数理解析研究所講究録
第 1985 巻 2016 年 116-137

116

On the Cattabriga problem appearing in
the two phase problem of the viscous fluid flows
Yoshihiro SHIBATA*

Abstract
In this paper, we report results concerning the two phase problem for the viscous fluid flows
without surface tension in a bounded region, which was announced in the RIMS Workshop on
Mathematical Analysis in Fluid and Gas Dynamics organized by Professor Takayuki Kobayashi of
Osaka University and Professor Tatsuo Iguchi of Keio University held at RIMS, Kyoto University,
July 8-10, 2015. Especially, we prove the unique existence theorem for the Cattabriga problem which
is obtained as a statinary problem for the linearized two phase problem system. Moreover, we proved
the unique existence theorem for some weak Dirichlet problem with jump condition on the interface.

Mathematics Subject Classification (2012). $35Q30,$ $76D27,$ $76N10,$
Keywords. two phase problem, viscous fluid flows, Cattabriga problem

1

Introduction, and letbe a subdomain
Let be a bounded domain in $N$-dimensional Euclidean space
occupied by some viscous fluids. Let -and be the boundary. The
of . Let. Assume that, respectively. Note tht
and are compact hypersurfaces ofand, respectively.
andand ,-be the time evolution of
claae $(N<r<\infty)$ . Let. Then, the two phase problem for the viscous fluids without
Set
,-and
surface tension is formulated mathematically as follows:$\mathbb{R}^{N}(N\geq 2)$$\Omega$$\Omega$$\Omega$$\Omega_{-}=\Omega\backslash \overline{\Omega_{+}}$$\Omega\pm are$$\Gamma_{-}\cap\Gamma=\emptyset$$\Omega_{+}$$W_{r}^{2-1/r}$$\Omega_{t,\pm},$$\dot{\Omega}_{t}=\Omega_{t,+}\cup\Omega_{t}$$\Omega_{+}$$\Gamma$$\Gamma$$\Gamma_{t}$$\Gamma_{-}$$\Gamma$$\Omega\pm,$$\Gamma_{\ell}$$\Gamma$$\Gamma_{-}$$\dot{\Omega}=\Omega_{+}\cup\Omega_{-}$(1.1)$\{\begin{array}{ll}\partial_{t}\rho+div(\rho v)=0 in\dot{\Omega}_{t,\pm},\rho(\partial_{t}v+(v\cdot\nabla)v)-Div(S(v)-\mathfrak{p}I)=0 in\dot{\Omega}_{t,\pm},{[}[(S(v)-\mathfrak{p}I)n_{t}]]=0, [[v]]=0 on\Gamma_{t},(S_{-}(v_{-})-\mathfrak{p}_{-}I)n_{t,-}|r,.-=0 on\Gamma_{t}(\rho, v)|_{t=0}=(\rho_{*}+\theta_{-}, v_{0}) in \dot{\Omega}\end{array}$for $t\in(0, T)$ . Here, $\rho_{*}=\rho_{*}(x, t)$ is a piece-wise constant function defined bydescribing the mass density of reference bodies $\Omega\pm;v=v(x, t)=$with some positive constants$(v_{1}(x, t), \ldots, v_{N}(x, t))$ denotes a velocity field;
a pressure field; and a density field. In the case of thearecompressible fluids, the mass field
are unknown functions; the pressure field, that is, the barotropic fluids are considered, where
functions of mass densitiesarefunctions defined for $r>0$ satisfying the conditions: $P_{\pm}’(r)>0$ for $r>0$ and $P_{\pm}(\rho_{*\pm})=0$ ; andare prescribed functions. In the case of the incompressible fluids, the mass fields
initial data and
, the
with
, so that the balance of mass is read as $divv\pm=0$ inis given bypressure term
is a prescribedandare unknown functions, and for the initial data$\rho_{*}(x, t)|_{\Omega_{t,\pm}}=\rho_{*\pm}$$\rho_{*\pm}$$\mathfrak{p}$$\rho$$\rho\pm=\rho|_{\Omega_{t,\pm}}$$\mathfrak{p}\pm=\mathfrak{p}|_{\Omega_{t,\pm}}$$P_{\pm}(r)$$\rho\pm as\mathfrak{p}\pm=P_{\pm}(\rho_{\pm})$$c\infty$$\theta_{0}$$v_{0}$$\rho|_{\Omega_{t,\pm}}=\rho_{*_{)}\pm}$$\mathfrak{p}\pm=\mathfrak{p}|_{\Omega\pm}$$\rho$$\Omega_{t,\pm}$$\theta_{0}=0$$v_{\pm}=v|_{\Omega_{\ell,\pm}}$$v_{0}$

function.
*Department of Mathematics and Research Institute of Science and Engineering, Waseda University,
Ohkubo 3-41, Shinjuku-ku, Tokyo 169-8555, Japan.
email address: yshibata@waseda.jp
Partially supported by Top Global University Project and JSPS Grant-in-aid for Scientific Research (S)$\#$24224004

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As for the remaining notation, I is the $N\cross N$ unit matrix; is the unit normal to pointing fromto ,-while ,-is the unit outer normal of
the $[[f]]$ denotes the jump quantity of along
defined by$\Gamma_{t}$$n_{t}$$\Omega_{t}$$\Omega_{t,+}$$\Gamma_{t}$$f$$\Gamma_{t}$$n_{t}$

for$[[f]](x_{0})=x \in\Omega xarrow x_{0}\lim_{+}f(x)-x\in\Omega\lim_{xarrow x_{0}}f(x)$and$x_{0}\in\Gamma_{t}$;is a stress tensor defined by$S$$S(u)|_{\Omega\pm}=\mu\pm D(u_{\pm})+(v\pm-\mu_{\pm})divu\pm I, D(u_{\pm})=\nabla u\pm+(\nabla u_{\pm})^{T}$, where, and $\mu\pm and\nu\pm are$ positive constants
withdenotes the transposed, for any matrix
describing the first and second viscosity coeﬃcinets. Furthermore,$K$$N$$(i,j)$$DivK$
components, the quantity
field
with
is an -vector with components, and for any vector of functions $w=(w_{1}, \ldots, w_{N})$ , we set
where,$(\nabla u_{\pm})^{T}$$u\pm=u|_{\Omega_{t,\pm}}$$\nabla u$$\rho_{t}=\partial_{t}\rho=\partial\rho/\partial t$$\sum_{j=1}^{N}\partial_{j}K_{ij},$$K_{\iota j}$$w_{t}=(\partial_{t}w_{1}, \ldots, \partial_{t}w_{N})$$\partial_{l}=\partial/\partial x_{j}$andLet $x=x(\xi, t)$ be a solution of the Cauchy problem$divw=\sum_{j=1}^{N}\partial_{j}w_{j}$$w\cdot\nabla w=(\sum_{j=1}^{N}w_{j}\partial_{j}w_{1}, \ldots, \sum_{j=1}^{N}w_{j}\partial_{J}w_{N})$$\frac{dx}{dt}=v(x, t)$.

with $x|_{t=0}=\xi.$

The kinematic condition is:$\Gamma_{t}=\{x=x(\xi, t)|\xi\in\Gamma\}, \Gamma_{t_{)}-}=\{x=x(\xi, t) |\xi\in\Gamma$

Notation. Throughout the paper, for any domain $D,$ $L_{q}(D)$ ,anddenote the usual
Lebesgue space, Sobolev space and Besov space, whileandare their
norms, where $1\leq p,$ $q\leq\infty,$
is any natural number and is any non-negative real number. Let$W_{q,0}^{1}(D)=\{u\in W_{q}^{1}(D)|u|_{\partial D}=0\}$ , whereis the boundary of $D$ . Given function defined onor , we setor. Given functions
on
defined on
is defined by$v(x)=v\pm(x)$ for. Letor $v(x)=v\pm(x)$ for$W_{q}^{n}(D)$$\Vert\cdot\Vert_{L_{q}(D)},$$n$$B_{q,p}^{s}(D)$$\Vert\cdot\Vert_{W_{q}^{n}(D)}$$\Vert\cdot\Vert_{B_{q.p}^{s}(D)}$$s$$\dot{\Omega}$$\partial D$$\dot{\Omega}_{t}$$v\pm=v|_{\Omega\pm}$$v$$v\pm=v|_{\Omega_{t,\pm}}$$\Omega\pm or$$v\pm$$x\in\Omega_{\pm}$$\Omega_{i,\pm},$$v$$x\in\Omega_{t,\pm}$$W_{q}^{n}(\dot{\Omega})=\{v\in L_{q}(\dot{\Omega})|v\pm=v|_{\Omega_{\pm}}\in W_{q}^{n}(\Omega_{\pm})\},$$\Vert v\Vert_{L_{q}(\dot{\Omega})}=\Vert v_{+}\Vert_{L_{q}(\Omega_{+})}+\Vert v_{-}\Vert_{L_{q}(\Omega-)},$$B_{q,p}^{s}(\dot{\Omega})=\{v\in L_{q}(\dot{\Omega})|v\pm=v|_{\Omega_{\pm}}\in B_{q,p}^{s}(\Omega_{\pm})\},$$\Vert v\Vert_{W_{q}^{n}(\dot{\Omega})}=\Vert v_{+}\Vert_{W_{q}^{n}(\Omega_{+})}+\Vert v_{-}\Vert_{W_{q}^{n}(\Omega-)},$$\Vert v\Vert_{B_{q.p}^{e}(\dot{\Omega})}=\Vert v_{+}\Vert_{B_{q.p}^{s}(\Omega_{+})}+\Vert v_{-}\Vert_{B_{q,p}^{\epsilon}(\Omega_{-})}.$

