数理解析研究所講究録 1235 巻 2001 年 132-159

132
ON ARESOLVENT ESTIMATE OF THE INTERFACE PROBLEM
FOR THE STOKES SYSTEM IN ABOUNDED DOMAIN良弘) \dag er

YOSHIHIRO SHIBATA (柴田Department of Mathematical Sciences, School of Science and Engineering,
Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan.
-mail address : yshibata@mn.waseda.ac.jp

Senjo SHIMIZU (清水 扇丈) \d ag er
Faculty of Engineering, Shizuoka University
Hamamatsu, Shizuoka 432-8561, Japane-mail address : tssshim@eng.shizuoka.ac.jp\S 1. Introduction,,
andbe bounded domains in Rn, $n\geqq 2$ ,, and
. We assume that and belong to .
is the unit outer normal to the boundary I1 of
andis the unit outer normal
to the boundaryof Q.
In this paper we consider the generalized Stokes resolvent problem in abounded
and with Dirichlet condition
domain with interface condition on the interface
on the boundary :Let$\Omega=\Omega^{1}\cup\Omega^{2}\cup\Gamma^{1}$$\Gamma^{1}\cup\Gamma^{2}=\emptyset$$\Gamma^{1}\cup\Gamma^{2}=\partial\Omega^{2}$$\Gamma^{1}=\partial\Omega^{1}$$\Omega^{2}$$\Omega^{1}$$C^{3}$$\Gamma^{2}$$\Gamma^{1}$$\nu^{1}$$\nu^{2}$$\Omega^{1}$$\Gamma^{2}$$\Gamma^{1}$$\Gamma^{2}$$\lambda u^{\ell}-\mathrm{D}\mathrm{i}\mathrm{v}T^{\ell}(u^{\ell}, \pi^{\ell})=f^{\ell}$

(1.1)

$\{$

$\nu^{1}\cdot$$T^{1}(u^{1}, \pi^{1})-\nu^{1}\cdot$,$\nabla\cdot$in$u^{\ell}=0$$T^{2}(u^{2},\pi^{2})=h^{1}-h^{2}$,$u^{1}=u^{2}$$u^{2}=0$

where
are unknown velocities inpressures in $\Omega^{\ell}(\ell=1,2)$ ,$(\ell=1,2)$ , defined by$u^{\ell}=$$(u_{1}^{\ell}, \cdots, u_{n}^{\ell})$$D_{jk}(u^{\ell})= \frac{1}{2}(\frac{\partial u_{j}^{\ell}}{\partial x_{k}}+\frac{\partial u_{k}^{\ell}}{\partial x_{j}})$,$\delta_{jk}=\{$$\Gamma^{1}$$\Gamma^{2}$,$\pi^{\ell}$$\Omega^{\ell}$$T_{jk}^{\ell}(u^{\ell}, \pi^{\ell})=2\mu\ell D_{jk}(u^{\ell})-\delta_{jk}\pi^{\ell}$

where

on
on

, $\ell=1,2$ ,,,are unknown
are the stress tensors in$\Omega^{\ell}(\ell=1,2)$$T^{\ell}(u^{\ell}, \pi^{\ell})=(T_{jk}^{\ell}(u^{\ell}, \pi^{\ell}))$$\Omega^{\ell}$

10,$j=k$ ,
$j\neq k$,

supported by Grant-in-Aid for Scientific Research (B) -12440055, Ministry of Education, Sciences, Sports and Culture, Japan.
tPartly supported by Grants in-Aid for Encouragement of Young Scientists (A) -12740088,
Ministry of Education, Science, Sports and Culture, Japan.
$MOS$ Subject Classification:.
-estimate, Interface problem, Bounded domain
Keywords: Stokes resolvent,$\uparrow \mathrm{P}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{l}\mathrm{y}$$35\mathrm{J}25,\mathit{7}\mathit{6}B\mathit{0}\mathit{3},42\mathrm{B}15$$L_{p}$

133
and $\mu_{\ell}(\ell=1,2)$ are viscous coefficients. Let
and I denote thematrices
whose $(j, k)$ components are, respectively. If we use the symbols
andand , then$D(u^{\ell})$$D_{jk}(u^{\ell})$$D(u^{\ell})$$n\cross n$$\delta_{jk}$$I$$T^{\ell}(u^{\ell}, \pi^{\ell})=2\mu\ell D(u^{\ell})-\pi^{\ell}I$.

The resolvent parameter Ais contained in the sectorial domain:$\Sigma_{\epsilon}=$$f^{\ell}=${A 6$\mathbb{C}|$$(f_{1}^{\ell}, \cdots, f_{n}^{\ell})(\ell=1,2)$$(\ell=1,2)$A$\neq 0$,$|\arg\lambda|\leqq\pi-\epsilon$},$0<\epsilon<\pi/2$

are the prescribed external forces,

are the prescribed boundary forces, where.

$f^{\ell}(x)$and.

$h^{\ell}=(h_{1}^{\ell}, \cdots, h_{n}^{\ell})$are defined at$h^{\ell}(x)$$x\in\Omega^{\ell}(\ell=1,2)$

We use the following symbols:$u^{1}(x)$$x\in\Omega^{1}$$u(x)=\{$$u^{2}(x)$$x\in\Omega^{2}$$f^{1}(x)$$x\in\Omega^{1}$,,$\pi(x)=\{$,,$h(x)=\{$$f(x)=\{$$f^{2}(x)$$x\in\Omega^{2}$,,$\pi^{1}(x)$$x\in\Omega^{1}$$\pi^{2}(x)$$x\in\Omega^{2}$$h^{1}(x)$$x\in\Omega^{1}$,$h^{2}(x)$$x\in\Omega^{2}$.

We are interested inestimates of the unknown velocities
and the pressures$\pi^{\ell}(\ell=1,2)$ .
We define the space
for the pressureby:$u^{\ell}$$L_{p}$$\pi^{\ell}$$\tilde{W}_{p}^{1}(\Omega)$

(1.2)

,

$\tilde{W}_{p}^{1}(\Omega)=\{\pi\in L_{p}(\Omega)|\int_{\Omega}\pi dx=0, \nabla\pi^{\ell}\in L_{p}(\Omega^{\ell}), \ell=1,2\}$$|| \pi||_{\overline{W}_{p}^{1}(\Omega)}=\sum_{\ell=1}^{2}||\pi^{\ell}||_{W_{p}^{1}(\Omega^{\ell})}$.

Our main result is stated in the following theorem.
Theorem 1.1. Let $1<p<\infty$ and $0<\epsilon<\pi/2$ . There exists a $\sigma>0$ such thatthe following assertion holds: For ever$ry$ A,,, $(1,1)$ admits a unique solution
withwhich satisfies the estimate:$\in\Sigma_{\epsilon}\cup\{\lambda\in \mathbb{C}||\lambda|\leqq\sigma\}$$h^{\ell}\in W_{p}^{1}(\Omega^{\ell})^{n}$$f\in L_{p}(\Omega)^{n}$$(u, \pi)\in W_{p}^{1}(\Omega)\cross\tilde{W}_{p}^{1}(\Omega)$$u^{\ell}\in$$W_{p}^{2}(\Omega^{\ell})^{n}$(1.3)$| \lambda|||u||_{L_{p}(\Omega)}+|\lambda\int^{\frac{1}{2}}||\nabla u||_{L_{p}(\Omega)}+\sum_{\ell=1}^{2}||u^{\ell}||_{W_{p}^{2}(\Omega^{\ell})}+||\pi||_{\overline{W}_{p}^{1}(\Omega)}$$\leqq C(||f||_{L_{p}(\Omega)}+|\lambda|^{\frac{1}{2}}||h||_{L_{p}(\Omega)}+\sum_{\ell=1}^{2}||h^{\ell}||_{W_{p}^{1}(\Omega^{\ell}))}$

for some constant$C$depending essentially only on ,

$n$

$p$

, ,

$\Omega$$\epsilon$

,

and .
$\sigma$, thenorm of is defined in the following way: Let$in W_{p}^{2}(\Omega)$ be asolution to the Neumann problem for $(-\Delta+1)$ in
:

Given

$\varphi\in L_{p}(\Omega)$$W_{p}^{-1}(\Omega)$$\varphi$$\Omega$(1.4)$(-\Delta+1)\Phi=\varphi$

in$\Omega$,$\frac{\partial\Phi}{\partial\nu}|_{\Gamma^{2}}=0$

which is uniquely solvable. Put(1.5)$||\varphi||_{W_{p}^{-1}(\Omega)}=||\nabla\Phi||_{L_{p}(\Omega)}$

The following theorem is akey of our argument

.

,

134

.

. Then there exists a positive
andTheorem 1.2. Let
1 depending only on p, n, e, and C such that for every A e
constant, \yen with, fE
and h’E
with |A|satisfy (1.1), thenu’ cE
$0<\mathrm{e}$$1<p<\mathrm{o}\mathrm{o}$$<\mathrm{x}/2$$\ovalbox{\tt\small REJECT} \mathrm{E}^{\ovalbox{\tt\small REJECT}}$$\mathrm{A}_{0}\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$\mathrm{A}_{0}$$L_{p}(\mathrm{O})^{\mathrm{n}}\rangle$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT} 3(\mathrm{O}^{\mathrm{t}})^{n}$$i\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}}\mathrm{p}\ovalbox{\tt\small REJECT}(\mathrm{O}’)$(1.6)$| \lambda|||u||_{L_{p}(\Omega)}+|\lambda|^{\frac{1}{2}}||\nabla u||_{L_{p}(\Omega)}+\sum_{\ell=1}^{2}||u^{\ell}||_{W_{p}^{2}(\Omega^{p})}+||\pi||_{\overline{W}_{p}^{1}(\Omega)}$$\leqq C(||f||_{L_{p}(\Omega)}+|\lambda|^{\frac{1}{2}}||h||_{L_{p}(\Omega)}+\sum_{\ell=1}^{2}||h||_{W_{p}^{1}(\Omega)}$$+|| \pi||_{L_{p}(\Omega)}+|\lambda|||u||_{W_{p}^{-1}(\Omega)}+|\lambda|^{\frac{1}{2}}||u||_{L_{p}(\Omega)}+\sum_{\ell=1}^{2}||\nabla u^{\ell}||_{L_{p}(\Omega^{p}))}$

where positive constant $C$ depends essentially only on ,
$p$

$n$

,

$\epsilon$,

and O.

We shall prove Theorem 1.2 by using the finite number of the partition of unity
and reducing (1.1) to the whole space problem, the half space Dirichilet problem,
and the interface problem with interface $x_{n}=0$ in the whole space. Since we use.
the cut off function , divergence free condition is broken such as
In order to reduce the problem to the divergence ffee case, we use asolution to
the Neumann problem for $(-\Delta+1)$ like (1.4). After this reduction, we solve thewhole space problem, the half space Dirichlet problem, and the interface problem byusing the Fourier transform. Applying the Fourier multiplier theorem to estimate
the solutions to such model problems and using the standard argument, we canprove Theorem 1.2. Once getting Theorem 1.2, we can prove Theorem 1.1 byusing the standard argument based on Banach’s closed range theorem and compact
perturbation method. Our idea is based on Farwig and Sohr [5] where they treatedthe Stokes resolvent problem with Dirichlet zero condition, and Shibata and Shimizu[8] where we treated the Stokes resolvent problem with Neumann condition.
Our problem is the one of the first step to consider aproblem with ffee boundary. Giga and Takahashi [7] constructed global weak solutions of the two phase
Stokes system, and Takahashi [9] constructed global weak solutions of the tw0phase Navier-Stokes system with inhomogeneous Dirichilet condiditon. Denisova[1] and Denisova and Solonnikov $[2, 3]$ investigated of the motion of two liquids inthe framework of the H\"older function space. We also refer to Tani [10], he studied
two hase problems for compressible viscous fluid motion in the ffamework of the
H\"older function space.
Throughout the paper we use the following symbols.
$\nabla\cdot(\varphi u)=(\nabla\varphi)\cdot u$$\varphi$$L_{p}( \Omega)^{n}=\{u=(u_{1}, \cdots, u_{n})|||u||_{L_{p}(\Omega)}=\sum_{j=1}^{n}||u_{j}||_{L_{p}(\Omega)}<\infty\}$$W_{p}^{k}( \Omega)=\{\pi\in L_{p}(\Omega)|||\pi||_{W_{p}^{k}(\Omega)}=\sum_{|\alpha|\leqq k}||\partial_{x}^{\alpha}\pi||_{L_{p}(\Omega)}<\infty\}$$W_{p}^{k}(\Omega)^{n}=\{u=(u_{1},$\cdots ,$u_{n})$

|

;

;

$||u||_{W_{p}^{k}(\Omega)}= \sum_{j=1}^{n}||u_{j}||_{W_{p}^{k}(\Omega)}<\infty\}$;135
for scalor valued , ;$( \pi, \theta)_{\Omega}=\int_{\Omega}\pi(x)\overline{\theta(x)}dx$$\pi$$(u, v)_{\Omega}=\mathrm{I}$ $\int_{\Omega}u_{j}(x)\overline{v_{j}(x)}dx$$\theta$for $u=(u_{1}, \cdots, u_{n})$ , $v=(v_{1}, \cdots, v_{n})$ ,being the surface element of$<u$ , $v>_{\Gamma^{\ell}}=\mathrm{I}$Q2. Weak Solutions in$L_{2}$$\Gamma^{\ell},\ell=1,2$.

Framework

In this section we investigate the weak solutions (1.1). We introduce the following
spaces:(2.1)$H_{0}^{1}(\Omega)=\{u\in W_{2}^{1}(\Omega)^{n}|u|_{\Gamma^{2}}=0\}$$D_{0}^{1}(\Omega)=${$u\in H_{0}^{1}(\Omega)^{n}|\nabla\cdot u=0$,in$\Omega$}.

By integration by parts, we have
(2.2)$(\lambda u-\mathrm{D}\mathrm{i}\mathrm{v}T(u, \pi),$$v)_{\Omega}+<\nu^{1}\cdot T^{1}(u^{1}, \pi^{1})-\nu^{1}\cdot T^{2}(u^{2}, \pi^{2})$$= \lambda(u, v)_{\Omega}+2\sum_{\ell=1}^{2}\mu^{\ell}(D(u^{\ell}), D(v^{\ell}))_{\Omega^{\ell}}-(\pi, \nabla\cdot v)_{\Omega}$

for any solution $(u, \pi)$ of (1.1) and$\langle D(u^{\ell})$,$v\in H^{1}(\Omega)^{n}$, where$D(v^{\ell}))_{\Omega^{\ell}}= \sum_{j,k=1}^{n}(D_{jk}(u^{\ell}), D_{jk}(v^{\ell}))_{\Omega^{\ell}}$.

In view of (2.2), we put
(2.3)

for ,
$u$

$B_{\lambda}[u, v]= \lambda(u, v)_{\Omega}+2\sum_{\ell=1}^{2}\mu^{\ell}(D(u^{\ell}), D(v^{\ell}))_{\Omega^{\ell}}$$v\in H_{0}^{1}(\Omega)$. Using the 1st Korn’s inequality (cf. [4]), we have

(2.4)

for every$||u||_{W_{2}^{1}(\Omega)}^{2}\leqq C(\Omega)[|D(u)||_{L_{2}(\Omega)}$$u\in H_{0}^{1}(\Omega)$with suitable constant $C(\Omega)>0$ , where$||u||_{W_{2}^{1}(\Omega)}^{2}=||u||_{L_{2}(\Omega)}^{2}+||\nabla u||_{L_{2}(\Omega)}^{2}$.

Employing the standard argument, we have the following lemma,$v>_{\Gamma^{1}}$136. Thenis a coercive bilinear form on
Lemma 2.1. Let $0<\epsilon<\pi/2$ and A$C=C(\epsilon,\Omega)>0$ such that
. In particular, there exists a constant$B_{\lambda}$$\in\Sigma_{\epsilon}$$H_{0}^{1}(\Omega)$(2.5)$|B_{\lambda}[u, u]|\geqq C(|\lambda|||u||_{L_{2}(\Omega)}^{2}+||\nabla u||_{L_{2}(\Omega)}^{2})$for every A$\in\Sigma_{\epsilon}$and u$\in H_{0}^{1}(\Omega)$

If we take $\sigma>0$ such as,
withA$\in \mathbb{C}$.

$\sigma C(\Omega)\leqq\min(\mu^{1}, \mu^{2})$, then by (2.4), we have for any$|\lambda|\leqq\sigma$(2.6)$|B_{\lambda}[u, u]| \geqq 2\sum_{\ell}^{2}\mu^{\ell}||D(u^{\ell})||_{L_{2}(\Omega^{p})}^{2}-|\lambda|||u||_{L_{2}(\Omega)}^{2}$$\geqq 2\min(\mu^{1}, \mu^{2})||D(u)||_{L_{2}(\Omega)}^{2}-|\lambda|C(\Omega)||D(u)||_{L_{2}(\Omega)}^{2}$$\geqq(2\min(\mu^{1}, \mu^{2})-\sigma C(\Omega))||D(u)||_{L_{2}(\Omega)}^{2}$$\geqq\min(\mu^{1}, \mu^{2})||D(u)||_{L_{2}(\Omega)}^{2}$$\geqq C(\Omega)\min(\mu^{1}, \mu^{2})||u||_{W_{2}^{1}(\Omega)}^{2}$for$\forall u\in H_{0}^{1}(\Omega)$.

