gives a preliminary a priori estimate for a solution (u, p) of S(u, p) = (f, g). Lemma 3.6 Assume the same for r, ω, α, β and λ as in Lemma 3.5. Then there exists an A r -consistent constant c = c(ε, r, β, Σ, A r (ω)) > 0 such that for every (u, p) D(S ω r,λ,η ), µ 2 +u, µ + u, 2 u, p, ηp r,ω c ( f, g, g, ξg r,ω + λ g; L r 0,ω + L r ω,1/η 0 ) + (3.6) u, ξu, p r,ω + λ u (W 1,r ω ), where µ + = λ + α + ξ 2 1/2, (f, g) = S(u, p) and (W 1,r ω (Σ). W 1,r ω ) denotes the dual space of Proof: The proof is devided into two parts, i.e, the case ξ 2 > β 2 and the other case ξ 2 β 2. The proof of the case ξ 2 > β 2 is based on a partition of unity in Σ and on the localization procedure reducing the problem to a finite number of problems of type (R λ,ξ ) in bent half spaces and in the whole space R n 1. Since Σ C 1,1, we can cover Σ by a finite number of balls B j, j 1, such that, after a translation and rotation of coordinates, Σ B j locally coincides with a bent half space Σ j = Σ σj where σ j C 1,1 (R n 1 ) has a compact support, σ j (0) = 0 and σ j (0) = 0. Choosing the balls B j small enough (and its number large enough) we may assume that σ j K 0 (ε, r, Σ, A r (ω)) for all j 1 where K 0 was introduced in Theorem 3.1 (ii). According to the covering Σ j 1 B j there are cut-off functions (ϕ j ) m j=0 such that such that 0 ϕ 0, ϕ j C (R n 1 ), j 0 ϕ j 1 in Σ, supp ϕ 0 Σ, supp ϕ j B j, j 1. (3.7) Given (u, p) D(S) and (f, g) = S(u, p), we get for each ϕ j, j 0, the local (R λ,ξ )-problems (λ + ξ 2 )(ϕ j u ) + (ϕ j p) = f j for (ϕ j u, ϕ j p), j 0, in R n 1 or Σ j ; here (λ + ξ 2 )(ϕ j u n ) + iξ(ϕ j p) = f jn (3.8) div ξ (ϕ j u) = g j f j = ϕ j f 2 ϕ j u ( ϕ j )u + (β 2 2iξ)(ϕ j u ) + ( ϕ j )p f jn = ϕ j f n 2 ϕ j u n ( ϕ j )u n + (β 2 2iξ)(ϕ j u n ) + β(ϕ j p) g j = ϕ j g + ϕ j u + βϕ j u n. (3.9) To control f j and g j note that u = 0 on Σ; hence Poincaré s ineuality for Muckenhoupt weighted space yields for all j 0 the estimate f j, g j, ξg j r,ω;σj c( f, g, g, ξg r,ω;σ + u, ξu, p r,ω;σ ), (3.10) 14

15 where Σ 0 R n 1 and c > 0 is A r -consistent. Moreover, let g = g 0 + g 1 denote any splitting of g L r 0,ω + L r ω,1/η. Defining the characteristic function χ j of Σ Σ j and the scalar 1 m j = (ϕ j g 0 + u ϕ j + βϕ j u n )dx Σ Σ j Σ Σ j 1 = (iξu n g 1 )ϕ j dx, Σ Σ j Σ Σ j we split g j in the form g j = g j0 + g j1 := (ϕ j g 0 + u ϕ j + βϕ j u n m j χ j ) + (ϕ j g 1 + m j χ j ). Concerning g j1 we get g j1 r r,ω;σ j = ϕ j g 1 + m j r ω dx Σ Σ j c(r) ( g 1 r r,ω;σ + m j r ω(σ Σ j ) ) c(r) ( g 1 rr,ω;σ + ω(σ Σ j) ω (Σ Σ j ) ) r/r ( ξu Σ Σ j r n r + g 1 r (W 1,r ω ) r,ω;σ) with c(r) > 0 independent of ω. Since we chose the balls B j for j 1 small enough, for each j 0 there is a cube Q j with Σ Σ j Q j and Q j < c(n) Σ Σ j where the constant c(n) > 0 is independent of j. Therefore ) g j1 r,ω;σj c(r) ( g 1 r,ω + c(n)ω(q j) 1/r ω (Q j ) 1/r Q j ( ξu n (W 1,r + g ω ) 1 r,ω ) c(r)(1 + A r (ω) 1/r ) ( ) (3.11) ξu n (W 1,r + g ω ) 1 r,ω;σ for j 0. Furthermore, for every test function Ψ C0 ( Σ j ) let 1 Ψ = Ψ Ψdx. Σ Σ j Σ Σ j By the definition of m j