Let$(u, v)_{\Omega\pm}= \int_{\Omega\pm}u(x)\overline{v(x)}dx, (u, v)_{\dot{\Omega}}=(u,v)_{\Omega_{+}}+(u,v)_{\Omega-},$$(u, v)_{\Gamma}= \int_{\Gamma}u(x)\overline{v(x)}d\sigma_{\Gamma}, (u, v)r_{-}=\int_{\Gamma}u(x)\overline{v(x)}d\sigma_{\Gamma-},$

where
and

denotes the complex conjugate of $v(x)$ , andanddenote the surface elements of, respectively. For any two $N$ vectors $a^{l}=(a_{1}^{l}, \ldots, a_{N}^{l})(i=1,2)$ , we set $a^{1}\cdot a^{2}=<a^{1},$ $a^{2}>=$. The $[[f]]$ denotes also the jump quantity of along defined by$\overline{v(x)}$$\Gamma_{-}$$d\sigma_{\Gamma}$$\sum_{j=1}^{N}a_{j}^{1}a_{j}^{2}$$f$$[[f]](x_{0})=x \in\Omega xarrow x\lim_{0,+}f(x)-x\in\Omega xarrow x\lim_{0}f(x)$$\Gamma$$d\sigma r_{-}$$\Gamma$for$x_{0}\in\Gamma.$

For two Banach spaces $X,$ $Y,$
denotes the set of all bounded linear operators from $X$ into $Y,$
while
denotes its norm. When $X=Y$ , we use the abbreviation:. The-product spaceis defined by of $X^{d}=\{u=(u_{1}, \ldots, u_{d})|u_{\iota}\in X(i=1, \ldots, d)\}$ , while its norm is
written byinstead offor short, whereis the norm of $X$ . The boldface letter is used$C$
representtovectors of functions. The letter is used to represent generic constants and the value of
may change from line to line.
$\mathcal{L}(X, Y)$$\mathcal{L}(X)=\mathcal{L}(X, X)$$\Vert\cdot\Vert_{\mathcal{L}(X,Y)}$$X^{d}$$d$$\Vert\cdot\Vert x$$\Vert\cdot\Vert_{X^{d}}$$\Vert\cdot\Vert_{X}$

$C$

118
be the Lagrangean description of the velocity field inStatement of main results. Let
coordinate are related by
Lagrangean
and then the Euler coordinate and the$\dot{\Omega},$$u(\xi, t)$

$\xi$

$x$

$x= \xi+\int_{0}^{t}u(\xi, s)ds=X_{u}(\xi,t)$for$\xi\in\dot{\Omega}.$

The Jacobi matrix of the transformation $x=X_{u}(\xi, t)$ is invertible, if(1.2)$\int_{0}^{T}\Vert\nabla u(\cdot, t)\Vert_{L_{\infty}(\dot{\Omega})}dt\leq\sigma_{0}$. By the Banach fixed point argument based on the maximal
does hold with some small
maximal regularity theorem for the linearized equations, we can prove the local well-posedness which isstated in$L_{r}-L_{q}$$\sigma_{0}>0$

Theorem 1.1. Let $N<q,$ $r<\infty,$ $2<p<\infty$ , and $R>0$ . Assume that $\max(q, q’)\leq r$ and that andclass. Then, there exists a positive time $T>0$ depending
are both compact hyper-surfaces ofwithandon such that for any initial data$\Gamma$$W_{r}^{2-1/r}$$\Gamma_{-}$$R$$\theta_{0,\pm}\in W_{q}^{1}(\Omega_{\pm})$$v_{0,\pm}\in B_{q,p}^{2(1-1/p)}(\Omega_{\pm})^{N}$$\Vert\theta_{0,\pm}\Vert_{W_{g}^{1}(\Omega\pm)}+\Vert v_{0,\pm}\Vert_{B_{q.p}^{2(1-1/p)}}(\Omega_{\pm})\leq R$