By Lemma 2.1 and (2.6), we have

Lemma 2.2.

$T/iere$ exist $\sigma=\sigma(\Omega, \epsilon)>0$(2.7)

and $C=C(\Omega, \epsilon)>0$ such that$|B_{\lambda}[u,u]|\geqq C(|\lambda|||u||_{L_{2}(\Omega)}^{2}+||u||_{W_{2}^{1}(\Omega)}^{2})$for every$\lambda\in\Sigma_{\epsilon}\cup\{\lambda\in \mathbb{C}||\lambda|\leqq\sigma\}$and$u\in H_{0}^{1}(\Omega)$.

By Lemma 2.2 and the Lax-Milgram theorem (cf. [11, III.7]), we have the following theorem.
Lemma 2.3. Let $0<\epsilon<\pi/2$ . There exists a constant $\sigma>0$ such that for every,, there eists a unique,Asatisfying the variational equation:$\in\Sigma_{\epsilon}\cup\{\lambda\in \mathbb{C}||\lambda|\leqq\sigma\}$$f\in \mathrm{L}2(\mathrm{f}\mathrm{i})$$h^{\ell}\in W_{2}^{1}(\Omega^{\ell})$$u\in D_{0}^{1}(\Omega)$(2.8)$\lambda(u, v)_{\Omega}+2\sum_{\ell=1}^{2}\mu^{\ell}(D(u^{\ell}), D(v^{\ell}))_{\Omega^{p}}$$=(f, v)_{\Omega}+<h^{1}-h^{2}$,$v>_{\Gamma^{1}}$for$\forall v\in D_{0}^{1}(\Omega)$.

Concerning the existence of the pressure, we know the following lemma (cf. [6,III, Theorem 5.2]):, then there exists
andLemma 2.4. If
for any
such that$\mathcal{F}$$\in H_{0}^{1}(\Omega)^{*}$$v\in D_{0}^{1}(\Omega)$$\mathcal{F}(v)=0$$a$$p\in\hat{L}_{2}(\Omega)$

(2.9)

where

$vdx$

$\mathcal{F}(v)=\int p\overline{\nabla}\cdot$$H_{0}^{1}(\Omega)^{*}$

is the dual space of$H_{0}^{1}(\Omega)$for$\forall v\in H_{0}^{1}(\Omega)$,and$\hat{L}_{2}(\Omega)=\{v\in L_{2}(\Omega)|\int_{\Omega}vdx=0\}$.

Combining Lemma 2.3 and Lemma 2.4, we have the main theorem in this section

137
Theorem 2.5. Let $0<\epsilon<\pi/2$ . There exists some positive constant, $f\in L_{2}(\Omega)$ ,,such that for every
there exisJ a uniquewith $\int_{\Omega}\pi dx=0$ which satisfies thevariational equation:$\sigma=\sigma(\Omega, \epsilon)>$$0$$h^{\ell}\in W_{2}^{1}(\Omega^{\ell})(\ell=1,2)$$\lambda\in\Sigma_{\epsilon}\cup\{\lambda\in \mathbb{C}||\lambda|\leqq\sigma\}$$(u, \pi)\in D_{0}^{1}(\Omega)\cross L_{2}(\Omega)$(2.10)$\lambda(u, v)_{\Omega}+2\sum_{\ell=1}^{2}\mu^{\ell}(D(u^{\ell}), D(v^{\ell}))_{\Omega^{p}}-(\pi, \nabla$$=(f, v)_{\Omega}+<h^{1}-h^{2}$

Proof. Let$u\in D_{0}^{1}(\Omega)$,$v>_{\Gamma^{1}}$for$\forall v\in H_{0}^{1}(\Omega)$.

$v)_{\Omega}$.

be asolution to (2.8). If we put$\mathcal{F}(v)=\lambda(u, v)_{\Omega}+2\sum_{\ell=1}^{2}\mu^{\ell}(D(u^{\ell}), D(v^{\ell}))_{\Omega^{\ell-}}(f, v)_{\Omega}-<h^{1}-h^{2}$, thenforLemma 2.4, there exists a$v\in H_{0}^{1}(\Omega)$

and $\mathcal{F}(v)=0$ for any
such that$\mathcal{F}\in H_{0}^{1}(\Omega)^{*}$$v\in D_{0}^{1}(\Omega)$,$v>_{\Gamma^{1}}$. Therefore by$\pi\in\hat{L}_{2}(\Omega)$$\mathcal{F}(v)=\int_{\Omega}\pi\overline{\nabla\cdot v}dx=(\pi, \nabla v)_{\Omega}$,which implies (2.10). This completes the proof the theorem.

$\square$\S 3. Resolvent estimates for the Stokes System in the whole space
In this section, we consider the Cattabriga problem:(3.1)$\lambda u-\mathrm{D}\mathrm{i}\mathrm{v}T(u, \pi)=f$,$\nabla$.u

As the class of the pressure , we set for any D
$\pi$

(3.2)
(3.3)

$=\nabla$.g in$\subseteq \mathbb{R}^{n}$,$\mathbb{R}^{n}$.

$\{\pi\in Ln(D)\overline{n}\vec{-\overline{p}}|\nabla\pi\in L_{p}(D)\}$$1<p<n$ ,$\{\pi\in L_{p,loc}(D)|\nabla\pi\in L_{p}(D)\}$$n\leqq p<\infty$$\hat{W}_{p}^{1}(D)=\{$$||\nabla\pi||_{L_{p}(D)}+||\pi||_{L(D)\overline{n},-\overline{\mathrm{p}}}n\mu$.

$1<p<n$ ,$||\pi||_{\hat{W}_{p}^{1}(D)}=\{$$n\leqq p<\infty$$||\nabla\pi||_{L_{p}(D)}$.

.
We note thatwith norm
is aclosure of
We shall show the uniqueness, existence and estimate of solutions to (3.1) (cf.
Shibata-Shimizu [8, Theorem 3.4] .
$\hat{W}_{p}^{1}(D)$$C_{0}^{\infty}(D)$$||\cdot$$||_{\hat{W}_{p}^{1}(D)}$$)$

Theorem 3.1. Let $1<p<\infty$ and $0<\epsilon<\pi/2$ .
(1) (Existence and Estimate) For everthere exists a solution,
and A6of (3.1) satisfying the estimate:$ryf\in L_{p}(\mathbb{R}^{n})^{n}$$(u, \pi)\in W_{p}^{2}(\mathbb{R}^{n})^{n}\cross\hat{W}_{p}^{1}(\mathbb{R}^{n})$(3.4)$g\in W_{p}^{2}(\mathbb{R}^{n})^{n}$$\Sigma_{\epsilon}$$|\lambda|||u||_{L_{p}(\mathbb{R}^{n})}+|\lambda|^{\frac{1}{2}}||\nabla u||_{L_{p}(\mathbb{R}^{n})}+||\nabla^{2}u||_{L_{p}(\mathbb{R}^{n})}$$+||\nabla\pi||_{L_{p}(\mathbb{R}^{n})}+||\pi(d_{p})^{-1}||_{L_{p}(\mathbb{R}^{n})}$$\leqq C(p,\epsilon, n)(||f||_{L_{p}(\mathrm{R}^{n})}+|\lambda|||g||_{L_{p}(\mathbb{R}^{n})}+|\lambda|^{\frac{1}{2}}||\nabla g||_{L_{p}(\mathbb{R}^{n})}+||\nabla^{2}g||_{L_{p}(\mathbb{R}^{n})})$

,

138

$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}"\ovalbox{\tt\small REJECT}${$(2+|x|)\log(2+|\mathrm{r}|)\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$1<p<\mathrm{o}\mathrm{o}$,$p^{\ovalbox{\tt\small REJECT}}$

Moreover, when $1<p<n$ ,(3.5)

when $pi^{-n}$ ,whenn.

$2+|\mathrm{r}|$$\pi\in L_{np/(n-p)}(\mathbb{R}^{n})$and$||\pi||_{L_{np/(n-p)}(\mathbb{R}^{n})}$$\leqq C(n,p,\epsilon)(||f||_{L_{p}(\mathrm{R}^{n})}+|\lambda|||g||_{L_{p}(\mathrm{R}^{n})}+|\lambda|^{\frac{1}{2}}||\nabla g||_{L_{p}(\mathrm{R}^{n})}+||\nabla^{2}g||_{L_{p}(\mathbb{R}^{n}))}$(2) (Uniqueness) Let A
homogeneous equation:(3.6)$\lambda u$then u $=0$ and$\pi$$\in\Sigma_{\epsilon}$. If uand$\in S’\cap L_{p}(\mathbb{R}^{n})$-Div $T(u, \pi)=0$ ,$\nabla$. u $=0$ in$\pi$$\in D’(\mathbb{R}^{n})$$\mathbb{R}^{n}$.

satisfy the,is a constant. In particular, $\dot{\iota}f\lim|x|arrow\infty\pi(x)=0$, then $\pi=0$ .

In order to get the interior estimate, we will use the following theorem (cf. [8,Theorem 3.5]).
Theorem 3.2. Let $1<p<\infty$ , $0<\epsilon<\pi/2$ andsuch that $\nabla\cdot u=0$ in O. Then, $/or$ every
andsolutionto the equation:$\varphi\in C_{0}^{\infty}(\Omega^{0})$$\lambda\in\Sigma_{\epsilon}$. Let$f\in L_{p}(\mathbb{R}^{n})^{n}$$u\in W_{p}^{1}(\Omega)^{n}$, there exists$(v, \pi)\in W_{p}^{2}(\mathrm{R}^{n})^{n}\cross\hat{W}_{p}^{1}(\mathbb{R}^{n})$(3.7)$\lambda v-\mathrm{D}\mathrm{i}\mathrm{v}T(v, \pi)=f$,$\nabla\cdot$$v=\nabla$. (pu) in$\mathrm{R}^{n}$.

Moreover, the $(v, \pi)$ satisfies the estimate:(3.8)$|\lambda|||v||_{L_{p}(\mathrm{R}^{n})}+|\lambda|^{\frac{1}{2}}||\nabla v||_{L_{p}(\mathrm{R}^{n})}+||\nabla^{2}v||_{L_{p}(\mathrm{R}^{n})}$$+||\nabla\pi||_{L_{p}(\mathrm{R}^{n})}+||\pi(d_{p})^{-1}||_{L_{p}(\mathrm{R}^{n})}\leqq C||f||_{L_{p}(\mathrm{R}^{n})}$$+C_{\varphi}(|\lambda|||u||_{W_{p}^{-1}(\Omega)}+|\lambda|^{1}f||u||_{L_{p}(\Omega)}+||u||_{W_{p}^{1}(\Omega)})$,$||\pi||_{L_{np/(n-p)}(\mathrm{R}^{n})}\leqq C||f||_{L_{p}(\mathrm{R}^{n})}$$+C_{\varphi}(|\lambda|||u||_{W_{p}^{-1}(\Omega)}+|\lambda|^{\}}||u||_{L_{p}(\Omega)}+||u||_{W_{p}^{1}(\Omega)})$with suitable constants $C=C(p,\epsilon, n)$ and $C_{\varphi}=C$ ($p,$$\epsilon$, , , Vp,$n$$\varphi$

if $1<p<n$ ,$\nabla^{2}\varphi$).

\S 4. Resolvent estimates for the Stokes System in the half space
In this section, we consider the following problem:(4.1)$\lambda u-\mathrm{D}\mathrm{i}\mathrm{v}T(u, \pi)=f$$\{$$u|_{x_{n}=0}=0$,$\nabla\cdot u=g$.

where $\mathbb{R}_{+}^{n}=\{x=(x_{1}, \cdots, x_{n})\in \mathbb{R}^{n}|x_{n}>0\}$ .

in$\mathbb{R}_{+}^{n}$

,

$a$

139
As the function class for , we adopt the following space for
$g$

(4.2)

$W_{p}^{-1}(D)$$=\hat{W}_{p}^{1},(D)^{*}$,$1<p<\infty$ ,$D=\mathbb{R}_{+}^{n}$or$D=\mathbb{R}^{n}$$1/p+1/p’=1$ .

Put
(4.3)

$||g||_{W_{p}^{-1}(D)}= \sup\{|<g, v>||v\in\hat{W}_{p}^{1},(D), ||\nabla v||_{L_{p’}(D)}=1\}$

for $g\in W_{p}^{-1}(D)$ . For $g\in L_{p}(D)$ with compact support, we put(4.4)$<g$ , $v>= \int_{D}g(x)\overline{v(x)}dx$

for$\forall v\in\hat{W}_{p}^{1},(D)$.

If there exists aconstant $C(g)>0$ such that
(4.5)$|<g$ , $v>|\leqq C(g)||\nabla v||_{L}$ $p1,(D),$ ’

then $g\in W_{p}^{-1}(D)$ and.
The following theorem was proved by Farwig-Sohr [5, Corollary 2.6].
$||g||_{W_{p}^{-1}(D)}\leqq \mathrm{C}(\mathrm{g})$Theorem 4.1. Let $1<p<\infty$ and $0<\epsilon<\pi/2$ . For every,,
having compact support, (4.1) admits a solution $(u, \pi)\in$
satisfying the estimate:$\lambda\in\Sigma_{\epsilon}$$f\in L_{p}(\mathbb{R}_{+}^{n})^{n}$$g\in W_{p}^{-1}(\mathbb{R}_{+}^{n})\cap W_{p}^{1}(\mathbb{R}_{+}^{n})$$W_{p}^{2}(\mathbb{R}_{+}^{n})^{n}\cross\hat{W}_{p}^{1}(\mathbb{R}_{+}^{n})$$|\lambda|||u||_{L_{p}(\mathbb{R}_{+}^{n})}+|\lambda|^{\frac{1}{2}}||\nabla u||_{L_{p}(\mathbb{R}_{+}^{n})}+||\nabla^{2}u||_{L_{p}(\mathbb{R}_{+}^{n})}+||\pi||_{\hat{W}_{p}^{1}(\mathbb{R}_{+}^{n})}$$\leqq C(p, \epsilon, n)(||f||_{L_{p}(\mathbb{R}_{+}^{n})}+|\lambda|||g||_{W_{p}^{-1}(\mathbb{R}_{+}^{n})}+|\lambda|^{\frac{1}{2}}||g||_{L_{p}(\mathbb{R}_{+}^{n})}+||\nabla g||_{L_{p}(\mathbb{R}_{+}^{n}))}$.

\S 5. Resolvent estimates for the Stokes System with interface condition
Let $\mathbb{R}_{\pm}^{n}=\{x=(x_{1}, \cdots, x_{n})=(x’, x_{n})\in \mathbb{R}^{n}|\pm x_{n}>0\}$ and IQ $=\{x=$. In this section, $\nu=(0, \cdots, 0, -1)$ denotes aunit outernormal of the boundary IQ of.
In this section, we consider the following problem:$(x’, x_{n})\in \mathbb{R}^{n}|x_{n}=0\}$$\mathbb{R}_{+}^{n}$$\lambda u^{\pm}-\mathrm{D}\mathrm{i}\mathrm{v}T^{\pm}(u^{\pm}, \pi^{\pm})=f^{\pm}$

(5.1)

,

in
on

$\nabla\cdot u^{\pm}=g^{\pm}$$\{$$\nu\cdot T^{+}(u^{+}, \pi^{+})-\nu\cdot T^{-}(u^{-}, \pi^{-})=h^{+}-h^{-}$,$u^{+}=u^{-}$$\mathbb{R}_{\pm}^{n}$$\mathfrak{W}$

,

.

where
is agiven function defined on
and $T^{\pm}(u^{\pm}, \pi^{\pm})=2\mu_{\pm}D(u^{\pm})-\pi^{\pm}I$.
As the function class for the pressure , we introduce the following space:$h^{\pm}$$\ovalbox{\tt\small REJECT}$

$\pi$

(5.2)
(4.3)

$X_{p}^{1}(\mathbb{R}_{\pm}^{n})=\{\pi=\Phi+\theta|\Phi\in\hat{W}_{p}^{1}(\mathbb{R}^{n}), \theta\in\tilde{X}_{p}^{1}(\mathbb{R}_{\pm}^{n})\}$$||\pi||_{X_{p}^{1}(\mathbb{R}_{\pm}^{n})}=$

inf,$(||\Phi||_{\hat{W}_{p}^{1}(\mathbb{R}^{n})}+||\theta||_{\overline{X}_{p}^{1}(\mathbb{R}_{\pm}^{n})})$$\Phi\in\hat{W}_{p}^{1}(\mathbb{R}^{n}),\theta\in\tilde{X}_{p}^{1}(\mathbb{R}_{\pm}^{n})\pi=\Phi+\theta$

(5.4)
(5.5)

,

$\tilde{X}_{p}^{1}(\mathrm{R}_{\pm}^{n})=\{\theta\in L_{\infty}(\mathbb{R}_{\pm;}L_{p}(\mathbb{R}^{n-1}))|\nabla\theta\in L_{p}(\mathrm{R}_{\pm}^{n})\}$$|| \theta||_{\tilde{X}_{p}^{1}(\mathbb{R}_{\pm}^{n})}=\sup_{\pm x_{n}>0}||\theta(\cdot, x_{n})||_{L_{p}(\mathbb{R}^{n-1})}+||\nabla\theta||_{L_{p}(\mathbb{R}_{\pm}^{n})}$

.