χ j we have Σ j g j0 dx = 0; hence by Poincaré s ineuality (see Proposition 2.9) Σ j g j0 Ψdx = Σ j g j0 Ψdx = Σ g 0(ϕ j Ψ)dx + Σ u ( ϕ j ) Ψdx + Σ βu nϕ j Ψ dx g 0 1,r,ω (ϕ j Ψ) r,ω + u (W 1,r ω ) ( ϕ j ) Ψ 1,r,ω + βu n (W 1,r ω ) ϕ j Ψ 1,r,ω c( g 0 1,r,ω + u (W 1,r ω ) ) Ψ r,ω ;Σ j, where c > 0 is A r -consistent. Thus g j0 1,r,ω;Σj c ( g 0 1,r,ω + u (W 1,r Summarizing (3.11) and (3.12), we get for j 0 ω ) ) for j 0. (3.12) g j ; Ŵ 1,r ω (Σ j ) + L r ω,1/ξ(σ j ) c ( u (W 1,r ω ) + g; L r 0,ω + L r ω,1/ξ 0 ) 15

16 with an A r -consistent c = c(r, A r (ω)) > 0, which yields in view of ξ 2 > β 2 that g j ; Ŵ ω 1,r (Σ j ) + L r ω,1/ξ(σ j ) c ( ) u (W 1,r + g; L r ω ) 0,ω + L r ω,1/η 0 (3.13) with an A r -consistent c = c(r, A r (ω)) > 0. To complete the proof, apply Theorem 3.1 (i) to (3.8), (3.9) when j = 0. Further use Theorem 3.1 (ii) in (3.8), (3.9) for j 1, but with λ replaced by λ + M with M = λ 0 + α 0, where λ 0 = λ 0 (ε, r, A r (ω)) is the A r -consistent constant indicated in Theorem 3.1 (ii). This shift in λ implies that f j has to be replaced by f j + Mϕ j u and that (3.2) will be used with λ replaced by λ + M. Summarizing (3.1), (3.2) as well as (3.10), (3.13) and summing over all j we arrive at (3.6) with the additional terms I = Mu r,ω + Mu (W 1,r ω ) + Mg; L r 0,ω + L r ω,1/η 0 on the right-hand side of the ineuality. Note that M = M(ε, r, A r (ω)) is A r - consistent, η max{ 2 ξ, 2β} and that g = div u + iηu n defines a natural splitting of g L r 0,ω(Σ) + L r ω(σ). Hence Poincaré s ineuality yields I M ( u r,ω;σ + div u 1,r,ω + u n r,ω;σ ) c 1 u r,ω;σ c 2 u r,ω;σ with A r -consistent constants c i = c i (ε, r, Σ, A r (ω)) > 0, i = 1, 2. Thus (3.6) is proved. and Next, consider the case ξ 2 β 2. Since S(u, p) = (f, g), we have (λ )u + p = f η 2 u, div u = g iηu n, in Σ, u Σ = 0, (λ )u n = f n η 2 u n iηp, in Σ, u n Σ = 0. (3.14) (3.15) Now, apply [16], Lemma 3.2 to (3.14). Then, in view of η 2β and Poincaré s ineuality, for all λ α + S ε, α (0, α 0 β 2 ) we have (λ + α)u, 2 u, p r,ω;σ c ( f, η 2 u r,ω;σ + λ g iηu n Ŵ 1,r ω (Σ) + g iηu n W 1,r ω (Σ) + ) λ u (W 1,r ω (Σ)) c ( f, u, p r,ω;σ + g W 1,r ω (Σ) + λ g iηu n Ŵ 1,r ω (Σ) + λ u ) (W 1,r ω (Σ)) with A r -consistent constant c = c(r, ε, α, β, Σ, A r (Ω)). In order to control g iηu n Ŵ 1,r ω (Σ), let us split g as g = g 0 + g 1, g 0 L r 0,ω(Σ), g 1 L r ω,1/η (Σ). Since g 1 iηu n has mean value zero in Σ, we get by poincaré s ineuality that g 1 iηu n, ψ = g 1 iηu n, ψ η g 1 /η r,ω ψ r,ω + η u n (W 1,r ψ ω (Σ)) W 1,r ω (Σ) c(r, Σ)( g 1 /η r,ω + u n (W 1,r ) ψ ω (Σ)) r,ω ;Σ, 16

17 for all ψ C ( Σ), where ψ = ψ 1 ψ Σ Σ dx. Therefore, g iηu n Ŵ 1,r Ω (Σ) g 0 Ŵ 1,r Ω (Σ) + c( g 1/η r,ω + u n (W 1,r ). ω (Σ)) Thus, for all λ α + S ε, α (0, α 0 β 2 ) we have (λ + α)u, 2 u, p r,ω;σ c ( f, u, p r,ω;σ + g W 1,r ω (Σ) + λ u + λ g : L r (W 1,r ω (Σ)) 0,ω + L r ω,1/η ) 0 (3.16) with A r -consistent constant c = c(r, ε, α, β, Σ, A r (Ω)). On the other hand, applying well-known results for the Laplace resolvent euations (cf. [16]) to (3.15), we get that (λ + α)u n, 2 u n r,ω;σ c( f n, u, p r,ω;σ (3.17) with c = c(r, ε, α, β, Σ, A r (Ω)). Thus, from (3.16) and (3.17) the assertion of the lemma for the case ξ 2 β 2 is proved. The proof of the lemma is complete. Lemma 3.7 Let 1 < r <, ω A r and ξ R, β (0, ᾱ), α (0, ᾱ β 2 ), ᾱ β ε (0, arctan 2 α ), λ α + S β ε. Then there is an A r -consistent constant c = c(α, ε, r, β, Σ, A r (ω)) such that for every (u, p) D(S) and (f, g) = S(u, p) the estimate µ 2 +u, µ + u, 2 u, p, ηp r,ω c ( f, g, g, ξg r,ω + ( λ + 1) g; L r 0,ω + L r ω,1/ξ ) (3.18) 0 holds; here µ + = λ + α + ξ 2 1/2. Proof: Assume that this lemma is wrong. Then there is a constant c 0 > 0, a seuence {ω j } j=1 A r with A r (ω j ) c 0 for all j, seuences {λ j } j=1 α + S ε, {ξ j } j=1 R and (u j, p j ) D(S ω j r,λ j,ξ j ) for all j N such that (λ j + α + ξ 2 j )u j, (λ j + α + ξ 2 j ) 1/2 u j, 2 u j, p j, η j p j r,ωj j ( f j, g j, g j, ξ j g j r,ωj + ( λ j + 1) g j ; L r m,ω j + L r ω j,1/η j 0 (3.19) where η j = ξ j + iβ, (f j, g j ) = S ω j r,λ j,η j (u j, p j ). Fix an arbitrary cube Q containing Σ. We may assume without loss of generality that A r (ω j ) c 0, ω j (Q) = 1 j N, (3.20) by using the A r -weight ω j := ω j (Q) 1 ω j instead of ω j if necessary. Note that (3.20) also holds for r, {ω j} in the following form: A r (ω j ) c r /r 0, ω j(q) c r /r 0 Q r. Therefore, by a minor modification of Proposition 2.7 (3), there exist numbers s, s 1 such that L r ω j (Σ) L s (Σ), L s 1 (Σ) L r ω (Σ), j N, (3.21) j 17

18 with embedding constants independent of j N. without loss of generality that Furthermore, we may assume (λ j + α + ξ 2 j )u j, (λ j + α + ξ 2 j ) 1/2 u j, 2 u j, p j, η j p j r,ωj = 1 (3.22) and conseuently that f j, g j, g j, ξ j g j r,ωj + ( λ j + 1) g j ; L r m,ω j + L r ω j,1/ξ j 0 0 as j. (3.23) From (3.21), (3.22) we have (λ j + α + ξ 2 j )u j, (λ j + α + ξ 2 j ) 1/2 u j, 2 u j, p j, η j p j s K, (3.24) with some K > 0 for all j N and f j, g j, g j, η j g j s 0 as j. (3.25) Without loss of generality let us suppose that as j, λ j λ α + S ε or λ j ξ j 0 or ξ j ξ 0 or ξ j. Thus we have to consider six possibilities. (i) The case λ j λ α + S ε, ξ j ξ <. Due to (3.24) {u j } W 2,s and {p j } W 1,s are bounded seuences. In virtue of the compactness of the embedding W 1,s (Σ) L s (Σ) for the bounded domain Σ, we may assume (suppressing indices for subseuences) that u j u, u j u in L s (strong convergence) 2 u j 2 u in L s (weak convergence) p j p in L s (strong convergence) p j p in L s (weak convergence) (3.26) for some (u, p) D(S s,λ,ξ ) as j. Therefore, S s,λ,ξ (u, p) = 0 and, conseuently, u = 0, p = 0 by Lemma 3.5. On the other hand we get from (3.22) that sup j N u j 2,r,ωj < and sup j N p j 1,r,ωj < which, together with the weak convergences u j 0 in W 2,s (Σ), p j 0 in W 1,s (Σ), yields u j 1,r,ωj 0, p j r,ωj 0 due to Proposition 2.8 (2). Moreover, since sup j N λ j u j r,ωj < and λ j u j λu = 0 in L s (Σ), Proposition 2.8 (3) implies that λ j u j (W 1,r 0. (3.27) ) Thus (3.6), (3.22) and (3.23) yield the contradiction 1 0. ω j 18