satisfying the range condition:$\rho_{*_{\rangle}\pm}/2<\rho_{*\pm}+\theta_{0,\pm}(x)\leq 2\rho_{*\pm}$and the compatibility conditions which is described as follows:$\bullet$compressible-compressible case:$[[(S(v_{0})-P(\rho_{0})I)n]]=0, [[v_{0}]]=0$$(S_{-}(v_{0,-})-P_{-}(\rho_{0,-})I)n_{-}|_{\Gamma}=0,$

where$\bullet$$\Omega_{+}$$\rho_{0,\pm}=\rho_{*\pm}+\theta_{0,\pm}$compressible and;incompressible case:$\Omega_{-}$$[[S(v_{0})n-<S(v_{0})n, n>n]]=0, [[v_{0}]]=0, divv_{0,-}=0,$$(S_{-}(v_{0,-})n_{-}-<S_{-}(v_{0,-})n_{-}, n_{-}>n_{-}|r_{-}=0,$$\bullet$$\Omega+$incompressible and$\Omega$-compressible case:$[[S(v_{0})n-<S(v_{0})n, n>n]]=0, [[v_{0}]]=0, divv_{0,+}=0,$$(S_{-}(v_{0,-})-P_{-}(\rho_{0,-})I)n_{-}|_{\Gamma_{-}}=0,$$\bullet$incompressible-incompressible case:$[[S(v_{0})n-<S(v_{0})n, n>n]]=0, [[v_{0}]]=0, divv_{0,\pm}=0,$$(S_{-}(v_{0,-})-<S_{-}(v_{0,-})n_{-}, n_{-}>n_{-})|_{\Gamma-}=0,$

is the unit outer normal tois the unit normal to pointing from $\Omega+into\Omega_{-}$ , while
uniquesolutionsadmit
equations (1.1) described in the Lagrange coordinate

where$n_{-}$$\Gamma$$n$compressible-compressible case: $\theta\pm andu\pm with$$\theta\pm\in W_{r}^{1}((0, T), W_{q}^{1}(\Omega_{\pm})) , u\pm\in W_{p}^{1}((0,T), L_{q}(\Omega_{\pm})^{N})\cap L_{p}((0, T), W_{q}^{2}(\Omega_{\pm})^{N})$$\bullet$$\Omega\pm\omega mp.-\Omega_{\mp}$incomp. case:$\theta\pm,$$\pi_{\mp}$

and $u\pm with$$\theta_{\pm}\in W_{p}^{1}((0, T), W_{q}^{1}(\Omega_{\pm})) , \pi_{\mp}\in L_{p}((0, T), W_{q}^{1}(\Omega_{\mp}))$$u\pm\in W_{p}^{1}((0, T), L_{q}(\Omega_{\pm})^{N})\cap L_{p}((0, T), W_{q}^{2}(\Omega_{\pm})^{N})$

;

,

;

$\Gamma_{-}$

, the

119
$\bullet$incompressible-incompressible case: $\pi\pm andu\pm with$$\pi\pm\in L_{p}((0, T), W_{q}^{1}(\Omega_{\pm})) , u\pm\in W_{p}^{1}((0, T), L_{q}(\Omega_{\pm})^{N})\cap L_{p}((0, T), W_{q}^{2}(\Omega_{\pm})^{N})$;satisfies (1.2). Here, $\theta\pm and\pi\pm$ denote the density fields and pressure fields in the Lagrange
coordinate, that is, $\rho\pm(X_{u}(\xi, t), t)=\theta_{\pm}(\xi, t)$ andfor$whe\tau eu$$\mathfrak{p}_{\pm}(X_{u}(\xi, t), t)=\pi\pm(\xi, t)$$\xi\in\Omega_{\pm}.$

Next theorem is concerned with the global well-posedness theorem for small initial data.
Theorem 1.2. Let $N<q,$ $r<\infty,$ $2<p<\infty$ , and $R>$ O. Assume that $\max(q, q’)\leq r$ , thatbe theclass, and that $2/p+N/q<1$ , Let
are both compact hyper-surfaces ofand$\mathcal{R}_{d}=\{u|D(u)=0\}$
with inner-productorthonormal basis of the rigid space$\Gamma$$W_{r}^{2-1/r}$$\Gamma_{-}$$\{p_{l}\}_{l=1}^{M}$$[u, v]=(\rho_{*},+u_{+}, v_{+})_{\Omega_{+}}+(\rho_{*-}u_{-}, v_{-})_{\Omega_{-}}.$