,

:

140
We use the following symbols:$u^{+}(x)$$x\in \mathbb{R}_{+}^{n}$$u(x)=\{$$u^{-}(x)$$f^{+}(x)$$x\in \mathbb{R}_{+}^{n}$$f(x)=\{$$f^{-}(x)$$x\in \mathbb{R}_{-}^{n}$$x\in \mathbb{R}_{-}^{n}$,$g(x)=\{$,,,$\pi^{+}(x)$$x\in \mathbb{R}_{+}^{n}$$\pi^{-}(x)$$x\in \mathbb{R}_{-}^{n}$$\pi(x)=\{$$g^{+}(x)g^{-}(x)$$x\in \mathbb{R}_{+}^{n}x\in \mathbb{R}_{-}^{n}’$,,,$h^{+}(x)$$x\in \mathbb{R}_{+}^{n}$$h^{-}(x)$$x\in \mathbb{R}_{-}^{n}$$h(x)=\{$,.

The following theorem is the main result in this section.

,,
Theorem 5.1. Let $1<p<\infty$ and $0<\epsilon<\pi/2$ . For every,(5.1)
admitshaving compact support, andsatisfying the estimate:a solution$\lambda\in\Sigma_{\epsilon}$$f\in L_{p}(\mathbb{R}^{n})^{n}$$h^{\pm}\in W_{p}^{1}(\mathbb{R}_{\pm}^{n})^{n}$$g\in W_{p}^{-1}(\mathbb{R}^{n})\cap W_{p}^{1}(\mathbb{R}^{n})$$(u^{\pm}, \pi^{\pm})\in W_{p}^{2}(\mathrm{E})^{n}\cross X_{p}^{1}(\ovalbox{\tt\small REJECT})$(5.6)$| \lambda|||u||_{L_{p}(\mathrm{R}^{n})}+|\lambda|^{1}2||\nabla u||_{L_{p}(\mathrm{R}^{n})}+\sum_{+-}(||\nabla^{2}u^{\pm}||_{L_{p}(\mathbb{R}_{\pm}^{n})}+||\pi^{\pm}||_{X_{p}^{1}(\mathbb{R}_{\pm}^{n}))}$$\leqq C(p, \epsilon,$n)$(||f||_{L_{p}(\mathrm{R}^{n})}+|\lambda|||g||_{W_{p}^{-1}(\mathrm{R}^{n})}+|\lambda|^{\frac{1}{2}}||g||_{L_{p}(\mathrm{R}^{n})}$$+|| \nabla g||_{L_{p}(\mathrm{R}^{n})}+|\lambda|^{1}2||h||_{L_{p}(\mathrm{R}^{n})}+\sum_{+-}||\nabla h^{\pm}||_{L_{p}(\mathbb{R}_{\pm}^{n})})$.

First we have to reduce the problem (5.1) to the divergence ffee case. To do this,we start with the following lemma.
having compact
Lemma 5.2. Let $1<p<\infty$ . For every, which satisfies the$V=g$ insuch that
support, there exists a
estimates:$g\in W_{p}^{-1}(\mathbb{R}^{n})\cap W_{p}^{1}(\mathbb{R}^{n})$$\mathbb{R}^{n}$$\nabla\cdot$$V\in W_{p}^{2}(\mathbb{R}^{n})^{n}$$||V||_{L_{p}(\mathrm{R}^{n})}\leqq C(p, n)||g||_{W_{p}^{-1}(\mathrm{R}^{n})}$$||\nabla V||_{L_{p}(\mathrm{R}^{n})}\leqq C(p, n)||g||_{L_{p}(\mathrm{R}^{n})}$Proof. Let

$E$

,

,

$||\nabla^{2}V||_{L_{p}(\mathrm{R}^{n})}\leqq C(p, n)||\nabla g||_{L_{p}(\mathrm{R}^{n})}$.

be afundamental solution of the Laplace operator given by(5.7)$\log|x|$$n=2$ ,$|x|^{-(n-2)}$$n\geq 3$$E(x)=c_{n}\{$.

. Therefore, if we put $V=\nabla\Phi$ , thenIf we put $\Phi=E*g$ , then $\Delta\Phi=g$ in$\nabla\cdot V=g$ . By the Fourier multiplier theorem, we see easily that$\mathbb{R}^{n}$$||\nabla^{2}\Phi||_{L_{p}(\mathrm{R}^{n})}\leqq C(p, n)||g||_{L_{p}(\mathrm{R}^{n})}$,$||\nabla\nabla^{2}\Phi||_{L_{p}(\mathrm{R}^{n})}\leqq C(p, n)||\nabla g||_{L_{p}(\mathbb{R}^{n})}$Below we shall show that
(5.8)$||\nabla\Phi||_{L_{p}(\mathrm{R}^{n})}\leqq C(p)||g||_{W_{p}^{-1}(\mathrm{R}^{n})}$.

It is sufficient to prove that
(5.9)$|(\nabla\Phi, \psi)_{\mathrm{R}^{n}}|\leqq C(p)||g||_{W_{p}^{-1}(\mathrm{R}^{n})}||\psi||_{L_{p’}(\mathrm{R}^{n})}$.

141
for any$\ovalbox{\tt\small REJECT} p$$(\ovalbox{\tt\small REJECT})(x)\ovalbox{\tt\small REJECT}$$\mathrm{c}_{\ovalbox{\tt\small REJECT}}.(|74(1\mathrm{J}\ovalbox{\tt\small REJECT})^{\mathrm{n}}$V .($\ovalbox{\tt\small REJECT} E*|$. Since$\mathrm{f}\#$is compactly supported, we put. v7 in. Moreover we haveav vtA). Then$\ovalbox{\tt\small REJECT}$(5.10)$\Psi(x)=O(|x|^{-(n-1)})$ ,(5.11)$\Phi(x)=\{$$\mathrm{V}(\mathrm{r})\ovalbox{\tt\small REJECT}$

E

(V .

$\ovalbox{\tt\small REJECT}($$1\ovalbox{\tt\small REJECT}$$O(\log|x|)$$n=2$ ,

$O(|x|^{-(n-2)})$$n\geq 3$,$\nabla\Psi(x)=O(|x|^{-n})$as$|x|arrow\infty$,$\nabla\Phi(x)=O(|x|^{-n})$as$|x|arrow\infty$.

By using (5.10) and (5.11), we have the identity$(\nabla\Phi, \psi)_{\mathbb{R}^{n}}=-(\Phi, \nabla\cdot\psi)_{\mathbb{R}^{n}}=-(\Phi, \Delta\Psi)_{\mathbb{R}^{n}}=-(\Delta\Phi, \Psi)_{\mathbb{R}^{n}}=(g, \Psi)_{\mathrm{R}^{n}}$

Since

,

$g\in W_{p}^{-1}(\mathbb{R}^{n})=\hat{W}_{p}^{1}$$(\mathbb{R}^{n})^{*}$

and

$g$

.

is compactly supported,$|(g, \Psi)_{\mathbb{R}^{n}}|\leqq||g||_{W_{p}^{-1}(\mathbb{R}^{n})}||\nabla\Psi||_{L_{p’}(\mathbb{R}^{n})}$.

By the Fourier multiplier theorem.

$||\nabla\Psi||_{L_{p’}(\mathbb{R}^{n})}\leqq||\nabla^{2}(E*\psi)||_{L_{p’}(\mathbb{R}^{n})}\leqq C(p)||\psi||_{L_{p’}(\mathbb{R}^{n})}$Thus we have (5.9), which completes the proof of the lemma.
Let
be arestriction of $V$ toreduced to$V^{\pm}$$\mathbb{R}_{\pm}^{n}$$\square$. If we put $u^{\pm}=v^{\pm}+V^{\pm}$ , then (5.1) is$\mathrm{i}_{\mathrm{V}}T^{\pm}(v^{\pm}, \pi^{\pm})=f^{\pm}+\mu\pm^{\nabla g^{\pm}-(\lambda}$$-\mu_{\pm}\Delta)V^{\pm}$,in$\nabla\cdot v^{\pm}=0$$\mathrm{E}$$+ \frac{\partial v_{k}^{+}}{\partial x_{n}})|_{x_{n}=0}-\mu-(\frac{\partial v^{-}}{\partial x_{k}}+\frac{\partial v_{k}^{-}}{\partial x_{n}})|_{x}$$n=0$$-h_{k}^{+}- \mu_{+}(\frac{\partial V_{n}^{+}}{\partial x_{k}}+\frac{\partial V_{k}^{+}}{\partial x_{n}})]|_{x_{n}=0}+[$ $h_{k}^{-}+ \mu-(\frac{\partial V^{-}}{\partial x_{k}}+\frac{\partial V_{k}^{-}}{\partial x_{n}})]|_{x_{n}=0}$$k=1$ ,$\{\begin{array}{l}\lambda v^{\pm}-\mathrm{D}\mu_{+}(\frac{\partial v^{+}}{\partial x_{k}}=[(2\mu_{+}\frac{\partial v}{\partial x}=-v^{+}|_{x_{n}=0}\end{array}$$\cdots$,, $n-1$ ,$+$$- \pi^{+})|_{x_{n}=0}-(2\mu_{-}\frac{\partial v^{-}}{\partial x_{n}}-\pi^{-})|_{x}$
$n$

$n=0$
$(h_{n}^{+}+2 \mu_{+}\frac{\partial V^{+}}{\partial x_{n}})|_{x_{n}=0}+(h_{n}^{-}+2\mu$ $- \frac{\partial V^{-}}{\partial x_{n}})|_{x_{n}=0}$$-v^{-}|_{x_{n}=0}=0$,.

Therefore it sufficies to solve$\lambda v^{\pm}-\mathrm{D}\mathrm{i}\mathrm{v}T^{\pm}(v^{\pm}, \pi^{\pm})=f^{\pm}$,$\nabla\cdot v^{\pm}=0$in$\mathrm{E}$,,$\mu_{+}(\frac{\partial v_{n}^{+}}{\partial x_{k}}+\frac{\partial v_{k}^{+}}{\partial x_{n}})|_{x_{n}=0}-\mu_{-}(\frac{\partial v^{-}}{\partial x_{k}}+\frac{\partial v_{k}^{-}}{\partial x_{n}})|_{x_{n}=0}=h_{k}^{+}|_{x_{n}=0}-h_{k}^{-}|_{x_{\mathfrak{n}}=0}$(5.12)$k=1$ ,$\{$$\cdot\cdot’$, $n-1$ ,$(2 \mu_{+}\frac{\partial v^{+}}{\partial x_{n}}-\pi^{+})|_{x_{n}=0}-(2\mu_{-}\frac{\partial v^{-}}{\partial x_{n}}-\pi^{-})|_{x_{n}=0}=h_{n}^{+}|_{x_{n}=0}-h_{n}^{-}|_{x_{n}=0}$$v^{+}|_{x_{n}=0}-v^{-}|_{x_{n}=0}=0$.

In order to prove Theorem 5.1, it sufficies to prove the following theorem

,

,

142
Theorem 5.3. Let $1<p<\infty$ and $0<\epsilon<\pi/2$ . For every, (5.12) admits a solution
andfying the estimate:,$f\in L_{p}^{2}(\mathbb{R}^{n})^{n}$$(u^{\pm}, \pi^{\pm})\in W_{p}^{2}(\mathrm{E})^{n}\cross X_{p}^{1}(\ovalbox{\tt\small REJECT})$$h\in W_{p}^{1}(\mathbb{R}^{n})^{n}$(5.13)$\lambda\in\Sigma_{\epsilon}$satis-$| \lambda|||u||_{L_{p}(\mathbb{R}^{n})}+|\lambda|^{\frac{1}{2}}||\nabla u||_{L_{p}(\mathbb{R}^{n})}+\sum_{+-}(||\nabla^{2}u^{\pm}||_{L_{p}(\mathbb{R}_{\pm}^{n})}+||\pi^{\pm}||_{X_{p}^{1}(\mathbb{R}_{\pm}^{n})})$$\leqq C(p, \epsilon, n)(|||f||_{L_{p}(\mathrm{R}^{n})}+|\lambda|^{z}||h||_{L_{p}(\mathrm{R}^{n})}+\sum_{+-}1||\nabla h^{\pm}||_{L_{p}(\mathbb{R}_{\pm}^{n}))}$

Below, we shall prove Theorem 5.3. Since. Put
assume that

$C_{0}^{\infty}(\mathbb{R}_{\pm}^{n})$is dense in$L_{p}(\mathrm{E})$.

, we may$f^{\pm}\in C_{0}^{\infty}(\mathrm{E})^{n}$$x_{n}>0$,$f_{j}^{+}(x’, -x_{n})x_{n}<0$,$f_{j}^{+}(x’, x_{n})$$f_{j}^{+e}(x)=\{$$f_{n}^{+\mathit{0}}(x)=\{\begin{array}{l}f_{n}^{+}(x’,x_{n})x_{n}>0-f_{n}^{-}(x,,-x_{n})x_{n}<0\end{array}$$f_{n}^{-\mathit{0}}(x)=\{$ $-f_{n}^{+}(x’,’-x_{n})f_{n}^{-}(x,x_{n})x_{n}>0x_{n}<0’$$f_{j}^{-e}(x)=\{\begin{array}{l}f_{j}^{-}(x’,-x_{n})x_{n}>0f_{j}^{-}(x,,x_{n})x_{n}<0\end{array}$where $j=1$ ,(5.14)$\ldots$, $n-1$ . Let$(U^{\pm}, \Phi^{\pm})$,

be asolution to the whole space problem:($\lambda-\mu\pm^{\Delta)U_{j}^{\pm}+\frac{\partial\Phi^{\pm}}{\partial x_{j}}=f_{j}^{\pm e}}$in$\mathbb{R}^{n}$,($\lambda-\mu\pm^{\Delta)U_{n}^{\pm}+\frac{\partial\Phi^{\pm}}{\partial x_{n}}=f_{n}^{\pm \mathit{0}}}$in$\mathrm{R}^{n}$,in$\mathrm{R}^{n}$.

$\nabla\cdot U^{\pm}=0$$j=1$ ,$\cdots$, $n-1$ ,Here we remark that $U_{n}^{\pm}(x’, 0)=0$ as was stated in Farwig-Sohr [5, Proof of, there exists asolution
Theorem 1.3]. By Theorem 3.1, for everyof (5.14) satisfying the estimate:$(U^{\pm}, \Phi^{\pm})\in$$\lambda\in\Sigma_{\epsilon}$$W_{p}^{2}(\mathbb{R}^{n})^{n}\cross\hat{W}_{p}^{1}(\mathbb{R}^{n})$$|\lambda|||U^{\pm}||_{L_{p}(\mathrm{R}^{n})}+|\lambda|^{\frac{1}{2}}||\nabla U^{\pm}||_{L_{p}(\mathrm{R}^{n})}+||\nabla^{2}U^{\pm}||_{L_{p}(\mathrm{R}^{n})}+||\nabla\Phi^{\pm}||_{L_{p}(\mathrm{R}^{n})}$$\leqq C(p, \epsilon, n)||f||_{L_{p}(\mathbb{R}^{n})}$Moreover when $1<p<n$ , it holds that$||\Phi^{\pm}||_{L_{np/(n-p)}(\mathrm{R}^{n})}\leqq C(n,p, \epsilon)||f||_{L_{p}(\mathrm{R}^{n})}$If we put $u^{\pm}=v^{\pm}+U^{\pm}$ ,$\pi^{\pm}=\theta^{\pm}+\Phi^{\pm}$.

, then (5.12) is reduced to(5.15)$\{\begin{array}{l}\lambda v^{\pm}-\mathrm{D}\mathrm{i}\mathrm{v}T^{\pm}(v^{\pm},\pi^{\pm})=f^{\pm},\nabla\cdot u^{\pm}=0\mathrm{i}\mathrm{n}\mathrm{R}_{\pm}^{n}\mu_{+}(_{x_{k}\hat{\partial x_{n}}}^{v_{\mathrm{A}}^{+}\partial v^{+}}\frac{\theta}{\partial}+)|_{x_{n}=0}-\mu_{-}(\frac{\partial v^{-}}{\partial x_{k}}+)\partial v^{-}\hat{\partial x_{n}}|_{x_{n}=0}=a_{k}^{+}|_{x_{n}=0}-a_{k}^{-}|_{x_{n}=0}k=1,\cdots,n-1(2\mu_{+}\frac{\partial v^{+}}{\theta x_{n}}-\pi^{+})|_{x_{n}=0}-(2\mu_{-}\frac{\theta v^{-}}{\theta x_{n}}-\pi^{-})|_{x_{n}=0}=a_{n}^{+}|_{x_{n}=0}-a_{n}^{-|_{x_{n}=0}}v_{k}^{+}|_{x_{n}=0}-v_{k}^{-}|_{x_{n}=0}=b_{k}^{+}|_{x_{n}=0}-b_{k}^{-}|_{x_{n}=0},k=\mathrm{l},\cdots,n-1v_{n}^{+}|_{x_{n}=0}-v_{n}^{-}|_{x_{n}=0}=0\end{array}$

.