19 (ii) The case λ j λ α + S ε, ξ j. From (3.22) we get u j, ξ j u j, p j r,ωj 0. On the other hand, since u j r,ωj 0 and u j 0 in L s as j, Proposition 2.8 (3) implies (3.27). Thus, from (3.6), (3.22) and (3.23) we get the contradiction 1 0. (iii) The case λ j, ξ j ξ <. By (3.22) u j, ξ j u j r,ωj 0 as j. (3.28) Further, (3.24) yields the convergence u j 0, u j 0 and 2 u j 0, λ j u j v, p j p and p j p, in L s, which, together with (3.25), leads to v + p = 0, v n + iηp = 0. (3.29) Let g j := g j0 + g j1, g j0 L r 0,ω j, g j1 L r ω j. Then, by (3.24) we have From (3.21), (3.30) we see that λ j g j0 1,r,ωj + λ j g j1 /η j r,ωj 0 (j ). (3.30) λ j g j, ϕ = λ j g j0, ϕ + λ j g j1, ϕ λ j g j0 1,r,ωj ϕ r,ω j + λ jg j1 r,ωj ϕ r,ω j c ( λ j g j0 1,r,ωj + λ j g j1 /η j r,ωj ) ϕ W 1,s 1 (Σ). Conseuently, λ j g j (W 1,s 1 (Σ)) and λ j g j (W 1,s 1 (Σ)) 0 as j. (3.31) Therefore, it follows from the divergence euation div η j u j = g j that for all ϕ C ( Σ) v, ϕ + iηv n, ϕ = lim j div λ j u j + iλ j ξ j u jn, ϕ = lim j λ j g j, ϕ = 0, yielding div v = iηv n, v N Σ = 0. Therefore (3.29) implies p + η 2 p = 0 in Σ, p N = 0 on Σ. (3.32) Here note that η 2 = ξ 2 β 2 + 2iξβ. Hence, if ξ 0 then p 0 since the all eigenvalues of the Neumann Laplacian in Σ is real; if ξ = 0, then η 2 = β 2 and hance p 0 due to the condition β 2 < ᾱ α 1. That is, we have p 0, and and also v 0. Now, due to Proposition 2.8 (2), (3), we get (3.27) and the convergence p j r,ωj 0, since λ j u j 0 in L s, p j 0 in W 1,s and sup j N λ j u j r,ωj <, 19

20 sup j N p j 1,r,ωj <. Thus (3.6), (3.22), (3.23) and (3.28) lead to the contradiction 1 0. (iv) The case λ j, ξ j. To come to a contradiction, it is enough to prove (3.27) since u j, ξ j u j, p j r,ωj 0 as j. From (3.22) we get the convergence u j 0, u j 0 and 2 u j 0, (λ j + η 2 j )u j v, p j 0 and p j 0, η j p j in L s with some v, L s. Therefore, (3.25) and (R λj,ξ j ) yield v = 0, v n + i = 0. Since λ j u j s c ε (λ j + η 2 j )u j s, there exists w = (w, w n ) L s such that, for a suitable subseuence, λ j u j w. Let g j = g j0 + g j1, j N, be a seuence of splittings satisfying (3.30). By (3.21) we get for all ϕ C ( Σ) λ j g j0, ϕ + λ jg j1, ϕ 0 as j, η j cf. (3.31) and (3.31). Hence, the divergence euation implies that for j λ j u jn, ϕ = 1 iη j λ j g j0, ϕ + λ jg j1 iη j, ϕ + 1 iη j λ j u j, ϕ 0 for all ϕ C ( Σ) yielding w n, ϕ = 0 and conseuently w n = 0. Obviously, η j u j 0 in L s as j. Therefore, by (3.25) and the boundedness of the seuence { η j u j r,ωj }, we get from the identity div (η j u j) + iη 2 j u jn = η j g j that η 2 j u jn 0 and hance ξ 2 j u jn 0 in L s as j. Thus we proved v n = 0. Now v = 0 together with the estimate (λ j + ξ 2 j )u j r,ωj 1 imply due to Proposition 2.8 (3) that (λ j + ξj 2 )u j 0 in (W 1,r ω ) as j. j Hence also (3.27) is proved. Now the proof of this lemma is complete. Theorem 3.8 Let 1 < r <, ω A r and ξ R, β (0, ᾱ), α (0, ᾱ β 2 ), ᾱ β ε (0, arctan 2 α ). Then for every λ α + S β ε, ξ R and f L r ω(σ), g W( ω 1,r (Σ) the parametrized resolvent problem (R λ,ξ,β ) has a uniue solution (u, p) W 2,r ω (Σ) W0,ω(Σ) ) 1,r Wω 1,r (Σ). Moreover, this solution satisfies the estimate (3.18) with an A r -consistent constant c = c(α, β, ε, r, Σ, A r (ω)) > 0. Proof: The existence is obvious since, for every λ α + S ε, ξ R and ω A r (R n 1 ), the range R(Sr,λ,ξ ω ) is closed and dense in Lr ω(σ) Wω 1,r (Σ) by Lemma 3.6 and by Lemma 3.5, respectively. Here note that for fixed λ C, ξ R the norm 20