Then, there exists ansuch that if initial dataandsatisfies smallness condition:$\epsilon>0$$)$$\theta_{0,\pm}$(in the incompressible case, we interpret$\theta_{0,\pm}=0$$v_{0,\pm}$$\Vert\theta_{0,\pm}\Vert_{W_{q}^{1}(\Omega\pm)}+\Vert v_{0}\Vert_{B_{q,p}^{2-1/p}}(\Omega)\leq\epsilon,$and orthogonal condihon:$((\rho_{*+}+\theta_{0,+})v_{0,+}) , p_{\ell})_{\Omega_{+}}+((\rho_{*-}+\theta_{0,-})v_{0,-}) , p_{\ell})_{\Omega_{-}}=0 (\ell=1, \cdots, M)$

as well as regularity condition, range condition and compatibility condition, then the equations (1.1),
described in the Lagrange coordinate admit unique solutions defined on the whole time interval$(0, \infty)$

which decay exponentially.

is the set of all $N$-vector of first order polynomials of the form: $Ax+b$
Remark 1.3. The rigid space$N\crossN$consists of all linear combiwith anti-symmetricmatrix and constant $N$ vector . Namely,$N$, where $e_{i}=(0, \ldots, 0,l, 0, \ldots, 0)$ .
nations of constant vectors and polynomials of the form:$\mathcal{R}_{d}$

$A$

$b$

$\mathcal{R}_{d}$$x_{\iota}e_{j}-x_{j}e_{\iota}$ith

To prove Theorem 1.2, the main tool is the exponential stability of semi-group associated with thelinearlized equations:(1.3)$\{\begin{array}{l}\partial_{t}\theta+\gamma_{0}divv=0 in\dot{\Omega}\cross(0, \infty) ,\partial_{t}v-\gamma_{1}Div(S(v)-\mathfrak{p}I)=0 in\dot{\Omega}\cross(0, \infty) ,{[}[(S(v)-\mathfrak{p}I)n]]=0, [[v]]=0,(S_{-}(v_{-})-\mathfrak{p}_{-}I)n_{-}|_{\Gamma-}=0,(\theta, v)|_{t=0}=(\theta_{0}, v_{0}) in \dot{\Omega},\end{array}$where $\gamma_{i}(i=0,1)$ are piece-wise constant functions defined by$\gamma_{\iota,\pm}$$\gamma_{i}|_{\Omega\pm}=\gamma_{i_{)}\pm}$with some positive constants. Moreover,$0$

the compressible-compressible case:with some positive constants$\mathfrak{p}=\gamma’\theta$$\gamma’|_{\Omega\pm}=\gamma’\pm$

comp.

incomp. case:$\bullet$the$\bullet$the incompressible-incompressible case:$\Omega\pm$$-\Omega_{\mp}$$\gamma’\pm$with some piece-wise constant function$\gamma’$defined by;$\mathfrak{p}\pm=\gamma’\pm\theta\pm$, while$\theta=\theta_{0}=0$$\theta_{\mp}=\theta_{0,\mp}=0$and$\mathfrak{p}$and$\mathfrak{p}_{\mp}$

is unknow function;is unknown function.

In fact, to prove Theorem 1.2, the key step is to prove the existence of, which is analytic. Here,with (1.3) on$C^{0}$

semigroup$\{T(t)\}_{t\geq 0}$

associated$\mathcal{H}_{q}(\Omega)$$\mathcal{H}_{q}(\Omega)=\{(\theta, v)\in W_{q}^{1}(\dot{\Omega})\cross L_{q}(\Omega)\}$in the compressible-compressible case,$\mathcal{H}_{q}(\Omega)=\{(\theta_{\pm}, v)\in W_{q}^{1}(\Omega_{\pm})\cross L_{q}(\Omega)|divv_{\mp}=0\}$in the$\mathcal{H}_{q}(\Omega)=\{v\in L_{q}(\Omega)|divv_{\mp}=0\}$

Moreover, if$v$$\{T(t)\}_{t\geq 0}$$\Omega\mp$satisfies the orthogonal condition:$(\gamma_{1}^{-1}v, p_{l})_{\dot{\Omega}}=0$

thenincomp. case,comp.in the incompressible-incompressible case.
$\Omega\pm$

is exponentially stable on$\mathcal{H}_{q}(\Omega)$

for all $\ell=1$ , . . . , $M$ ,, that is,

(1.4)