143

$a_{k}^{\pm}=h_{k}^{\pm}- \mu\pm(\frac{\partial U_{n}^{\pm}}{\partial x_{k}}+\frac{\partial U_{k}^{\pm}}{\partial x_{n}})$$a_{n}^{\pm}=h_{n}^{\pm}-(2\mu\pm^{\frac{\partial U_{n}^{\pm}}{\partial x_{n}}-\Phi^{\pm)}}$$b_{k}^{\pm}=-U_{k}^{\pm}$,$k=1$ ,$\cdots$$b^{\pm}=(b_{1}^{\pm},$$a_{n}^{\pm})$$k=1$ ,$\cdots$, $n-1$ ,,, $n-1$ .

We putand\cdots ,\cdots ,5.3, it is sufficient to show that for every$a^{\pm}=(a_{1}^{\pm},$,0). In order to prove Theorem, the following estimate holds:$b_{n-1}^{\pm},$$\lambda\in\Sigma_{\epsilon}$$| \lambda|||v||_{L_{p}(\mathbb{R}^{n})}+|\lambda|^{\frac{1}{2}}||\nabla v||_{L_{p}(\mathbb{R}^{n})}+\sum_{+-}(||\nabla^{2}v^{\pm}||_{L_{p}(\mathbb{R}_{\pm}^{n})}+||\theta^{\pm}||_{\overline{X}_{p}^{1}(\mathbb{R}_{\pm}^{n})})$$\leqq C(p, \epsilon, n)\sum(|\lambda|^{\frac{1}{2}}||a^{\pm}||_{L_{p}(\mathbb{R}_{\pm}^{n})}+||\nabla a^{\pm}||_{L_{p}(\mathbb{R}_{\pm}^{n})}$$+-$$|\lambda|||b^{\pm}||_{L_{p}(\mathbb{R}_{\pm}^{n})}+|\lambda|^{\frac{1}{2}}||\nabla b^{\pm}||_{L_{p}(\mathbb{R}_{\pm}^{n})}+||\nabla^{2}b^{\pm}||_{L_{p}(\mathbb{R}_{\pm}^{n})})$

By the scaling argument, it is sufficient to show that for every A6$|\lambda|=1$ ,, (5.15) admits asolution
andsatisfying the estimate:(5.16)$a^{\pm}\in W_{p}^{1}(\mathbb{R}_{\pm}^{n})^{n}$$\Sigma_{\epsilon}$

.

with

$(v^{\pm}, \theta^{\pm})\in$$b^{\pm}\in W_{p}^{2}(\mathrm{E})^{n}$$W_{p}^{2}(\mathbb{R}_{\pm}^{n})^{n}\cross\tilde{X}_{p}^{1}(\mathbb{R}_{\pm}^{n})$$\sum_{+-}(||v^{\pm}||_{W_{p}^{2}(\mathbb{R}_{\pm}^{n})}+||\theta^{\pm}||_{\tilde{X}_{p}^{1}(\mathbb{R}_{\pm}^{n})})\leqq C(p, \epsilon, n)\sum_{+-}(||a^{\pm}||_{W_{p}^{1}(\mathbb{R}_{\pm}^{n})}+||b^{\pm}||_{W_{p}^{2}(\mathbb{R}_{\pm}^{n})})$.

Taking the divergence of the first formula of (5.15) and using the condition, we havein. Applying the Laplace operator to the -th component of
the first formula of (5.15), we have
. By using,infinally we arrive at the following equations for:(5.17)$\nabla\cdot v^{\pm}=$$0$$\Delta\theta^{\pm}=0$$\mathbb{R}_{\pm}^{n}$$\mathrm{n}$$(\lambda-\mu\pm\Delta)\Delta v_{n}^{\pm}=0$$\mathrm{E}$$\nabla\cdot v^{\pm}=0$$(v_{n}^{\pm}, \theta^{\pm})$$\{\begin{array}{l}(\lambda-\mu\pm^{\Delta)\Delta v_{n}^{\pm}=\mathrm{o}},\Delta\theta^{\pm}=0\mathrm{i}\mathrm{n}\mathbb{R}_{\pm}^{n}v_{n}^{+}|_{x_{n}=0}-v_{n}^{-}|_{x_{n}=0}=0(2\mu_{+}\frac{\partial v^{+}}{\partial x_{n}}-\pi^{+})|_{x_{n}=0}-(2\mu_{-}\frac{\partial v^{-}}{\partial x_{n}}-\pi^{-})|_{x_{n}=0}=a_{n}^{+}|_{x_{n}=0}-a_{n}^{-}|_{x_{n}=0}\frac{\partial v_{n}^{+}}{\partial x_{n}}||_{x_{n}=0}=-\sum_{j=1\vec{\partial x_{\mathrm{j}}}}^{n-1}\partial b^{+}||_{x_{n}=0}\mu_{+}(\frac{\partial^{2}v_{n}^{+}}{\partial x_{n}^{2}}-\sum_{j=1}^{n-1}\frac{\partial^{2}v_{n}^{+}}{\partial x_{j}^{2}})|_{x_{n}=0}-\mu_{-}(\frac{\partial^{2}v^{-}}{\partial x_{n}^{2}}-\sum_{j=1}^{n-1}\frac{\partial^{2}v_{n}^{-}}{\partial x_{\mathrm{j}}^{2}})|_{x_{n}=0}=-\sum_{j=1}^{n-1}\frac{\partial a_{\mathrm{j}}^{+}}{\partial x_{j}}||_{x_{n}=0}[(\lambda-\mu\pm^{\Delta)v_{n}^{\pm}+\frac{\partial\theta^{\pm}}{\partial x_{n}}]}|_{x_{n}=0}=0\end{array}$

144
After solving (5.17), we shall solve the equations for$v_{k}^{\pm}$, $k=1$ ,$\cdots$, $n-1$ ,(5.18)$\{\begin{array}{l}(\lambda-\mu\pm^{\Delta)v_{k}^{\pm}=-\frac{\partial\theta^{\pm}}{\partial x_{k}}}\mathrm{i}\mathrm{n}\mathbb{R}_{\pm}^{n}\mu_{+}\frac{\partial v_{k}^{+}}{\partial x_{n}}|_{x_{n}=0}-\mu_{-}\frac{\partial v_{k}^{-}}{\partial x_{n}}|_{x_{n}=0}=(a_{k}^{+}-\mu_{+}\frac{\theta v^{+}}{\theta x_{k}})|_{x_{n}=0}-(a_{k}^{-}-\mu_{-}\frac{\partial v^{-}}{\partial x_{k}})|_{x_{n}=0}v_{k}^{+}|_{x_{n}=0}-v_{k}^{-}|_{x_{n}=0}=b_{k}^{+}|_{x_{n}=0}-b_{k}^{-}|_{x_{n}=0}\end{array}$

Now we solve (5.17). Applying the partial Fourier multiplier theorem with respect
to to (5.17), we have$x’$

(5.19)

where$|$$\mu_{+}(\partial_{n}^{2}v_{n}^{+},+|\xi’|_{n}^{2+})|_{x_{n}=0}’-\mu-(,\partial_{nn}^{2}\hat{v}^{-}+|\xi’,|^{2}v_{n}^{-)1_{x_{n}=0}}\partial_{n}\hat{v}_{n}^{+}|x_{n}=0-\partial_{n}^{\frac{1}{\hat{v}n}1x_{n}=0=-i\xi’\cdot\hat{b}^{+}’|_{x_{n}=0+i\xi’\cdot\hat{b}^{-}|_{x_{n}=0}}}\hat{v}\hat{v}^{+}|_{x_{n}=0-\hat{v}_{n}^{-}1_{x_{n}=0=0}}(\lambda+\mu_{\pm}|\xi’|^{2}-\mu\pm\partial_{n}^{2})(-|\xi’|^{2}+\partial_{n}^{2})\hat{v}_{n}^{\pm}=0,(|\xi’|^{2}-\partial_{n}^{2},)\hat{\theta}^{\pm},=0\mathrm{i}\mathrm{n}\mathbb{R}_{\pm}[(\dagger\pm(2\mu_{+}\partial_{n}\hat{v}_{n}^{+}-\hat{\theta}^{+})-(2\mu-0\partial_{n}\hat{v}_{n}^{-}-\hat{\theta}^{-})|_{x_{n}=0}x_{n}==\hat{a}_{n}^{+}|_{x_{n}=0-\hat{a}_{n}^{-1_{x_{n}=0}}}n_{\lambda\mu|\xi|^{2}-\mu\pm^{\partial_{n}^{2})\hat{v}_{n}^{\pm}+\partial_{n}\hat{\theta}^{\pm}]1_{x_{n}=0}=0}}=-i\xi\cdot\hat{a}^{+}’|_{x_{n}=0+i\xi’\cdot\hat{a}^{-1_{x_{n}=0}}}’,"$$\sqrt{(\mu\pm)^{-1}\lambda+|\xi’|^{2}}$of the form:(5.20).

If we put $A=|\xi’|$ and $B_{\pm}=$with ${\rm Re} B\pm>0$ , we shall seek the solution
to (5.19)$\hat{v}_{n}^{\pm}=\hat{v}_{n}^{\pm}(\xi’, x_{n})$and$\hat{\theta}_{n}^{\pm}=\hat{\theta}_{n}^{\pm}(\xi’, x_{n})$$(\hat{v}_{n}^{\pm},\hat{\theta}^{\pm})$$\hat{v}_{n}^{+}=\alpha^{+}(e^{-Ax_{n}}-e^{-B_{+}x_{n}})+\beta e^{-Bx_{n}}+$$\hat{v}_{n}^{-}=\alpha^{-}(e^{Ax_{n}}-e^{B_{-}x_{n}})+\beta e^{B_{-}x_{n}}$,,$\hat{\theta}^{+}=\gamma^{+}e^{-Ax_{n}}$$\hat{\theta}^{-}=\gamma^{-}e^{Ax_{n}}$,.

iRom the boundary condition in (5.19), we have$L$$(\begin{array}{l}\alpha^{+}\alpha^{-}\beta\end{array})=(_{-A(\hat{a}_{n}^{+}(\xi’,0)-\hat{a}_{n}^{-}(\xi’,0))}^{-iA\tilde{\xi}’\cdot(\hat{b}^{+}’(\xi’,0)-\hat{b}^{-}(\xi’,0))}-iA\tilde{\xi}’\cdot(\hat{a}^{+’}(\xi’,0)-\hat{a}^{-}’,(\xi’,0)))$,$\gamma^{+}=-A^{-1}\mu_{+}(A^{2}-B_{+}^{2})\alpha^{+}\gamma^{-}=A^{-1}\mu_{-}(A^{2}-B_{-}^{2})\alpha^{-}$

where$L=(_{-\mu_{+}(A-B_{+})^{2}}^{B_{+}-A}\mu_{+}(A^{2}-B_{+}^{2})$$-\mu-(A-B_{-}^{2}-\mu-(A^{2}-B_{\frac{2}{)}})B_{-}-A \mu_{+}(A^{2}+B_{+}^{2})-\mu-(A^{2}+B_{-}^{2})2(\mu_{+}AB_{+}+\mu-AB_{-})-(B_{+}+B_{-}))$

By direct calculation, we have
(5.21)(5.22)$\det L=(A-B_{+})(A-B_{-})f(A, B_{+}, B-)$ ,$f(A, B_{+}, B_{-})=-(\mu_{+}-\mu_{-})^{2}A^{3}$$+\{(3\mu_{+}^{2}-\mu_{+}\mu_{-})B_{+}+(3\mu_{-}^{2}-\mu_{+}\mu_{-})B_{-}\}A^{2}$$+\{(\mu_{+}B_{+}+\mu_{-}B_{-})^{2}+\mu_{+}\mu_{-}(B_{+}+B_{-})^{2}\}A$$+B_{+}^{2}(\mu_{+}^{2}B_{+}+\mu_{+}\mu_{-}B_{-})+B_{-}^{2}(\mu_{-}^{2}B_{+}+\mu_{+}\mu_{-}B_{+})$

To verify the invertibility of , we use the following lemma
$L$

.

.

,

Lemma 5.4. Let. For every A C
have the following two inequalities:$0<\mathrm{e}$$<\mathrm{v}/2$$\ovalbox{\tt\small REJECT} \mathrm{E}$.

with |A|(5.23)$|f(A, B_{+}, B_{-})|\geqq c(\epsilon, \mu_{\pm})(1+|\xi’|^{2})^{\frac{3}{2}}$(5.24)${\rm Re} B_{\pm}\geqq c(\epsilon, \mu\pm)(1+|\xi’|^{2})^{\frac{1}{2}}$with some positive number$c(\epsilon, \mu\pm)$

1 and

.

$\ovalbox{\tt\small REJECT} 71^{n-1}$, we,, which implies that$(\mu\pm)^{-1}\lambda+|\xi’|^{2}=(\mu\pm)^{-1}|\lambda+\mu\pm|\xi’|^{2}|e^{:\theta}$

provided that$\cos(\theta/2)\geqq\sin(\epsilon/2)$ . Combining this with$-\pi+\epsilon\leqq\theta\leqq\pi-\epsilon$$\lambda\in\Sigma_{\epsilon}$

and$\xi’\in \mathbb{R}^{n-1}$$| \lambda+\mu_{\pm}|\xi|^{2}|\geqq\sin(\epsilon/2)\min(1, \mu\pm)(|\lambda|+|\xi|^{2})$we have for every A$\langle^{\mathrm{I}}$.

Proof. First we shall show (5.24). If we put

then$\ovalbox{\tt\small REJECT}$

145,$\in\Sigma_{\epsilon}$${\rm Re} B_{\pm}=(\mu_{\pm})^{-\frac{1}{2}}|\lambda+\mu_{\pm}|\xi’|^{2}|^{\frac{1}{2}}\cos(\theta/2)$$\geqq(\mu_{\pm})^{-\frac{1}{2}}\min(1,$$(\mu\pm)^{\frac{1}{2}})(\sin(\epsilon/2))^{\frac{3}{2}}(|\lambda|+|\xi’|^{2})^{\frac{1}{2}}$which implies (5.24) for $|\lambda|=1$ .
Next we shall show (5.23). First we consider the casethat${\rm Im}\lambda\neq 0$(5.25)A$f(A, B_{+}, B_{-})\neq 0$

for$\forall\lambda\in\sigma_{\epsilon}$with$|\lambda|=1$,${\rm Im}$$\neq 0$,. We shall show

and V4’$\in \mathbb{R}^{n-1}$.

by using the uniqueness of the solution to ordinary differential equation (5.19). Let
be asolution to$(\hat{v}_{n}^{\pm}(\xi’, x_{n}),\hat{\theta}^{\pm}(\xi’, x_{n}))$,$(\partial_{n}^{2}-A^{2})\hat{\theta}^{\pm}=0$$(\partial_{n}^{2}-B_{\pm}^{2})(\partial_{n}^{2}-A^{2})\hat{v}_{n}^{\pm}=0$$\hat{v}_{n}^{+}|_{x_{n}=0}-\hat{v}_{n}^{-}|_{x_{n}=0}=0$

in

,

,

$(2\mu_{+}\partial_{n}\hat{v}_{n}^{+}-\hat{\theta}^{+})|_{x_{n}=0}-(2\mu_{-}\partial_{n}\hat{v}_{n}^{-}-\hat{\theta}^{-})|_{x_{n}=0}=0$(5.26)$\mathbb{R}_{\pm}$$\{$$\partial_{n}\hat{v}_{n}^{+}|_{x_{n}=0}-\partial_{n}\hat{v}_{n}^{-}|_{x_{n}=0}=0$,,$\mu_{+}(\partial_{n}^{2}v_{n}^{+}+|\xi’|^{2}\hat{v}_{n}^{+})|_{x_{n}=0}-\mu_{-}(\partial_{n}^{2}\hat{v}_{n}^{-}+|\xi’|^{2}\hat{v}_{n}^{-})|_{x_{n}=0}=0$,.