21 g, g, ξg r,ω + (1 + λ ) g; L r m,ω + L r ω,1/ξ 0 is euivalent to the norm of Wω 1,r (Σ). The uniueness of solutions is obvious from Lemma 3.5. by Now, for fixed ω A r, 1 < r <, define the operator-valued functions a 1 : R L(L r ω(σ); W0,ω(Σ) 2,r Wω 1,r (Σ)), b 1 : R L(L r ω(σ); Wω 1,r (Σ)) a 1 (ξ)f := u 1 (ξ), b 1 (ξ)f := p 1 (ξ), (3.33) where (u 1 (ξ), p 1 (ξ)) is the solution to (R λ,ξ,β ) corresponding to f L r ω(σ) and g = 0. Further, define a 2 : R L(Wω 1,r (Σ); W0,ω(Σ) 2,r Wω 1,r (Σ)), b 2 : R L(Wω 1,r (Σ); Wω 1,r (Σ)) by a 2 (ξ)g := u 2 (ξ), b 2 (ξ)g := p 2 (ξ). (3.34) with (u 2 (ξ), p 2 (ξ)) the solution to (R λ,ξ,β ) corresponding to f = 0 and g Wω 1,r (Σ). Corollary 3.9 Assume the same for α, β, ξ, λ as in Theorem 3.8. Then, the operator-valued functions a 1, b 1 and a 2, b 2 defined by (3.33), (3.34) are Fréchet differentiable in ξ R. Furthermore, their derivatives w 1 = d a dξ 1(ξ)f, 1 = d b dξ 1(ξ)f for fixed f L r ω(σ) and w 2 = d a dξ 2(ξ)g, 2 = d b dξ 2(ξ)g for fixed g Wω 1,r (Σ) satisfy the estimates (λ + α)ξw 1, ξ 2 w 1, ξ 3 w 1, ξ 1, ξη 1 r,ω c f r,ω (3.35) and (λ + α)ξw 2, ξ 2 w 2, ξ 3 w 2, ξ 2, ξη 2 r,ω c ( g, g, ξg r,ω + ( λ + 1) g; L r 0,ω + L r ω,1/η ) (3.36) 0, with an A r -consistent constant c = c(α, β, r, ε, Σ, A r (ω)) independent of λ α+s ε and ξ R. Proof: Since ξ enters in (R λ,ξ ) in a polynomial way, it is easy to prove that a j (ξ), b j (ξ), j = 1, 2, are Fréchet differentiable and their derivatives w j, j solve the system (λ + η 2 )w j + j = 2ηu j (λ + η 2 )w jn + iη j = 2ηu jn ip j div w j + iηw jn = iu jn, (3.37) where (u 1, p 1 ), (u 2, p 2 ) are the solutions to (R λ,ξ,β ) for f L r ω(σ), g = 0 and f = 0, g Wω 1,r (Σ), respectively. 21

22 We get from (3.37) and Theorem 3.8 for j = 1, 2, (λ + α)ξw j, ξ 2 w j, ξ 3 w j, ξ j, ξη j r,ω c ( ξηu j, ξp j, ξ u jn, ξ 2 u jn r,ω + ( λ + 1) iηu jn ; L r 0,ω + L r ω,1/η ) 0 c ( ) ξ 2 u j, ξp j, ξ u j r,ω + ( λ + 1) u j r,ω (3.38) c u j, (λ + α + ξ 2 )u j, λ + α + ξ 2 u j, ξp j r,ω c (λ + α + ξ 2 )u j, λ + α + ξ 2 u j, 2 u j, ξp j r,ω, with an A r -consistent constant c = c(α, r, ε, Σ, A r (ω)); here we used the fact that ξ 2 + λ + α c(ε, α) λ + α + ξ 2 for all λ α + S ε, ξ R and u j r,ω c(a r (ω)) 2 u j r,ω (see [16], Corollary 2.2). Thus Theorem 3.8 and (3.38) prove (3.35), (3.36). 