120
$\bullet$compressible-compressible case:$\Vert T(t)(\theta,v)\Vert_{W_{q}^{1}(\dot{\Omega})xL_{q}(\Omega)}\leq Ce^{-ct}\Vert(\theta,v)\Vert_{W_{q}^{1}(\dot{\Omega})xL_{q}(\Omega)}$$\Omega\pm$

compressible-$\Omega_{\mp}$;incompressible case:$\Vert T(t)(\theta_{\pm}, v)\Vert_{W_{q}^{1}(\Omega\pm)xL_{q}(\Omega)}\leq Ce^{-ct}\Vert(\theta_{\pm}, v)\Vert_{W_{q}^{1}(\Omega\pm)xL_{q}(\Omega);}$$\bullet$incompressible-incompressible case:$\Vert T(t)v\Vert_{L_{q}(\Omega)}\leq Ce^{-ct}\Vert v\Vert_{L_{q}(\Omega)}$with some positive constants and for any $t>0.$
To prove the exponential stability, one of key steps is to prove the unique existence theorem for thefollowing problem:
$C$

$c$

(1.5)

$\{\begin{array}{ll}\gamma_{0}divv=f in \dot{\Omega},-\gamma_{1}(DivS(v)-\nabla \mathfrak{p})=g in St,{[}[(S(v)-\mathfrak{p}I)n]]=[[h]], [[v]]=0 on \Gamma,(S_{-}(v_{-})-\mathfrak{p}_{-}I)n_{-}|r_{-}=h_{-}|r_{-} on \Gamma_{-}.\end{array}$Dividing the first equation in (1.5) byconcerned with problem (1.5) with$\gamma_{0}$$\gamma_{0}=1$, we may assume that, and we prove$\gamma_{0}=1$

in the following. This paper is

Theorem 1.4. Let $1<q<\infty$ and $N<r<\infty$ . Assume that $r \geq\max(q, q’)$ with $q’=q/(q-1)$ and thatbe the orthonormal basis of theclass. Letandare both compact hyper-surfaces of$[u,v]$. Then,on
rigid space $\mathcal{R}_{d}=\{u|D(u)=0\}$ with respect to the inner-product:
orthogonal,,thecondition:satisfying
any
and
for
$\Gamma$$W_{r}^{2-1/r}$$\Gamma_{-}$$\{p_{\ell}\}_{\ell=1}^{M}$$:=(\gamma_{1}^{-1}u,v)_{\dot{\Omega}}$$f\in W_{q}^{1}(\dot{\Omega})$$g\in L_{q}(\dot{\Omega})$$h\in W_{q}^{1}(\dot{\Omega})$$h_{-}\in W_{q}^{1}(\Omega_{-})$$(\gamma_{1}^{-l}g, p_{\ell})_{\Omega}+(h, p_{\ell})_{\Gamma}+(h_{-},p_{\ell})r_{-}=0$

then problem (1.5) admits a unique solution$L_{q}(\dot{\Omega})$$v\in W_{q}^{2}(\dot{\Omega})$$(\gamma_{1}^{-1}v, p\ell)_{\Omega}=0$

for $all\ell=1$ , . .. ,

$M$

,

(1.6)

satisfying the orthogonal condition:

for $alle=1$ , . . . ,

(1.7)

$M$

and the estimate:$\Vert v\Vert_{W_{q}^{2}(\dot{\Omega})}\leq C(\Vert f\Vert_{W_{q}^{1}(\dot{\Omega})}+\Vert g\Vert_{L_{q}(\dot{\Omega})}+\Vert h\Vert_{W_{q}^{1}(\dot{\Omega})}+\Vert h_{-}\Vert_{W_{q}^{1}(\Omega-)})$

.

(1.8)

Moreover, we discuss the unique solvability of the weak Dirichlet problem:$(\gamma_{1}\nabla u, \nabla\varphi)_{\dot{\Omega}}=(f, \nabla\varphi)_{\dot{\Omega}}$

for any$\varphi\in W_{q,0}^{1}(\Omega)$

.

(1.9)

For problem (1.9) we prove
Theorem 1.5. Let $1<q<\infty$ and $N<r<\infty$ . Assume that $r \geq\max(q, q’)$ and that both of and, problem (1.9) admits a unique solution
dass. Then, for any
are hyper-surfaces ofsatisfying the estimate:$\Gamma$$W_{r}^{2-1/r}$$u\in W_{q,0}^{1}(\Omega)$2$f\in L_{q}(\Omega)^{N}$$1u\Vert_{W_{q}^{1}(\Omega)}\leq C\Vert f\Vert_{L_{q}(\Omega)}.$On the weak Dirichlet problem

In this section, we discuss the weak Dirichlet problem (1.9).