$[\mu\pm(B_{\pm}^{2}-\partial_{n}^{2})\hat{v}_{n}^{\pm}+\partial_{n}\hat{\theta}^{\pm}]|_{x_{n}=0}=0$

Let $\hat{v}_{k}^{\pm}(\xi’, x_{n})(k=1, \cdots, n-1)$ be asolution to$\mu\pm(B_{\pm}^{2}-\partial_{n}^{2})v_{k}^{\pm}=-i\xi_{k}\hat{\theta}^{\pm}$(5.27)$\{$

in$\mathbb{R}_{\pm}$,,$\mu_{+}\partial_{n}v_{k}^{+}|_{x_{n}=0}-\mu_{-}\partial_{n}v_{k}^{-}|_{x_{n}=0}=-i\xi_{k}(\mu_{+}\hat{v}_{n}^{+}-\mu_{-}\hat{v}_{n}^{-})|_{x_{n}=0}$$v_{k}^{+}|_{x_{n}=0}-v_{k}^{-}|_{x_{n}=0}=0$.

By the first, the second and the 6th formula of (5.26),

in R3 ,$(\partial_{n}^{2}-A^{2})[\mu\pm(B_{\pm}^{2}-\partial_{n}^{2})\hat{v}_{n}^{\pm}+\partial_{n}\hat{\theta}^{\pm}]$

$\{$

,

$[\mu\pm(B_{\pm}^{2}-\partial_{n}^{2})\hat{v}_{n}^{\pm}+\partial_{n}\hat{\theta}^{\pm}]|_{x_{n}=0}=0$

146
so we have
(5.28)$\mu\pm(B_{\pm}^{2}-\partial_{n}^{2})\hat{v}_{n}^{\pm}+\partial_{n}\hat{\theta}^{\pm}=0$in$\mathbb{R}\pm\cdot$of (5.28), multiplying the first formula of (5.27) by
Takingsecond formula of (5.26), we have$\partial_{n}$$\mu\pm(B_{\pm}^{2}-\partial_{n}^{2})[\partial_{n}\hat{v}_{n}^{\pm}+\sum_{k=1}^{n-1}i\xi_{k}\hat{v}_{n}^{\pm}]=0$in$i\xi_{k}$and using$\mathbb{R}\pm\cdot$

So we have
(5.29)$0=( \mu_{+}(B_{+}^{2}-\partial_{n}^{2})(\partial_{n}\hat{v}_{n}^{+}+\sum_{k=1}^{n-1}i\xi_{k}\hat{v}_{n}^{+}),\partial_{n}\hat{v}_{n}^{+}+\sum_{k=1}^{n-1}i\xi_{k}\hat{v}_{n}^{+})_{\mathrm{R}_{+}}$$+( \mu_{-}(B_{-}^{2}-\partial_{n}^{2})(\partial_{n}\hat{v}_{n}^{-}+\sum_{k=1}^{n-1}i\xi_{k}\hat{v}_{n}^{-}),$$\partial_{n}\hat{v}_{n}^{-}+\sum_{k=1}^{n-1}i\xi_{k}\hat{v}_{n}^{-})_{\mathrm{R}_{-}}$

Using the 5th formula of (5.26) and the 3rd formula of (5.27), we can proceed
$n-1$

$n-1$

$0=- \langle\mu+(\partial_{n}^{2}\hat{v}_{n}^{+}+\sum i\xi_{k}\partial_{k}\hat{v}_{k}^{+})-\mu_{-}(\partial_{n}^{2}\hat{v}_{n}^{-}+\sum i\xi_{k}\partial_{k}\hat{v}_{k}^{-})$

$k=1$

,

$k=1$

$\partial_{n}\hat{v}_{n}^{+}+\sum_{k=1}^{n-1}i\xi_{k}\hat{v}_{k}^{+}\rangle_{ox_{n}=0}$$- \mu_{+}||\partial_{n}(\partial_{n}\hat{v}_{n}^{+}+\sum_{k=1}^{n-1}i\xi_{k}\hat{v}_{k}^{+})||_{\mathrm{R}}^{2}+-\mu_{+}B_{+}^{2}||\partial_{n}\hat{v}_{n}^{+}+\sum_{k=1}^{n-1}i\xi_{k}\hat{v}_{k}^{+}||_{\mathrm{R}}^{2}+$$- \mu_{-}||\partial_{n}(\partial_{n}\hat{v}_{n}^{-}+\sum_{k=1}^{n-1}i\xi_{k}\hat{v}_{k}^{-})||_{\mathrm{R}_{-}}^{2}-\mu_{-}B_{-}^{2}||\partial_{n}\hat{v}_{n}^{-}+\sum_{k=1}^{n-1}i\xi_{k}\hat{v}_{k}^{-}||_{\mathrm{R}_{-}}^{2}$.

By the 6th formula of (5.26) and the second formula of (5.27), it holds that$\mu+(\partial_{n}^{2}\hat{v}_{n}^{+}+\sum_{k=1}^{n-1}i\xi_{k}\partial_{n}\hat{v}_{k}^{+})|_{x_{n}=0}-\mu-(\partial_{n}^{2}\hat{v}_{n}^{-}+\sum_{k=1}^{n-1}i\xi_{k}\partial_{n}\hat{v}_{k}^{-})|_{x_{n}=0}$$=(- \mu_{+}A^{2}\hat{v}_{n}^{+}+\mu_{-}A^{2}\hat{v}_{n}^{-})|_{x_{n}=0}+\sum_{k=1}^{n-1}i\xi_{k}(-i\xi_{k})(\mu_{+}\hat{v}_{n}^{+}-\mu_{-}\hat{v}_{n}^{-})|_{x_{n}=0}$$=-A^{2}(\mu_{+}\hat{v}_{n}^{+}-\mu_{-}\hat{v}_{n}^{-})|_{x_{n}=0}+A^{2}(\mu_{+}\hat{v}_{n}^{+}-\mu_{-}\hat{v}_{n}^{-})|_{x_{n}=0}=0$.

Therefore we have
(5.30)$0= \mu_{+}||\partial_{n}(\partial_{n}\hat{v}_{n}^{+}+\sum_{k=1}^{n-1}i\xi_{k}\hat{v}_{k}^{+})||_{\mathrm{R}}^{2}++\mu_{-}||\partial_{n}(\partial_{n}\hat{v}_{n}^{-}+\sum_{k=1}^{n-1}i\xi_{k}\hat{v}_{k}^{-})||_{\mathbb{R}_{-}}^{2}$$+ \mu_{+}B_{+}^{2}||\partial_{n}\hat{v}_{n}^{+}+\sum_{k=1}^{n-1}i\xi_{k}\hat{v}_{k}^{+}||_{\mathrm{R}_{+}}^{2}+\mu_{-}B_{-}^{2}||\partial_{n}\hat{v}_{n}^{-}+\sum_{k=1}^{n-1}i\xi_{k}\hat{v}_{k}^{-}||_{\mathbb{R}_{-}}^{2}$.

147
We note that$\mu\pm^{B_{\pm}^{2}=\mu\pm(\lambda}/\mu_{\pm}+|\xi’|^{2})=\lambda+\mu_{\pm}|\xi’|^{2}$.

Taking the imaginary part of (5.34), we obtain(5.31)$\partial_{n}\hat{v}_{n}^{\pm}+\sum_{k=1}^{n-1}i\xi_{k}\hat{v}_{k}^{\pm}=0$in$\mathbb{R}\pm\cdot$

By (5.28) and the first equation of (5.27), we obtain(5.32)$0= \sum_{+-}[((\lambda+\mu\pm|\xi’|^{2}-\mu\pm^{\partial_{n}^{2})\hat{v}_{n}^{\pm}+\partial_{n}\hat{\theta}^{\pm},\hat{v}_{n}^{\pm})_{\mathbb{R}}}\pm$$+ \sum_{k=1}^{n-1}((\lambda+\mu_{\pm}|\xi’|^{2}-\mu_{\pm}\partial_{n}^{2})\hat{v}_{k}^{\pm}+i\xi_{k}\hat{\theta}^{\pm},\hat{v}_{k}^{\pm})_{\mathbb{R}}\pm]$$=<(\mu_{+}\partial_{n}\hat{v}_{n}^{+}-\hat{\theta}^{+})-(\mu_{-}\partial_{n}\hat{v}_{n}^{-}-\hat{\theta}^{-}),\hat{v}_{n}^{+}>_{x_{n}=0}$$+ \sum_{k=1}^{n-1}<\mu_{+}\partial_{n}\hat{v}_{k}^{+}-\mu_{-}\partial_{n}\hat{v}_{k}^{-},\hat{v}_{k}^{+}>_{x_{n}=0}$$+ \sum_{+-}[(\lambda+\mu_{\pm}|\xi’|^{2})||\hat{v}_{n}^{\pm}||_{\mathbb{R}}^{2}\pm+\mu\pm||\partial_{n}\hat{v}_{n}^{\pm}||_{\mathbb{R}}^{2}\pm$$+ \sum_{k=1}^{n-1}\{(\lambda+\mu\pm|\xi’|^{2})||\hat{v}_{k}^{\pm}||_{\mathbb{R}}^{2}\pm+\mu\pm||\partial_{n}\hat{v}_{k}^{\pm}||_{\mathbb{R}}^{2}\pm\}$

where we use thaton $x_{n}=0$ , $k=1$ ,of (5.26) and (5.27), and (5.31), we have$\hat{v}_{k}^{+}=\hat{v}_{k}^{-}$$\cdots$$-( \hat{\theta}^{\pm}, \partial_{n}\hat{v}_{n}^{\pm}+\sum_{k=1}^{n-1}i\xi_{k}\hat{v}_{k}^{\pm})_{\mathbb{R}}\pm]$, $n-1$ . By the boundary conditions$<(\mu_{+}\partial_{n}\hat{v}_{n}^{+}-\hat{\theta}^{+})-(\mu_{-}\partial_{n}\hat{v}_{n}-\hat{\theta}^{-}),\hat{v}_{n}^{+}>_{x_{n}=0}$$+ \sum_{k=1}^{n-1}<\mu_{+}\partial_{n}\hat{v}_{k}^{+}-\mu_{-}\partial_{n}\hat{v}_{k}^{-},\hat{v}_{k}^{+}>_{x_{n}=0}$$=<- \mu_{+}\partial_{n}\hat{v}_{n}^{+}+\mu_{-}\partial_{n}\hat{v}_{n}^{-},\hat{v}_{n}^{+}>_{x_{n}=0}+<\mu_{+}\hat{v}_{n}^{+}-\mu_{-}\hat{v}_{n}^{-},\sum_{k=1}^{n-1}i\xi_{k}\hat{v}_{k}^{+}>_{x_{n}=0}$$=-(\mu_{+}-\mu_{-})<\partial_{n}\hat{v}_{n}^{+},\hat{v}_{n}^{+}>_{x_{n}=0}+(\mu_{+}-\mu_{-})<\hat{v}_{n}^{+},$ $-\partial_{n}\hat{v}_{n}^{+}>_{x_{n}=0}$$=-2(\mu_{+}-\mu_{-})\mathrm{R}e<\partial_{n}\hat{v}_{n}^{+},\hat{v}_{n}^{+}>_{x_{n}=0}$.

Therefore by (5.31) and (5.32)(5.33)$0=-2(\mu_{+}-\mu_{-})\mathrm{R}e<\partial_{n}\hat{v}_{n}^{+},\hat{v}_{n}^{+}>_{x_{n}=0}$$+ \sum_{+-}[(\lambda+\mu\pm|\xi’|^{2})(\sum_{k=1}^{n-1}||\hat{v}_{k}^{\pm}||_{\mathbb{R}}^{2}\pm+||\hat{v}_{n}^{\pm}||_{\mathbb{R}}^{2}\pm)+\mu\pm(\sum_{k=1}^{n-1}||\partial_{n}\hat{v}_{k}^{\pm}||_{\mathrm{R}}^{2}\pm+||\partial_{n}\hat{v}_{n}^{\pm}||_{\mathbb{R}}^{2}\pm)]$.

148
Taking the imaginary part of (5.33), we obtain$\hat{v}_{k}^{\pm}=0$in$\mathbb{R}\pm$,$k=1$ ,$\cdots$

,

$n$

.

Thus we prove (5.25). (5.24) is showed in the similar way to [8, Proof of Lemma
4.4].
, namely $\lambda=1$ . In this case we calculate
Next we consider the case$f(A, B_{+}, B_{-})$ directly. Now we assume that, and then $A<B_{+}\leqq B_{-}$ .
Since${\rm Im}\lambda=0$$\mu_{+}\geqq\mu_{-}$$\{(3\mu_{+}^{2}-\mu_{+}\mu_{-})B_{+}+(3\mu_{-}^{2}-\mu_{+}\mu_{-})B_{-}\}A^{2}$$+\{(\mu_{+}B_{+}+\mu_{-}B_{-})^{2}+\mu_{+}\mu_{-}(B_{+}+B_{-})^{2}\}A$$\geqq 4(\mu_{+}B_{+}+\mu_{+}\mu_{-}B_{-}+\mu_{-}^{2}B_{-})A^{2}$,

it holds that$f(A, B_{+}, B_{-})\geqq[-(\mu_{+}-\mu_{-})^{2}B_{+}+4(\mu_{+}^{2}B_{+}+\mu_{+}\mu_{-}B_{-}+\mu_{-}^{2}B_{-})]A^{2}$$+B_{+}^{2}(\mu_{+}^{2}B_{+}+\mu_{+}\mu_{-}B_{-})+B_{-}^{2}(\mu_{-}^{2}B_{-}+\mu_{+}\mu_{-}B_{+})$$\geqq\{3\mu_{+}^{2}B_{+}+2\mu_{+}\mu_{-}B_{+}+4\mu_{+}\mu_{-}B_{+}+\mu_{-}^{2}(4B_{-}-B_{+})\}A^{2}+(\mu_{+}+\mu_{-})^{2}B^{+3}$$\geqq(\mu_{+}+\mu_{-})^{2}(1+|\xi’|^{2})^{3}2$.

This completes the proof of the lemma.

$\square$

By direct calculation, we have
(5.34)$(\begin{array}{l}\alpha^{+}\alpha^{-}\sqrt\end{array})=L^{-1}(_{-A(\hat{a}_{n}^{+}(\xi’,0)-\hat{a}_{n}^{-}(\xi’,0))}^{-iA\tilde{\xi}’\cdot(\hat{b}^{+}(\xi’,0)-\hat{b}^{-}(\xi’,0))}-iA\tilde{\xi}’\cdot(\hat{a}^{+}’,(\xi’,0)-\hat{a}^{-}’,(\xi’,0)))$,$L^{-1}=(_{L_{1}^{\frac{}{3}1}}^{L_{1}^{1}}L_{1}^{\frac{-1}{2}1}$$L_{2}^{1}L_{2}^{\frac{-1}{\frac{2}{3}}1}L_{2}^{1}$$L_{3}^{\frac{}{3}1}L_{3}^{\frac{-1}{2}1}L_{3}^{1})$where$L_{11}^{-1}= \frac{-\mu-}{(A-B_{+})f(A,B_{+},B_{-})}\cross[(\mu_{+}-\mu_{-})A^{3}+(3\mu_{-}-\mu_{+})A^{2}B_{-}$$+2\mu_{+}AB_{+}(A+B_{-})+A(\mu_{+}B_{+}^{2}+\mu_{-}B_{-}^{2})+B_{-}(\mu_{-}B_{-}^{2}-\mu_{+}B_{+}^{2})]$,$L_{12}^{-1}= \frac{AB_{+}(\mu_{+}-\mu_{-})+A(\mu_{+}B_{+}+\mu_{-}B_{-})+\mu_{-}B_{-}(B_{+}+B_{-})}{(A-B_{+})f(A,B_{+},B_{-})}$$L_{13}^{-1}= \frac{-\mu_{+}(A^{2}+B_{+}^{2})+\mu_{-}A(A-B_{+})-\mu_{-}B_{-}(A+B_{+})}{(A-B_{+})f(A,B_{+},B_{-})}$,$L_{21}^{-1}= \frac{-\mu_{+}}{(A-B_{-})f(A,B_{+},B_{-})}\cross[\mu_{+}(-A^{3}+3A^{2}B_{+}+AB_{+}^{2}+B^{+3})$$+2\mu_{-}AB_{-}(A+B_{+})+\mu_{-}(A^{2}+B_{+}^{2})(A-B_{+})]$$L_{22}^{-1}= \frac{-[(2\mu_{-}-\mu_{+})AB_{-}+\mu_{+}B_{+}(A+B_{+}+B_{-})]}{(A-B_{-})f(A,B_{+},B_{-})}$,$L_{23}^{-1}= \frac{\mu_{+}A(A-B_{+})-\mu_{-}B_{-}(A+B_{+})-\mu_{-}(A^{2}+B_{-}^{2})}{(A-B_{-})f(A,B_{+},B_{-})}$

,

,

149
2p.