4 Proof of the Main Results 4.1 Proof of Theorem 2.1 Theorem 2.3 The proof of Theorem 2.1 is based on the theory of operator-valued Fourier multipliers. The classical Hörmander-Michlin theorem for scalar-valued multipliers for L (R k ), (1, ), k N, extends to an operator-valued version for Bochner spaces L (R k ; X) provided that X is a UMD space and that the boundedness condition for the derivatives of the multipliers is strengthened to R-boundedness. Definition 4.1 A Banach space X is called a UMD space if the Hilbert transform Hf(t) = 1 f(s) π PV t s ds for f S(R; X), where S(R; X) is the Schwartz space of all rapidly decreasing X-valued functions, extends to a bounded linear operator in L (R; X) for some (1, ). It is well known that, if X is a UMD space, then the Hilbert transform is bounded in L (R; X) for all (1, ) (see e.g. [32], Theorem 1.3) and that weighted Lebesgue spaces L r ω(σ), 1 < r <, ω A r, are UMD spaces. Definition 4.2 Let X, Y be Banach spaces. An operator family T L(X; Y ) is called R-bounded if there is a constant c > 0 such that for all T 1,..., T N T, x 1,..., x N X and N N N ε j (s)t j x L j c N ε (0,1;Y ) j (s)x L j (4.1) (0,1;X) j=1 for some [1, ), where (ε j ) is any seuence of independent, symmetric { 1, 1}- valued random variables on [0, 1]. The smallest constant c for which (4.1) holds is denoted by R (T ), the R-bound of T. j=1 22

23 Remark 4.3 (1) Due to Kahane s ineuality ([5]) N ε j (s)x L j 1 (0,1;X) c( 1, 2, X) j=1 N ε j (s)x L j 2 (0,1;X), 1 1, 2 <, (4.2) j=1 the ineuality (4.1) holds for all [1, ) if it holds for some [1, ). (2) If an operator family T L(L r ω(σ)), 1 < r <, ω A r (R n 1 ), is R- bounded, then R 1 (T ) CR 2 (T ) for all 1, 2 [1, ) with a constant C = C( 1, 2 ) > 0 independent of ω. In fact, introducing the isometric isomorphism I ω : L r ω(σ) L r (Σ), I ω f = fω 1/r, for all T L(L r ω(σ)) we have T ω = I ω T Iω 1 L(L r (Σ)) and T L(L r ω (Σ)) = T ω L(L r (Σ)). Then it is easily seen that T ω := {I ω T Iω 1 : T T } L(L r (Σ)) is R-bounded and R ( T ω ) = R (T ) for all [1, ). Thus the assertion follows. Definition 4.4 (1) Let X be a Banach space and (x n ) n=1 X. A series n=1 x n is called unconditionally convergent if n=1 x σ(n) is convergent in norm for every permutation σ : N N. (2) A seuence of projections ( j ) j N L(X) is called a Schauder decomposition of a Banach space X if i j = 0 for all i j, j x = x for each x X. j=1 A Schauder decomposition ( j ) j N is called unconditional if the series j=1 jx converges unconditionally for each x X. Remark 4.5 (1) If ( j ) j N is an unconditional Schauder decomposition of a Banach space Y, then for each p [1, ) there is a constant c = c (p) > 0 such that for all x j in the range R( j ) of j the ineualities c 1 k Y x j j=l k L k Y ε j (s)x j c x j (4.3) p (0,1;Y ) j=l j=l are valid for any seuence (ε j (s)) of independent, symmetric { 1, 1}-valued random variables defined on (0, 1) and for all l k Z, see e.g. [4], (3.8). (2) Let Y = L (R; L r ω(σ)) and assume that each j commutes with the isomorphism I ω introduced in Remark 4.3 (2). Then the constant c is easily seen to be independent of the weight ω. (3) In the previous definitions and results the set of indices N may be replaced by Z without any further changes. (4) Let X be a UMD space and χ [a,b) denote the characteristic function for the interval [a, b). Let R s = F 1 χ [s, ) F and j := R 2 j R 2 j+1, j Z. 23