$\Gamma’$121
The weak Dirichle problem in

2.1$\mathbb{R}^{N}$

Let$\mathbb{R}_{\pm}^{N}=\{x=(x_{1}, \ldots, x_{N})\in \mathbb{R}^{N}|\pm x_{N}>0\}, \mathbb{R}_{0}^{N}=\{x=(x_{1}, \ldots, x_{N})\in \mathbb{R}^{N}|x_{N}=0\},$and set$\dot{\mathbb{R}}^{N}=\mathbb{R}_{+}^{N}\cup \mathbb{R}_{-}^{N}$. First of all, we consider the variational equation:
for any$\lambda(u, \varphi)_{\dot{\mathbb{R}}^{N}}+(\gamma_{1}\nabla u, \nabla\varphi)_{\dot{\mathbb{R}}^{N}}=(f, \nabla\varphi)_{R^{N}}$where is a piece-wise constant function defined bysolve (2.1), we consider the strong form of (2.1):$\gamma_{1}$$\gamma|_{R_{\pm}^{N}}=\gamma_{1,\pm}$$\varphi\in W_{q}^{1},(\mathbb{R}^{N})$

(2.1)

,

with some positive constants$\gamma_{1,\pm}$(2.2)$\{\begin{array}{l}\lambda u\pm-\gamma_{1,\pm}\Delta u\pm=divf\pm in \mathbb{R}_{\pm}^{N},\gamma_{1,+}\partial_{N}u_{+}|_{xN^{=0+}}-\gamma_{1,-}\partial_{N}u_{-}|_{xN^{=0-}}=g,u_{+}|_{xN^{=0+}}=u_{-}|_{xN}=0-\cdot\end{array}$If$f\in L_{q}(\mathbb{R}^{N})^{N}$$f\pm\in C_{0}^{\infty}(\mathbb{R}_{\pm}^{N})^{N}$, then. Sinceis dense in. First of all, we construct solutions of (2.2). For any functions$L_{q}(\mathbb{R}_{\pm}^{N})$$C_{0}^{\infty}(\mathbb{R}_{\pm}^{N})$$f\pm=f|_{\mathbb{R}_{\pm}^{N}}\in L_{q}(\Omega_{\pm})^{N}$. To, we may assume thatdefined on $\pm xN>0,$$h\pm$

let$h_{\pm}^{e}(x)=\{$$h_{\pm}(x’, x_{N})-h_{\pm}(x’,-x_{N})$$\pm x_{N}<0\pmx_{N}>0,$\pm x_{N}>0,h_{\pm}(x’,$h_{\pm}^{o}(x)=\{\begin{array}{ll}h_{\pm}(x’,x_{N})-x_{N}) \pm x_{N}<0,\end{array}$where $x’=(x_{1}, \ldots, x_{N-1})$ . Letby$\mathcal{F}$and$\mathcal{F}_{\xi}^{-1}$be Fourier transform and Fourier inverse transform defined$\mathcal{F}[f](\xi)=\int_{\mathbb{R}^{N}}e^{-\iota x\cdot\xi}f(x)dx, \mathcal{F}_{\xi}^{-1}[f](x)=\frac{1}{(2\pi)^{N}}\int_{\mathbb{R}^{N}}e^{\iota x\cdot\xi}f(\xi)d\xi.$Since$( divf_{\pm})^{o}=\sum_{j=1}^{N-1}\partial_{j}(f_{j}^{o})+\partial_{N}(f_{N}^{e})$with $f\pm=(f_{\pm 1}, \ldots, f_{\pm N})$ , we have$\mathcal{F}[(divf_{\pm})^{o}](\xi)=\sum_{j=1}^{N-1}i\xi_{j}\mathcal{F}[f_{\pm j}^{o}](\xi)+i\xi_{N}\mathcal{F}[f_{\pm N}^{e}](\xi)$.