$p(A^{2}-\mathit{1}\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}^{B}}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}})$$\yen$$L\ovalbox{\tt\small REJECT}^{1}.\ovalbox{\tt\small REJECT}$f($L\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}^{1}}\ovalbox{\tt\small REJECT}$$A_{\mathrm{t}}$

B.,

fC4,

$\ovalbox{\tt\small REJECT}$$B_{+}$,)$i^{t}-(A B_{-})$’$p_{+}(A B_{+})$$f(A_{7}B_{+\rangle}B_{-})$’$B_{-})$

By inserting the formula (5.34) into (5.20), we obtain the explicit expression of thesolutionsand$\hat{\theta}^{\pm}:$$\hat{v}_{n}^{\pm}$(5.35)$\hat{v}_{n}^{+}(\xi’, x_{n})=\frac{e^{-B}+^{x_{n}}-e^{-Ax_{n}}}{B_{+}-A}A\frac{i\mu_{-}}{f(A,B_{+},B_{-})}[(\mu_{+}-\mu_{-})A^{3}+(3\mu_{-}-\mu_{+})A^{2}B_{-}$$+2\mu_{+}AB_{+}(A+B_{-})+A(\mu_{+}B_{+}^{2}+\mu_{-}B_{-}^{2})+B_{-}(\mu_{-}B_{-}^{2}-\mu_{+}B_{+}^{2})]$$\cross\tilde{\xi}’\cdot(\hat{b}^{+’}(\xi’, 0)-\hat{b}^{-}’(\xi’, 0))$$+ \frac{e^{-Bx_{n}}+-e^{-Ax_{n}}}{B_{+}-A}A\frac{AB_{+}(\mu_{+}-\mu_{-})+A(\mu_{+}B_{+}+\mu_{-}B_{-})+\mu_{-}B_{-}(B_{+}+B_{-})}{f(A,B_{+},B_{-})}$$\cross(-i)\tilde{\xi}’\cdot(\hat{a}^{+’}(\xi’, 0)-\hat{a}^{-}’(\xi’, 0))$$+ \frac{e^{-Bx_{n}}+-e^{-Ax_{n}}}{B_{+}-A}A\frac{\mu_{+}(A^{2}+B_{+}^{2})-\mu_{-}A(A-B_{+})+\mu_{-}B_{-}(A+B_{+})}{f(A,B_{+},B_{-})}$$\cross(\hat{a}_{n}^{+}(\xi’, 0)-\hat{a}_{n}^{-}(\xi’, 0))$$+e^{-B_{\dagger}x_{n}} \frac{-2i\mu_{+}\mu_{-}A(A^{2}-B_{+}B_{-})}{f(A,B_{+},B_{-})}\tilde{\xi}’\cdot(\hat{b}^{+^{l}}(\xi’, 0)-\hat{b}^{-}’(\xi’, 0))$$+e^{-Bx_{n}}+ \frac{-iA[\mu_{-}(A-B_{-})-\mu_{+}(A-B_{+})]}{f(A,B_{+},B_{-})}\tilde{\xi}’\cdot(\hat{a}^{+’}(\xi’, 0)-\hat{a}^{-^{l}}(\xi’, 0))$$+e^{-Bx_{n}}+ \frac{-A[\mu_{+}(A+B_{+})+\mu_{-}(A+B_{-})]}{f(A,B_{+},B_{-})}(\hat{a}_{n}^{+}(\xi’, 0)-\hat{a}_{n}^{-}(\xi’, 0))$(5.36)$\hat{v}_{n}^{-}(\xi’, x_{n})=\frac{e^{B_{-}x_{n}}-e^{Ax_{n}}}{B_{-}-A}A\frac{i\mu_{+}}{f(A,B_{+},B_{-})}$$\cross[\mu_{+}(-A^{3}+3A^{2}B_{+}+AB_{+}^{2}+B^{+3})+2\mu_{-}AB_{-}(A+B_{+})$$+\mu_{-}(A^{2}+B_{+}^{2})(A-B_{+})]\tilde{\xi}’\cdot(\hat{b}^{+}’(\xi’, 0)-\hat{b}^{-}’(\xi’, 0))$$+ \frac{e^{B_{-}x_{n}}-e^{Ax_{n}}}{B_{-}-A}A\frac{(2\mu_{-}-\mu_{+})AB_{-}+\mu_{+}B_{+}(A+B_{+}+B_{-})}{f(A,B_{+},B_{-})}$$\cross i\tilde{\xi}’\cdot(\hat{a}^{+}’(\xi’, 0)-\hat{a}^{-}’(\xi’, 0))$$+ \frac{e^{B_{-}x_{n}}-e^{Ax_{n}}}{B_{-}-A}A\frac{-\mu_{+}A(A-B_{+})+\mu_{-}B_{-}(A+B_{+})+\mu_{-}(A^{2}+B_{-}^{2})}{f(A,B_{+},B_{-})}$$\cross(\hat{a}_{n}^{+}(\xi’, 0)-\hat{a}_{n}^{-}(\xi’, 0))$$+e^{B_{-}x_{n}} \frac{-2i\mu_{+}\mu_{-}A(A^{2}-B_{+}B_{-})}{f(A,B_{+},B_{-})}\tilde{\xi}’\cdot(\hat{b}^{+’}(\xi’,0)-\hat{b}^{-}’(\xi’, 0))$

150

.

$\ovalbox{\tt\small REJECT}[/’(A\ovalbox{\tt\small REJECT} \mathrm{j},\ovalbox{\tt\small REJECT}^{B_{\ovalbox{\tt\small REJECT}})} \ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}^{+(\ovalbox{\tt\small REJECT}_{+})]}\ovalbox{\tt\small REJECT}^{7}\cdot(\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}^{-1^{1}}-(4^{\mathrm{z}}\rangle 0) \ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}^{-^{1}}(4^{\mathrm{z}}\rangle 0))$$+e^{B}x_{\ovalbox{\tt\small REJECT}}$$f(\ovalbox{\tt\small REJECT}/1, B_{+}, B_{-})$$A[\mathrm{P}+(A+\ovalbox{\tt\small REJECT} B_{+})+p-(A+B_{-})]$$\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}}1)\mathrm{t}_{-\}(_{\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} 1\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} 4\ovalbox{\tt\small REJECT} 7\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}^{-\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}}}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}(\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}(\cdot \mathit{7}\ovalbox{\tt\small REJECT}) \ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}(\cdot \mathit{1}\ovalbox{\tt\small REJECT}))$$\ovalbox{\tt\small REJECT}- e^{B_{-}}"$$f(A, B_{+}, B_{-})$(5.37)$\hat{\theta}^{+}(\xi’,x_{n})=e^{-Ax_{n}}[\frac{-i\mu_{+}\mu_{-}(A+B_{+})}{f(A,B_{+},B_{-})}[(\mu_{+}-\mu_{-})A^{3}+(3\mu_{-}-\mu_{+})A^{2}B_{-}$$+2\mu_{+}AB_{+}(A+B_{-})+A(\mu_{+}B_{+}^{2}+\mu-B_{-}^{2})+B_{-}(\mu_{-}B_{-}^{2}-\mu_{+}B_{+}^{2})]$$\mathrm{x}\tilde{\xi}’\cdot(\hat{b}^{+’}(\xi’, 0)-\hat{b}^{-}’(\xi’, 0))$$+ \frac{\mu+(A+B_{+})\{AB_{+}(\mu_{+}-\mu-)+A(\mu_{+}B_{+}+\mu-B_{-})+\mu-B_{-}(B_{+}+B_{-})\}}{f(A,B_{+},B_{-})}$$\cross$$i\tilde{\xi}’\cdot(\hat{a}^{+}’(\xi’, 0)-\hat{a}^{-}’(\xi’, 0))$$+ \frac{\mu-(A+B_{+})\{-\mu+(A^{2}+B_{+}^{2})+\mu_{-}A(A-B_{+})-\mu-B_{-}(A+B_{+})\}}{f(A,B_{+},B_{-})}$$\cross(\hat{a}_{n}^{+}(\xi’, 0)-\hat{a}_{n}^{-}(\xi’, 0))]$(5.38)$\hat{\theta}^{-}(\xi’,x_{n})=e^{Ax_{n}}[\frac{i\mu_{+}\mu_{-}(A+B_{+})}{f(A,B_{+},B_{-})}[\mu+(-A^{3}+3A^{2}B_{+}+AB_{+}^{2}+B^{+3})$$+2\mu-AB_{-}(A+B_{+})+\mu-(A^{2}+B_{+}^{2})(A-B_{+})]\tilde{\xi}’\cdot(\hat{b}^{+’}(\xi’, 0)-\hat{b}^{-}’(\xi’, 0))$$+ \frac{i\mu_{-}(A+B_{-})[(2\mu_{-}-\mu_{+})AB_{-}+\mu_{+}B_{+}(A+B_{+}+B_{-})]}{f(A,B_{+},B_{-})}$$\tilde{\xi}’\cdot(\hat{a}^{+’}(\xi’, 0)-\hat{a}^{-}’(\xi’, 0))$$- \frac{\mu-(A+B_{-})[\mu_{+}A(A-B_{+})-\mu-B_{-}(A+B_{+})-\mu-(A^{2}+B_{-}^{2})]}{f(A,B_{+},B_{-})}$$(\hat{a}_{n}^{+}(\xi’, 0)-\hat{a}_{n}^{-}(\xi’, 0))]$.

If we put$v_{n}^{\pm}(x)=\mathcal{F}_{\xi}^{-1},[\hat{v}_{n}^{\pm}(\xi’, x_{n})](x’)$,$\theta^{\pm}(x)=\mathcal{F}_{\xi’}^{-1}[\hat{\theta}_{n}^{\pm}(\xi’,x_{n})](x’)$,

where , denotes the inverse partial Fourier transform with respect to , thensatisfy (5.37). By using the Fourier multiplier theorem and the Agmonand
Douglis-Nirenberg theorem, we can show$\xi’$$\mathcal{F}_{\xi}^{-1}$$v_{n}^{\pm}$(5.39)(5.40)$\theta^{\pm}$$||v_{n}^{\pm}||_{W_{p}^{2}(\mathrm{R}_{\pm}^{n})} \leqq c(p, \epsilon, n)\sum_{+-}(||a^{\pm}||_{W_{p}^{1}(\mathrm{R}_{\pm}^{n})}+||b^{\pm}||_{W_{p}^{2}(\mathrm{R}_{\pm}^{n})})$,$|| \theta^{\pm}||_{\tilde{X}_{p}(\mathrm{R}_{\pm}^{n})}\leqq c(p, \epsilon, n)\sum_{+-}(||a^{\pm}||_{W_{p}^{1}(\mathrm{R}_{\pm}^{n})}+||b^{\pm}||_{W_{p}^{2}(\mathrm{R}_{\pm}^{n})})$,

151
i. Since the argument is now well-known (cf. Farwi
for every A E I. with
Sohr [5, Section 2]), we may omit the proof of (5.39) and (5.40).
Finally we shall solve (5.18). Put$|_{\ovalbox{\tt\small REJECT}}\mathrm{X}|\ovalbox{\tt\small REJECT}$$- \frac{\partial\theta^{+}}{\partial x_{k}}$$g_{k}^{+}(x)=\{$

00$x_{n}>0$$x_{n}<0$,$x_{n}>0$$g_{k}^{+}(x)=\{$$x_{n}<0$$- \frac{\partial\theta^{-}}{\partial x_{k}}$and$w_{k}^{\pm}(x)=\mathcal{F}_{\xi}^{-1}[(\lambda+\mu\pm|\xi’|^{2})^{-1}\hat{g}_{k}^{\pm}(\xi)](x)$,.

Then the solutions $v_{k}^{\pm}(k=1, \cdots, n-1)$ are given by$v_{k}^{\pm}(x)=w_{k}^{\pm}(x)+z_{k}^{\pm}(x)$where$z_{k}^{\pm}$,

are the solutions to((5.41)$\{$$\lambda-\mu\pm^{\Delta)z_{k}^{\pm}}=0$in$\mathbb{R}_{\pm}^{n}$,,$\mu_{+}\partial_{n}z_{k}^{+}|_{x_{n}=0}-\mu-\partial_{n}z_{k}^{-}|_{x_{n}=0}=l_{k}^{+}|_{x_{n}=0}-l_{k}^{+}|_{x_{n}=0}$$z_{k}^{+}|_{x_{n}=0}-z_{k}^{-}|_{x_{n}=0}=h_{k}^{+}|_{x_{n}=0}-h_{k}^{+}|_{x_{n}=0}$

,

where
$l_{k}^{\pm}=a_{k}^{\pm}- \mu\pm(\frac{\partial v_{n}^{\pm}}{\partial x_{k}}+\frac{\partial w_{k}^{\pm}}{\partial x_{n}})$,$h_{k}^{\pm}=b_{k}^{\pm}-w_{k}^{\pm}$,$k=1$ ,$\cdots$, $n-1$ .

Applying the partial Fourier transform with respect to $x’$ to (5.41), we have(
(5.42)$\{$$\lambda+\mu\pm|\xi’|^{2}-\mu\pm^{\partial_{n}^{2})\hat{z}_{k}^{\pm}(\xi’,x_{n})=0}$

in$\mathbb{R}_{\pm}$,$\mu_{+}\partial_{n}\hat{z}_{k}^{+}(\xi’, 0)-\mu_{-}\partial_{n}\hat{z}_{k}^{-}(\xi’, 0)=\hat{l}_{k}^{+}(\xi’, 0)-\hat{l}_{k}^{-}(\xi’, 0)$$\hat{z}_{k}^{+}(\xi’, 0)-\hat{z}_{k}^{-}(\xi’, 0)=\hat{h}_{k}^{+}(\xi’, 0)-\hat{h}_{k}^{-}(\xi’, 0)$,$k=1$ ,

, $k=1$ ,$\cdots$$\cdots$, $n-1_{1}$, $n-1$ .

By the first equation of (5.42),$\hat{z}_{k}^{+}(\xi’, x_{n})=c^{+}(\xi’)e^{-Bx_{n}}+(x_{n}>0),\hat{z}_{k}^{-}(\xi’, x_{n})=c^{-}(\xi’)e^{B_{-}x_{n}}(x_{n}<0)$.

By the interface conditions of (5.42), we have
(5.43)$\hat{z}_{k}^{\pm}(\xi’, x_{n})=e^{-B(\pm x_{n})}\pm[-(\mu_{+}B_{+}+\mu_{-}B_{-})^{-1}(\hat{l}_{k}^{+}(\xi’, 0)-\hat{l}_{k}^{-}(\xi’, 0))$$\pm\mu\pm B_{\pm}(\mu_{+}B_{+}+\mu_{-}B_{-})^{-1}(\hat{h}_{k}^{+}(\xi’, 0)-\hat{h}_{k}^{-}(\xi’, 0))]$

By Theorem 3.1, (5.40) and the Fourier multiplier theorem, we have
(5.44)$||w_{k}^{\pm}||_{W_{p}^{2}(\mathbb{R}^{n})} \leqq C(p, n)||g_{k}^{\pm}||_{L_{p}(\mathbb{R}_{\pm}^{n})}\leqq C(p, n)||\frac{\partial\theta^{\pm}}{\partial x_{k}}||_{L_{p}(\mathrm{R}_{\pm}^{n})}$$\leqq C(p, n)\sum_{+-}(||a^{\pm}||_{W_{p}^{1}(\mathbb{R}_{\pm}^{n})}+||b^{\pm}||_{W_{p}^{2}(\mathbb{R}_{\pm}^{n})})$

.

.

152
By (5.39), (5.44) and the Fourier multiplier theorem, we have$||z_{k}^{\pm}||_{W_{p}^{2}(\mathbb{R}_{\pm}^{n})} \leqq C(p, n)\sum_{+-}(||l^{\pm}||_{W_{p}^{1}(\mathbb{R}_{\pm}^{n})}+||h^{\pm}||_{W_{p}^{2}(\mathbb{R}_{\pm}^{n})})$$\leqq C(p, n)\sum_{+-}(||a^{\pm}||_{W_{p}^{1}(\mathbb{R}_{\pm}^{n})}+||b^{\pm}||_{W_{p}^{2}(\mathbb{R}_{\pm}^{n})})$

when A$\in\Sigma_{\epsilon}$and$|\lambda|=1$,. Therefore we obtain$||v_{k}^{\pm}||_{W_{p}^{2}(\mathbb{R}_{\pm}^{n})} \leqq C(p, n)\sum_{+-}(||a^{\pm}||_{W_{p}^{1}(\mathbb{R}_{\pm}^{n})}+||b^{\pm}||_{W_{p}^{2}(\mathbb{R}_{\pm}^{n})})$,

which completes the proof of Theorem 5.3.
Now we shall discuss the uniqueness of solutions to (5.1). To do this we use thefollowing lemma (cf. Galdi [6, III]).
Lemma 5.5. Let $1<p<\infty$ ,Put$\pi\in X_{p}^{1}(\mathrm{E})$, and v,$\psi(t)=\{$

10

with $1/p+1/p’=1$ .

for any $R>1$ ,$\phi_{R}(x)=\psi(\ln(\ln|x|)/\ln(\ln R))$$\psi(t)\in C_{0}^{\infty}(\mathbb{R})$$\in L_{p},(\mathrm{E})$,$|t|\leqq 1/2$$|t|\geqq 1$.