24 It is well known that the Riesz projection R 0 is bounded in L (R; X) and that the set {R s R t : s, t R} is R-bounded in L(L (R; X)) for each (1, ). In particular, { j : j Z} is R-bounded in L(L (R; X)) and an unconditional Schauder decomposition of R 0 L (R; X), the image of L (R; X) by the Riesz projection R 0, see [4], proof of Theorem We recall an operator-valued Fourier multiplier theorem in Banach spaces. Let D 0 (R; X) denote the set of C -functions f : R X with compact support in R. Theorem 4.6 ([4], Theorem 3.19, [36], Theorem 3.4) Let X and Y be UMD spaces and 1 < <. Let M : R L(X, Y ) be a differentiable function such that Then the operator R ( {M(t), tm (t) : t R } ) A. T f = ( M( ) ˆf( ) ), f D0 (X), extends to a bounded operator T : L (R; X) L (R; Y ) with operator norm T L(L (R;X);L (R;Y )) CA where C > 0 depends only on, X and Y. Remark 4.7 Checking the proof of [4], Theorem 3.19, one can see that the constant C in Theorem 4.6 euals C = R(P) (c ) 2 where R(P) is the R-bound of the operator family P = {R s R t : s, t R} in L(L (R; X)) and c is the unconditional constant of the Schauder decomposition { j : j Z} of the space R 0 L (R; X); see [4], Section 3, for details. In particular, for X = L r ω(σ), 1 < r <, ω A r, using the isometry I ω of Remark 4.3 (2), we get that the constants R(P), see Remark 4.3 (2), and c do not depend on the weight ω; concerning c we again use that I ω commutes with each j. Theorem 4.8 (Extrapolation Theorem) Let 1 < r, s <, ω A r (R n 1 ) and Σ R n 1 be an open set. Moreover let T be a family of linear operators with the property that there exists an A s -consistent constant C T = C T (A s (ν)) > 0 such that for all ν A s T f s,ν C T f s,ν for all T T and all f L s ν(σ). Then every T T can be extended to L r ω(σ) and T is R-bounded in L(L r ω(σ)) with an A r -consistent R-bound c T (, r, A r (ω)), i.e., R (T ) c T (, r, A r (ω)) for all (1, ). (4.4) Proof: From the proof of [16], Theorem 4.3, it can be deduced that T is R-bounded in L(L r ω(σ)) and that (4.4) is satisfied for = r. Then, Remark 4.3 yields (4.4) for every 1 < <. Now we are in a position to prove Theorem

25 Proof of Theorem 2.1: Let f(x, x n ) := e βxn F (x, x n ) for (x, x n ) Σ R and let us define u, p in the cylinder Ω = Σ R by u(x) = F 1 (a 1 ˆf)(x), p(x) = F 1 (b 1 ˆf)(x), where a 1, b 1 are the operator-valued multiplier functions defined in (3.33). We will show that (U, P ) = (e βxn u, e βxn p) is the uniue solution to (R λ ) with g = 0 such that (u, p) ( Wω 2;,r (Ω) W 1;,r 0,ω (Ω) ) Ŵ ω 1;,r (Ω) (4.5) and the estimate (2.1) holds. Obviously, (U, P ) solves the resolvent problem (R λ ) with g = 0. For ξ R define m λ (ξ) : L r ω(σ) L r ω(σ) by m λ (ξ)f := ( (λ + α)a 1 (ξ) ˆf, ξ a 1 (ξ) ˆf, 2 a 1 (ξ) ˆf, ξ 2 a 1 (ξ) ˆf, b 1 (ξ) ˆf, ξb 1 (ξ) ˆf ). Theorem 3.8 and Corollary 3.9 show that the operator family {m λ (ξ), ξm λ (ξ) : ξ R } satisfies the assumptions of Theorem 4.8, e.g., with s = r. Therefore, this operator family is R-bounded in L(L r ω(σ)); to be more precise, R ( {mλ (ξ), ξm λ(ξ) : ξ R } ) c(, r, α, β, ε, Σ, A r (ω)) <. Hence Theorem 4.6 and Remark 4.7 imply that (m λ ˆf) L (L r ω ) C