Then we have
(5.45)$\lim_{Rarrow\infty}\int_{\mathrm{R}_{\pm}^{n}}|\frac{\partial}{\partial x_{j}}\phi_{R}(x)||\pi(x)||v(x)|dx=0$

Theorem 5.6. Let $1<p<\infty$ , $0<\epsilon<\pi/2$ $and$ )satisfies the homogeneous equation:

j $=1$ , \cdots , n.

,$\in\Sigma_{\epsilon}$. If$(u^{\pm}, \pi^{\pm})\in W_{p}^{2}(\mathbb{R}_{\pm}^{n})\cross$$X_{p}^{1}(\mathrm{E})$$\lambda u^{\pm}-DivT^{\pm}(u^{\pm}, \pi^{\pm})=0$(5.46),$\nabla\cdot$$u^{\pm}=0$

in

,

on

$\{$

$\nu\cdot T^{+}(u^{+}, \pi^{+})-\nu\cdot T^{-}(u^{-}, \pi^{-})=0$

then $u^{+}=u^{-}=0$ and there exists a constant

Proof. Let

such that$\phi_{R}$$c$$u^{+}=u^{-}$$\mathrm{R}_{0}^{n}$,,such that $\pi^{\pm}=c$ .

be the same function as in Lemma 5.5. For every, we haveinand $v^{+}=v^{-}$ on$\nabla\cdot v^{\pm}=0$$\ovalbox{\tt\small REJECT}$$v^{\pm}\in W_{p}^{2}(\mathbb{R}_{\pm}^{n})$$\mathfrak{W}$$\mathrm{E}$$0=$ $(\lambda u^{+}-\mathrm{D}\mathrm{i}\mathrm{v}T^{+}(u^{+}, \pi^{+})$,$\phi_{R}v^{+})_{\mathrm{R}_{+}^{n}}+(\lambda u^{-}-\mathrm{D}\mathrm{i}\mathrm{v}T^{-}(u^{-}, \pi^{-}),$ $\phi_{R}v^{-})_{\mathrm{R}_{-}^{n}}$$=\lambda(u^{+}, \phi_{R}v^{+})_{\mathrm{R}_{+}^{n}}+\lambda(u^{-}, \phi_{R}v^{-})_{\mathrm{R}_{-}^{n}}$$-<\nu\cdot T^{+}(u^{+}, \pi^{+})$,$+ \sum_{+-}[(T^{\pm}(u^{\pm}, \pi^{\pm})$

Since $v^{+}=v^{-}$ on$\mathrm{R}_{0}^{n}$$\phi_{R}v^{+}>_{\mathrm{R}_{0}^{n}}+<\nu\cdot T^{-}(u^{-}, \pi^{-})$,,$\phi_{R}v^{-}>_{\mathrm{R}_{0}^{n}}$$(\nabla\phi_{R})v^{\pm})_{\mathrm{R}_{\pm}^{n}}+2\mu\pm(D(u^{\pm}), D(v^{\pm})\phi_{R})_{\mathrm{R}_{\pm}^{n}}]$.

, by the interface condition,$-<\nu\cdot T^{+}(u^{+}, \pi^{+})$,$\phi_{R}v^{+}>_{\mathrm{R}_{0}^{n}}+<\nu\cdot T^{-}(u^{-}, \pi^{-})$,$\phi_{R}v^{-}>_{\mathrm{R}_{0}^{n}}=0$

.

153
Since

$u^{\pm}$,$v^{\pm}\in W_{p}^{2}(\ovalbox{\tt\small REJECT})$

and

$\pi$

, by Lemma 5.5 we have$\in X_{p}^{1}(\ovalbox{\tt\small REJECT})$$\lim_{Rarrow\infty}(T^{\pm}(u^{\pm}, \pi^{\pm})$,$(\nabla\phi_{R})v^{\pm})_{\mathbb{R}_{\pm}^{n}}=0$.

Therefore by letting $Rarrow\infty$ , we have
(5.47)$0=\lambda(u, v)_{\mathbb{R}^{n}}+2\mu_{+}(D(u^{+}), D(v^{+}))_{\mathbb{R}_{+}^{n}}+2\mu_{-}(D(u^{-}), D(v^{-}))_{\mathbb{R}_{-}^{n}}$Theorem 5.1 implies for any,
which satisfies$X_{p}^{1}$$f^{\pm}\in C_{0}^{\infty}(\mathbb{R}_{\pm}^{n})$, there exists

.

, (E)

$(v^{\pm}, \theta^{\pm})\in W_{p}^{2}$$\cross$$(\ovalbox{\tt\small REJECT})$,$\nabla\cdot v^{\pm}=0$$\overline{\lambda}v^{\pm}-\mathrm{D}\mathrm{i}\mathrm{v}T^{\pm}(v^{\pm}, \theta^{\pm})=f^{\pm}$$\{$$\nu\cdot T^{+}(v^{+}, \theta^{+})-\nu\cdot T^{-}(v^{-}, \theta^{-})=0$,$v^{+}=v^{-}$

inon$\ovalbox{\tt\small REJECT}$$\mathrm{R}_{0}^{n}$,,In the same manner as above$(u, f)= \lim_{Rarrow\infty}\{(u^{+}, f^{+}\phi_{R})_{\mathbb{R}_{+}^{n}}+(u^{-}, f^{-}\phi_{R})_{\mathbb{R}_{-}^{n}}\}$$= \lim_{Rarrow\infty}\{(\phi_{R}u^{+},\overline{\lambda}v^{+}-\mathrm{D}\mathrm{i}\mathrm{v}T^{+}(v^{+}, \Phi^{+}))_{\mathbb{R}_{+}^{n}}+(\phi_{R}u^{-},\overline{\lambda}v^{-}-\mathrm{D}\mathrm{i}\mathrm{v}T^{-}(v^{-}, \Phi^{-}))_{\mathbb{R}_{-}^{n}}\}$$=\lambda(u, v)_{\mathbb{R}^{n}}+2\mu_{+}(D(u^{+}), D(v^{+}))_{\mathbb{R}_{+}^{n}}+2\mu_{-}(D(u^{-}), D(v^{-}))_{\mathbb{R}_{-}^{n}}$By (5.47) we have
$u=0$ . By (5.46) we have$(u, f)_{\mathbb{R}^{n}}=0$.

. The arbitrariness of the choice of$f$implies that

, $\nu(\pi^{+}-\pi^{-})=0$ inin,$\pi^{\pm}=c$which implies that there exists aconstant such that
. This completes theproof of the theorem.
$\nabla\pi^{\pm}=0$$\mathbb{R}_{\pm}^{n}$$\mathbb{R}_{0}^{n}$$c$$\square$56. The bended space for the Stokes System with interface conditionLet :are all bounded in$\omega$$H^{0}$$\mathbb{R}^{n-1}arrow \mathbb{R}$be abounded function inclass whose derivative up to 3
. Let
be the bended space defined as$C^{3}$$H^{\pm}$$\mathbb{R}^{n-1}$$H^{+}=\{x=(x’, x_{n})\in \mathbb{R}^{n}|x_{n}>\omega(x’), x’\in \mathbb{R}^{n-1}\}$,$H^{-}=\{x=(x’, x_{n})\in \mathbb{R}^{n}|x_{n}<\omega(x’), x’\in \mathbb{R}^{n-1}\}$.

denotes the interface of$H^{+}$

and$H^{-}$, which is given by.

$H^{0}=\{x=(x’, x_{n})\in \mathbb{R}^{n}|x_{n}=\omega(x’), x’\in \mathbb{R}^{n-1}\}$$\nu H(x)$denotes the unit outer normal to$\nu_{H}(x)=(\nabla’\omega, -1)/\sqrt{1+|\nabla’\omega|^{2}}$$H^{0}$,

of$H^{+}$, namely$\nabla’\omega=(\partial\omega/\partial x_{1}, \cdots, \partial\omega/\partial x_{n-1})$.

Put$X_{p}^{1}(H^{\pm})=\{u^{\pm}(x)\}\tilde{u}^{\pm}(y)--u^{\pm}(y’, y_{n}+\omega(y’))\in X_{p}^{1}(\mathbb{R}_{\pm}^{n})\}$,.

$W_{p}^{-1}(H^{\pm})=\{u^{\pm}(x)|\tilde{u}^{\pm}(y)=u^{\pm}(y’, y_{n}+\omega(y’))\in W_{p}^{1}(\mathbb{R}_{\pm}^{n})\}$We use the following symbles in this section:$x\in H^{+}$ ,$u^{+}(x)$$u(x)=\{$$\pi(x)=\{$$x\in H^{-}$ ,$u^{-}(x)$$f(x)=\{$$f^{+}(x)$$x\in H^{+}$$g^{+}(x)$$g(x)=\{$,For the resolvent problem in$f^{-}(x)$$x\in H^{-}$$\pi^{-}(x)$,$x\in H^{-}$ ,$x\in H^{+}$$x\in H^{+}$$h^{+}(x)$$x\in H^{+}$$h^{-}(x)$$x\in H^{-}$$x\in H^{-},’ h(x)=\{$$g^{-}(x)$$H^{\pm}$$\pi^{+}(x)$, we shall show the following theorem.

,

154
Theorem 6.1. Let $1<p<\infty$ and $0<\epsilon<\pi/2$ . Then there exist constants
and $K_{0}=K_{0}(p, \epsilon, n)$ with $0<K_{0}\leqq 1$ such that
, $f^{\pm}\in L_{p}(H^{\pm})^{n}$ ,with, then for every
if, tftere exists
Aavtny compact support andto the equation:a$\lambda_{0}=\lambda_{0}(p, \epsilon, ||\omega||_{B^{3}(\mathbb{R}^{n-1})}, n)\geqq 1$$|\lambda|\geqq\lambda_{0}$$\lambda\in\Sigma_{\epsilon}$$||\nabla’\omega||_{L_{\infty}(\mathrm{R}^{n-1})}\leqq \mathrm{K}\mathrm{o}$$h^{\pm}\in W_{p}^{1}(H^{\pm})^{n}$$g\in W_{p}^{1}(\mathbb{R}^{n})\cap W_{p}^{-1}(\mathbb{R}^{n})$$s\mathrm{o}/utton$$(u^{\pm}, \pi^{\pm})\in W_{p}^{2}(H^{\pm})^{n}\cross X_{p}^{1}(H^{\pm})$$\lambda u^{\pm}-DivT^{\pm}(u^{\pm}, \pi^{\pm})=f^{\pm}$

(6.1)

,

$\nabla\cdot$$u^{\pm}=g^{\pm}$$\{$$\nu\cdot T^{+}(u^{+}, \pi^{+})-\nu\cdot$ $T^{-}(u^{-},

Moreover, the$(u^{\pm}, \pi^{\pm})$\pi^{-})=h^{+}-h^{-}$,$u^{+}=u^{-}$

in$H^{\pm}$,on$H^{0}$.

satisfies the estimate:$\sum_{+-}(|\lambda|||u^{\pm}||_{L_{p}(H^{\pm})}+|\lambda|^{\frac{1}{2}}||\nabla u^{\pm}||_{L_{p}(H)}\pm+||\nabla^{2}u^{\pm}||_{L_{p}(H^{\pm})}+||\pi^{\pm}||_{X_{p}^{1}(H^{\pm}))}$$\leqq C(||f||_{L_{p}(\mathrm{R}^{n})}+|\lambda|||g||_{W_{p}^{-1}(\mathrm{R}^{n})}+|\lambda|^{1}||g||_{L_{p}(\mathrm{R}^{n})}$$+|| \nabla g||_{L_{p}(\mathrm{R}^{n})}+|\lambda|^{\frac{1}{2}}||h||_{L_{p}(\mathrm{R}^{n})}+\sum_{+-}||\nabla h^{\pm}||_{L_{p}(H^{\pm}))}$

with some constant$C=C(p, \epsilon, ||\omega||_{B^{3}(\mathrm{R}^{n-1})}, n)>0$.

$\# ene$, we set$|| \omega||_{B^{3}(\mathrm{R}^{n-1})}=,\sum_{|\alpha|\leqq 3}||\partial_{x}^{\alpha’},\omega||_{L_{\infty}(\mathrm{R}^{n-1})}$.

By using the diffeomorphism, we reduce (6.1) to (5.1). Therefore by Theorem
5.1 we can prove Theorem 6.1.
Now we shall show the uniqueness of (6.1). To do this we start with the following
lemma.
is theLemma 6.2. let $1<p<\infty$ . Put $\rho_{R}(x)=\phi_{R}(x’, x_{n}-\omega(x’))$ where
and $v\in L_{p},(H^{\pm})^{n}$ withsame as in Lemma 5.5. Then for every$1/p+1/p’=1$ we have$\phi_{R}$$\pi\in X_{p}^{1}(H^{\pm})$

,

(6.2)

$R \lim_{arrow\infty}\int_{H^{\pm}}|\frac{\partial}{\partial x_{j}}\rho_{R}(x)||\pi(x)||v(x)|dx=0$

j $=1$ , \cdots , n.

Proof. By the change of variables: $x’=y’$ , $x_{n}=y_{n}+\omega(y’)$ ,, $j=1,\cdots,n-1$ ,$\frac{}{\partial x_{n}},\rho R(x)=\frac{\partial y_{j}\partial\partial}{\partial y_{n}}\rho_{R}(y)\frac{\partial}{\partial x_{j},\partial}\rho_{R}(x)=\frac\rho_{R}(y),-\frac{\partial\omega}{\partial x_{j}}\frac{\partial}{\partial y_{n}}\rho_{R}(y)$so we have
(6.3)$\int_{H^{\pm}}|\frac{\partial}{\partial x_{j}}\rho_{R}(x)||\pi(x)||v(x)|dx$$\leqq C\int_{\mathrm{R}_{\pm}^{n}}|\nabla\phi_{R}(y)||\tilde{\pi}(y)||\tilde{v}(y)|dy$,$j=1$ ,$\cdots$

, ,
$n$

155
where $\tilde{\pi}(y)=\pi(x),\tilde{v}(y)=v(x)$ , and $C$ is apositive number. By the definition ofand. Therefore by Lemma 5.5,
and
we have$X_{p}^{1}(H^{\pm})$$L_{p’}(H^{\pm}),\tilde{\pi}\in X_{p}^{1}(\mathbb{R}_{\pm}^{n})$(6.4)$\tilde{v}\in L_{p’}(\mathbb{R}_{\pm}^{n})$$\int_{\mathbb{R}_{\pm}^{n}}|\nabla\phi_{R}(y)||\tilde{\pi}(y)||\tilde{v}(y)|dy=0$.

Combining (6.3) and (6.4) implies (6.2). We have thus proved the lemma.
Theorem 6.3. Let $1<p<\infty$ , $0<\epsilon<\pi/2$ andsatisfies the homogeneous equation:$\lambda\in\Sigma_{\epsilon}$. If$\square$$(u^{\pm}, \pi^{\pm})\in W_{p}^{2}(H^{\pm})\cross$$X_{p}^{1}(H^{\pm})$,$\nabla\cdot u^{\pm}=0$$\lambda u^{\pm}-DivT^{\pm}(u^{\pm}, \pi^{\pm})=0$

(6.5)

$\{$

$T^{+}(u^{+}, \pi^{+})-\nu\cdot$ $T^{-}(u^{-}, \pi^{-})=0$$\nu\cdot$

then $u^{+}=u^{-}=0$ in

Proof. Let$H^{\pm}$,$u^{+}=u^{-}$

in$H^{\pm}$on$H^{0}$,,and there exists a constant c such that $\pi^{\pm}=c$ in$v^{\pm}=0$
withinmanner as the proof of Theorem 5.6, we have$v^{\pm}\in W_{p}^{2},(H^{\pm})$$\nabla\cdot$$H^{\pm}\mathrm{a}\mathrm{n}\mathrm{d}v^{+}=v^{-}$

on$H^{0}$$H^{\pm}$. In the same

$0=\lambda(u, v)_{\mathbb{R}^{n}}+2\mu_{+}(D(u^{+}), D(v^{+}))_{H}++2\mu_{-}(D(u^{-}), D(v^{-}))_{H}-$,

where we have used Lemma 6.2 and the interface condition of (6.1). For, let
be asolution to$C_{0}^{\infty}(H^{\pm})$.

$f^{\pm}\in$$(v^{\pm}, \theta^{\pm})\in W_{p}^{2},(H^{\pm})^{n}\cross X_{p}^{1}(H^{\pm})$$\overline{\lambda}v^{\pm}-\mathrm{D}\mathrm{i}\mathrm{v}T^{\pm}(v^{\pm}, \theta^{\pm})--f^{\pm}$,$\nabla\cdot v^{\pm}=0$$\{$$\nu\cdot$$T^{+}(v^{+}, \theta^{+})-\nu\cdot T^{-}(v^{-}, \theta)=0$,$v^{+}=v^{-}$

in$H^{\pm}$,on$H^{0}$.

Since$(u, f)=(u^{+}, f^{+})_{H+}+(u^{-}, f^{-})_{H}-$$=\lambda(u, v)_{\mathbb{R}^{n}}+2\mu_{+}(D(u^{+}), D(v^{+}))_{H+}+2\mu_{-}(D(u^{-}), D(v^{-}))_{H}-$

we have$(u, f)_{\mathbb{R}^{n}}=0$

by (6.5),$\pi^{\pm}=c$. The arbitrariness of the choice of$f$. This completes the proof of the theorem.