f L (L r ω ) with an A r -consistent constant C = C(, r, α, β, ε, Σ, A r (ω)) > 0 independent of the resolvent parameter λ α + S ε. Note that, due to the definition of the multiplier m λ (ξ), we have (λ + α)u, 2 u, p L (L r ω) and (λ + α)u, 2 u, p L (L r ω) (m λ ˆf) L (L r ω). Thus the existence of a solution satisfying (2.1) is proved. The uniueness of solutions is obvious by the uniueness result for β = 0 of [12], Theorem 2.1. Now the proof of Theorem 2.1 is complete. Proof of Corollary 2.2: Defining the Stokes operator A = A,r;β,ω by (2.2), due to the Helmholtz decomposition of the space L β (Lr ω) on the cylinder Ω, see [9], we get that for F L β (Lr ω) σ the solvability of the euation (λ + A)U = F in L β (Lr ω) σ (4.6) is euivalent to the solvability of (R λ ) with right-hand side G 0. By virtue of Theorem 2.1 for every λ α + S ε there exists a uniue solution U = (λ + A) 1 F D(A) to (4.6) satisfying the estimate (λ + α)u L β (Lr ω )σ = (λ + α)u L (L r ω) C f L (L r ω) = C F L β (Lr ω )σ with C = C(, r, α, β, ε, Σ, A r (ω)) independent of λ, where u = e βxn U, f = e βxn F. Hence (2.3) is proved. Then (2.4) is a direct conseuence of (2.3) using semigroup theory. 25

26 Proof of Theorem 2.3: The proof will be done if we show that the operator family T = {λ(λ + A,r;β,ω ) 1 : λ ir} is R-bounded in L(L β (Lr ω) σ ). By the way, since L β (Lr ω) σ is isomorphic to a closed subspace X of L (L r ω) with isomorphism I β F := e βxn F, it is enough to show R- boundedness of T = {I β λ(λ + A,r;β,ω ) 1 I 1 β In the following we write shortly : λ ir} L(X). H β I β λ(λ + A,r;β,ω ) 1 I 1 β. For ξ R and λ S ε, let m λ (ξ) := λa 1 (ξ) where a 1 (ξ) is the solution operator for (R λ,ξ,β ) with g = 0 defined by (3.33). Then, we have H β f = I β λu = λi β U = (m λ (ξ) ˆf), f S(R; L r ω(σ)) X, with U the solution to (R λ ) with F = I 1 β f, G = 0. Note that S(R; Lr ω(σ)) is dense in L (R; L r ω(σ)) and hence S(R; L r ω(σ)) X is dense in X. Hence, in view of Definition 4.2 and Remark 4.3, R-boundedness of T in L(X) is proved if there is a constant C > 0 such that N ε i (m λi ˆfi ) C N L ε (0,1;L (R:L r i f L i (4.7) ω(σ))) (0,1;L (R:L r ω(σ))) i=1 for any independent, symmetric and { 1, 1}-valued random variables (ε i (s)) defined on (0, 1), for all (λ i ) ir and (f i ) S(R; L r ω(σ)) X. Without loss of generality we may assume that supp ˆf i [0, ), i = 1,..., N, since R 0 f := (χ [0, ) (ξ) ˆf) is continuous in L (R; L r ω(σ)) and f i (x, x n ) = (χ [0, ) ˆfi (ξ)) (x, x n ) + (χ [0, ) ˆfi ( ξ)) (x, x n ). Note that, if supp ˆf [0, ), then supp(m λ ˆf) [0, ) as well. Therefore, instead of (4.7) we shall prove the estimate i=1 N ε i (m λi ˆfi ) C N L ε (0,1;Y ) i f L i (4.8) (0,1;Y ) i=1 for (f i ) S(R; L r ω(σ)) X Y. Obviously m λ (ξ) = m λ (2 j ) + ξ m 2 j λ (τ) dτ for ξ [2j, 2 j+1 ), j Z, and ( mλ (2 j ) j f ) = mλ (2 j ) j f for f S(R; L r ω(σ)) X Y. Furthermore, ( ξ m λ(τ) dτ ) ( 2 j+1 j f(ξ) = m λ(τ)χ [2 j,ξ)(τ) j f(ξ) dτ 2 j 2 ( 1 j = 2 j m λ(2 j (1 + t))χ [2 j,ξ)(2 j (1 + t))χ [2 j,2 j+1 )(ξ) ˆf(ξ) dt = j m λ(2 j (1 + t))b j,t j f dt. 26 i=1 ) )