,

implies that $u=0$ , and$\square$

We end this section with outline of the proof of Theorem 1.2. Let, and then (1.1) is reduced to the equation:
in$\varphi$

be afunction$C_{0}^{\infty}(\mathbb{R}^{n})$$\lambda(\varphi u^{\ell})-\mathrm{D}\mathrm{i}\mathrm{v}T^{\ell}(\varphi u^{\ell}, \varphi\pi^{\ell})=f_{\varphi}^{\ell}$,$\nabla\cdot(\varphi u^{\ell})=g_{\varphi}^{\ell}$$\nu^{1}\cdot T^{1}(\varphi u^{1}, \varphi\pi^{1})-\nu^{1}\cdot T^{2}(\varphi u^{2}, \varphi\pi^{2})=h_{\varphi}^{1}-h_{\varphi}^{2}$

(6.6)

$\{$

$\varphi u^{1}=\varphi u^{2}$(6.7)$f_{\varphi}^{\ell}=(f_{\varphi_{1}}^{\ell},$\cdots ,$\Omega^{\ell}$$\Gamma^{1}$$\varphi u^{2}=0$

where

in , $\ell=1,2$ ,on I 1,on ,on ,$\Gamma^{2}$$f_{\varphi_{n}}^{\ell})$,\cdots ,$h_{\varphi}^{\ell}=(h_{\varphi_{1}}^{\ell},$$h_{\varphi_{n}}^{\ell})$,$f_{\varphi k}^{\ell}= \varphi f_{k}^{\ell}-\mu\ell\sum_{j=1}^{n}[2\frac{\partial\varphi}{\partial x_{j}}D_{jk}(u^{\ell})+\frac{\partial}{\partial x_{j}}(\frac{\partial\varphi}{\partial x_{k}}u_{j}^{\ell}+\frac{\partial\varphi}{\partial x_{j}}u_{k}^{\ell})]+\frac{\partial\varphi}{\partial x_{k}}\pi^{\ell}$$h_{\varphi_{k}}^{\ell}= \varphi h_{k}^{\ell}+\mu\ell\sum_{j=1}^{n}\nu_{j}^{1}(\frac{\partial\varphi}{\partial x_{j}}u_{k}^{\ell}+\frac{\partial\varphi}{\partial x_{k}}u_{j}^{\ell)}$$g_{\varphi}^{\ell}=\nabla$.

$(\varphi u^{\ell})=(\nabla\varphi)\cdot u^{\ell}$

,

,

156
for \yen is suitably extended into
as avector of functions inhaving the compact supports. Applying the standard argument to (6.6) by using
Theorem 3.2, Theorem 4.1 and Theorem 6.1 we shall derive Theorem 1.2.
$\mathrm{R}^{n}$$C^{3}(\mathrm{R}^{n})$\S 7. Aproof of Theorem 1.1
In this section, we shall prove Theorem 1.1. By Theorem 6.1, we obtain thefollowing lemma.
Lemma 7.1. Let $1<p<\infty$ , $0<\epsilon<\pi/2$ , A
and
. If

$(u, \pi)\in D_{0}^{1}(\Omega)^{n}\cross L_{p}(\Omega)$$h^{\ell}\in W_{p}^{1}(\Omega^{\ell})^{n}$(7.1)

satisfies$( \lambda u, v)_{\Omega}+2\sum_{\ell=1}^{2}\mu\ell(D(u^{\ell}), D(v^{\ell}))_{\Omega^{p}}-(\pi, \nabla$.

$\pi)\in W_{p}^{1}(\Omega)^{n}\cross\tilde{W}_{p}^{1}(\Omega)$with$u^{\ell}\in W_{p}^{2}(\Omega^{\ell})^{n}$$\int_{\Omega}\pi dx=0$$f\in L_{p}(\Omega)^{n}$and$v)_{\Omega}$$=(f,v)_{\Omega}+<h^{1}-h^{2}$then (u,equation (1.1).

,$\in\Sigma_{\epsilon}\cup\{\lambda\in \mathbb{C}||\lambda|\leqq\sigma\}$,for$v>_{\Gamma^{1}}$. Moreover$\forall v\in W_{p}^{1},(\Omega)$,satisfies the$(u^{\ell}, \pi^{\ell})$

By Lemma 7.1 and the Sobolev imbedding theorem, we have the following lemma.

Lemma 7.2. Let $1<p<\infty$ , $0<\epsilon<\pi/2$ , A,, (1.1) admits a solution
any
.
with

$\in\Sigma_{\epsilon}\cup\{\lambda\in \mathbb{C}||\lambda|\leqq\sigma\}$$f\in C_{0}^{\infty}(\Omega)^{n}$$h\in C^{\infty}(\overline{\Omega})^{n}$. TAen for$(u, \pi)\in W_{p}^{1}(\Omega)^{n}\cross\tilde{W}_{p}^{1}(\Omega)$$u^{\ell}\in W_{p}^{2}(\Omega^{\ell})^{n}$

By Lemma 7.2, we obtain the following lemma.
Lemma 7.3. Let $1<p<\infty$ , $0<\epsilon<\pi/2$ , A
. If
with
homogeneous
equation:thesatisfies$\in\Sigma_{\epsilon}\cup\{\lambda\in \mathbb{C}||\lambda|\leqq\sigma\}$$(u, \pi)\in W_{p}^{1}(\Omega)^{n}\cross\tilde{W}_{p}^{1}(\Omega)$(7.2)$u^{\ell}\in W_{p}^{2}(\Omega^{\ell})^{n}$$\{\begin{array}{l}\lambda u^{\ell}-DivT^{\ell}(u^{\ell},\pi^{\ell})=0,\nabla\cdot u^{\ell}=0in\Omega^{\ell},l=1,2\nu^{1}\cdot T^{1}(u^{1},\pi^{1})-\nu^{1}\cdot T^{2}(u^{2},\pi^{2})=0,u^{1}=u^{2}on\Gamma^{\mathrm{l}}u^{2}=0on\Gamma^{2}\end{array}$then $u^{\ell}=0$ and$\pi^{\ell}=0$.

Now we shall show the aprioi estimate, following Farwig-Sohr [4, Lemma 4.2].
Lemma 7.4. Let $1<p<\infty$ , $0<\epsilon<\pi/2$ , A
satisfywith$u^{2}=0$, andon . Put$\in\Sigma_{\epsilon}\cup\{\lambda\in \mathbb{C}||\lambda|\leqq\sigma\}$$(u, \pi)\in W_{p}^{1}(\Omega)^{n}\cross\tilde{W}_{p}^{1}(\Omega)$$\Gamma^{1}$$u^{\ell}\in W_{p}^{2}(\Omega^{\ell})^{n}$$\nabla\cdot$$u^{\ell}=0$in$\Omega^{\ell}$

,

. Let

$u^{1}=u^{2}$on$\Gamma^{2}$$f^{\ell}=\lambda u^{\ell}-DivT^{\ell}(u^{\ell}, \pi^{\ell})$,$h^{\ell}=\nu^{1}\cdot$$T^{\ell}(u^{\ell}, \pi^{\ell})$,$\ell=1,2$.

TAen (1.3) holds

Proof. By Theorem 1.2, there exists a
when Awith. If we taken,such that (1.6) holds, thenso large that$\lambda_{0}=\lambda_{0}(p,$$\in\Sigma_{\epsilon}$$|\lambda|\geqq\lambda 0$$\lambda_{1}\geqq\lambda_{0}$$\epsilon, \Omega)\geqq 1$$\lambda_{1}^{-1/2}C\leqq 1/2$

157
from (1.6) we have
(7.3)$| \lambda|||u||_{L_{p}(\Omega)}+|\lambda|^{\frac{1}{2}}||\nabla u||_{L_{p}(\Omega)}+\sum_{\ell=1}^{2}||u^{\ell}||_{W_{p}^{2}(\Omega^{\ell})}+||\pi||_{\tilde{W}_{p}^{1}(\Omega)}$$\leqq C(||f||_{L_{p}(\Omega)}+|\lambda|^{\frac{1}{2}}||h||_{L_{p}(\Omega)}+\sum_{\ell=1}^{2}||h^{\ell}||_{W_{p}^{1}(\Omega^{p})}|\lambda|||u||_{W_{p}^{-1}(\Omega)}+||\pi||_{L_{p}(\Omega))}$for A6equation:$\Sigma_{\epsilon}$with$|\lambda|\geqq\lambda_{1}$. When$0<|\lambda|\leqq\lambda \mathrm{i}$$\lambda u^{\ell}-\mathrm{D}\mathrm{i}\mathrm{v}T^{\ell}(u^{\ell}, \pi^{\ell})=f^{\ell}+(\lambda_{1}-\lambda)u^{\ell}$$\{$, observing that $(u, \pi)$ satisfies the,$\nabla\cdot u^{\ell}=0$$\nu^{1}\cdot T^{1}(u^{1}, \pi^{1})-\nu^{1}\cdot T^{2}(u^{2}, \pi^{2})=h^{1}-h^{2}$,$u^{1}=u^{2}$$u^{2}=0$i$\Omega^{\ell}$$\mathrm{n}$onon, $\ell=1,2$ ,,,$\Gamma^{1}$$\Gamma^{2}$and noting that
, we have (7.3) for.
To prove the lemma, it suficies to show that there exists aconstant $C=$
such that$||(\lambda_{1}-\lambda)u^{\ell}||_{L_{p}(\Omega^{p})}\leqq 2\lambda_{1}||u^{\ell}||_{L_{p}(\Omega^{\ell})}(\ell=1,2)$$0<|\lambda|\leqq\lambda_{1}$$C(p, \epsilon, n, \Omega, \sigma)$(7.4)$|\lambda|||u||_{W_{p}^{-1}(\Omega)}+||u||_{L_{p}(\Omega)}+||\pi||_{L_{p}(\Omega)}$$\leqq C(||f||_{L_{p}(\Omega)}$

十$| \lambda|^{\frac{1}{2}}||h||_{L_{p}(\Omega)}+\sum_{\ell=1}^{2}||h’||_{W_{p}^{1}(\Omega^{\ell}))}$for A. To show (7.4), it sufficies to derive acontradiction
from the following condition: For every integer there exist
with,, $(\ell =1,2)$ ,in
on , $u_{k}^{2}=0$ on ,andsuch that if we put$\in\Sigma_{\epsilon}\cup\{\lambda\in \mathbb{C}||\lambda|\leqq\sigma\}$$k$$u_{k}^{\ell}=0$$\Omega^{\ell}$$u_{k}^{\ell}\in W_{p}^{2}(\Omega^{\ell})^{n}$$\nabla\cdot$$\pi_{k}\in\tilde{W}_{p}^{1}(\Omega)$$\lambda_{k}\in\Sigma_{\epsilon}\cup\{\lambda\in \mathbb{C}||\lambda|\leqq\sigma\}$(7.3)$\lambda u_{k}^{\ell}-\mathrm{D}\mathrm{i}\mathrm{v}T^{\ell}(u_{k}^{\ell}, \pi_{k}^{\ell})=f_{k}^{\ell}$$u_{k}^{1}=u_{k}^{2}$$u_{k}\in W_{p}^{1}(\Omega)^{n}$$\Gamma^{1}$$\Gamma^{2}$,$\nu^{1}\cdot T^{1}(u_{k}^{1}, \pi_{k}^{1})|_{\Gamma^{1}}-\nu^{1}\cdot T^{2}(u_{k}^{2}, \pi_{k}^{2})|_{\Gamma^{1}}=h^{1}|_{\Gamma^{1}}-h^{2}|_{\Gamma^{1}}$

,

then
(7.6)

(7.7)

$||f_{k}||_{L_{p}(\Omega)}+| \lambda_{k}|^{\frac{1}{2}}||h_{k}||_{L_{p}(\Omega)}+\sum_{\ell=1}^{2}||\nabla h_{k}^{\ell}||_{L_{p}(\Omega^{\ell})}<1/k$$|\lambda_{k}|||u_{k}||_{W_{p}^{-1}(\Omega)}+||u_{k}||_{L_{p}(\Omega)}+||\pi_{k}||_{L_{p}(\Omega)}=1$,.

Combining (7.6), (7.7) and (7.3) we have
(7.8)$| \lambda_{k}|||u_{k}||_{L_{p}(\Omega)}+|\lambda_{k}|^{\frac{1}{2}}||\nabla u_{k}||_{L_{p}(\Omega)}+\sum_{\ell=1}^{2}||u_{k}^{\ell}||_{W_{p}^{2}(\Omega^{p})}+||\pi_{k}||_{\tilde{W}_{p}^{1}(\Omega)}$$\leqq C(1/k+1)\leqq 2C$ ,.

$\forall k\in \mathrm{N}$158
Since Q is bounded, passing to the subsequence if necessary, we may assume that, and vrEsuch that, u cEwiththere exist vE$L_{p}(\mathrm{O})^{n}$(7.9)$u_{k}^{\ell}arrow u^{\ell}$$\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}\mathrm{l}\mathrm{y}*$$\lambda_{k}u_{k}arrow v$$\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}\mathrm{l}\mathrm{y}*$$\pi_{k}arrow\pi$(7.10)$u\cdot \mathrm{I}/\mathrm{y}\mathrm{p}2(\mathrm{g}/)^{\mathrm{n}}$$W\ovalbox{\tt\small REJECT}((l)^{n}$$\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}\mathrm{l}\mathrm{y}*$inin$W_{p}^{2}(\Omega^{\ell})^{n}$instrongly instrongly in$u_{k}arrow u$$\pi_{k}arrow\pi$,$W_{p}^{1}(\Omega)$,$W_{p}^{1}(\Omega)^{n}$$L_{p}(\Omega)$strongly in$\lambda_{k}u_{k}arrow v$$L_{p}(\Omega)^{n}$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}3(0)$,$\ell=1,2$

,

,

,

$W_{p}^{-1}(\Omega)^{n}$.

In particular the last assertion of (7.10) was showed in [8, Proof of Lemma 7.4].
By (7.7) and (7.10), we have
(7.11)$||v||_{W_{p}^{-1}(\Omega)}+||u||_{L_{p}(\Omega)}+||\pi||_{L_{p}(\Omega)}=1$.

Now studying two cases, we shall derive acontradiction to (7.11).

Case 1.

$|\lambda_{k}|arrow\infty$

as k$arrow\infty$.

, and therefore $u_{k}arrow 0$ strongly in
By (7.8), we have
which conbined with (7.9) and (7.11) implies u $=0$ in and$||u_{k}||_{L_{p}(\Omega)}\leqq|\lambda_{k}|^{-1}2C$$L_{p}(\Omega)$$\Omega$(7.12)$||v||_{W_{p}^{-1}(\Omega)}+||\pi||_{L_{p}(\Omega)}=1$

Letting k$arrow\infty$.

, by (7.5), (7.6) and (7.9) with u $=0$ , we have$v+\nabla\pi=0$(7.13),$\{$$\nu^{2}\cdot v|_{\Gamma^{2}}=0$$\nabla\cdot$

$v=0$

in

$\Omega$.

By the uniqueness of the Helmholtz decomposition to (7.13), we obtain $v=0$ and$\nabla\pi=0$ .
, we have $\pi=0$ . This leads to acontradiction to (7.11).
Since$\int_{\Omega}\pi dx=0$

Case 2.

$\lambda_{k}arrow\lambda$

as k$arrow\infty$.

, we have. Letting$karrow\infty$ , by (7.5), (7.6) and (7.9) we see thatwithsatisfy the homogeneous equation (7.2). By lemma 7.3, we have$\pi=0$$=0$ strongly in, which$u=0$ and. By (7.10), we have
$v=0$
. Therefore we havecombined with the last assertion of (7.10) implies that$u=0$ , $v=0$ and $\pi=0$ , which leads to acontradiction to (7.11). This completes
the proof of the Lemma.

Since$\lambda_{k}\in\Sigma_{\epsilon}\cup\{\lambda\in \mathbb{C}||\lambda|\leqq\sigma\}$$\lambda\in\Sigma_{\epsilon}\cup\{\lambda\in \mathbb{C}||\lambda|\leqq\sigma\}$$(u, \pi)\in W_{p}^{1}(\Omega)^{n}\cross\tilde{W}_{p}^{1}(\Omega)$$u^{\ell}\in W_{p}^{2}(\Omega^{\ell})^{n}$$\lambda_{k}u_{k}arrow\lambda u$$W_{p}^{1}(\Omega)^{n}$$\square$

,
and
and
are dense in
A proof of Theorem 1.1. Since
respectively, by Lemma 7.2, Lemma 7.3 and Lemma 7.4 we can show Theorem 1.1.
$C_{0}^{\infty}(\Omega)$$C^{\infty}(\overline{\Omega})$$L_{p}(\Omega)$$W_{p}^{1}(\Omega)$

159
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