Thus, if we set(2.3)$u \pm 1=\mathcal{F}_{\xi}^{-1}[\frac{\mathcal{F}[(divf_{\pm})^{o}](\xi)}{\lambda+\gamma_{1,\pm}|\xi|^{2}}]=\mathcal{F}_{\xi}^{-1}[\frac{\sum_{J}^{N-1}=1i\xi_{J}\mathcal{F}[f_{\pm j}^{o}](\xi)+i\xi_{N}\mathcal{F}[f_{\pm N}^{e}](\xi)}{\lambda+\gamma_{1,\pm}|\xi|^{2}}]$

we havein$\lambda u\pm-\gamma_{1,\pm}\triangle u\pm=divf\pm$

In the following, we calculate $u_{\pm 1}(x’, 0)$ and$f_{\pm N}(x’, 0)=0$ with $f\pm=(f_{\pm 1}, \ldots, f_{\pm N})$ . Let$\hat{g}(\xi’, x_{N})=\int_{R^{N-1}}e^{-ix’\cdot\xi’}g(x’, x_{N})dx’,$$(\partial_{N}u\pm 1)(x’, 0)$$\mathbb{R}_{\pm}^{N}$.

. Recall that$f\pm\in C_{0}^{\infty}(\mathbb{R}_{\pm}^{N})^{N}$. Especially,$\mathcal{F}_{\xi}^{-1}[g(\cdot, x_{N})](x’)=\frac{1}{(2\pi)^{N-1}}\int_{R^{N-1}}e^{tx’\cdot\xi’}g(\xi’, x_{N})d\xi’.$

The anddenote the partial Fourier transform with respect to‘, we haverespect to $\xi’=(\xi_{1}, \ldots, \xi_{N-1})$ . Writing$\hat{9}$(2.4)$\mathcal{F}_{\xi}^{-1}[9]$$x’$and its inversion formula with$\omega\pm=\sqrt{\lambda}/\gamma_{1,\pm}+|\xi$$|^{2}$$\hat{u}_{+1}(\xi’, 0)=0,$$( \partial_{N}\hat{u}_{+1})(\xi’, 0)=-\sum_{j=1}^{N-1}\frac{\xi_{j}}{\gamma_{1,+}}\int_{0}^{\infty}e^{-y_{N}\omega_{+}}\hat{f}_{+j}(\xi’, y_{N})dy_{N}+\frac{\omega+}{i\gamma_{1,+}}\int_{0}^{\infty}e^{-y_{N}\omega_{+}}\hat{f}_{+N}(\xi’, y_{N})dy_{N}$In fact, by the residue theorem$\hat{u}_{+1}(\xi’, 0)=\sum_{j=1}^{N-1}\frac{1}{2\pi\gamma_{1,+}}\int_{0}^{\infty}(\int_{-\infty}^{\infty}\frac{i\xi_{J}(e^{\iota y_{N}\xi_{N}}-e^{-\iotay_{N}\xi_{N}})}{\lambda/\gamma_{1,+}+|\xi’|^{2}+\xi_{N}^{2}}d\xi_{N})\hat{f}_{+j}(\xi’, y_{N})dy_{N}$

.

(2.5)

122
$+ \frac{1}{2\pi\gamma_{1,+}}\int_{0}^{\infty}(\int_{-\infty}^{\infty}\frac{i\xi_{N}(e^{iy_{N}\xi_{N}}+e^{-iy_{N}\xi_{N}})}{\lambda/\gamma_{1,+}+|\xi’|^{2}+\xi_{N}^{2}}d\xi_{N})\hat{f}_{+N}(\xi’,y_{N})dy_{N}$$= \sum_{j=1}^{N-1}\frac{1}{\gamma_{1,+}}\int_{0}^{\infty}[\frac{i\xi_{j}e^{-y_{N}\omega+}}{2i\omega+}-(-\frac{i\xi_{j}e^{-y_{N}\omega_{+}}}{-2i\omega+})]\hat{f}_{+j}(\xi’)y_{N})dy_{N}$$+ \frac{1}{\gamma_{1,+}}\int_{0}^{\infty}[\frac{i(i\omega_{+})e^{-y_{N}\omega_{+}}}{2i\omega+}+(-\frac{i(-i\omega_{+})e^{-y_{N}\omega_{+}}}{-2i\omega+})]\hat{f}_{+N}(\xi’, y_{N})dy_{N}$$=0.$Analogously,$( \partial_{N}\hat{u}_{+1})(\xi’, 0)=\sum_{j=1}^{N-1}\frac{-1}{2\pi\gamma_{1,+}}\int_{0}^{\infty}(\int_{-\infty}^{\infty}\frac{\xi_{j}\xi_{N}(e^{ly_{N}\xi_{N}}-e^{-iy_{N}\xi_{N}})}{\lambda/\gamma_{1,+}+|\xi’|^{2}+\xi_{N}^{2}}d\xi_{N})\hat{f}_{+J}(\xi’,y_{N})dy_{N}$$- \frac{1}{2\pi\gamma_{1,+}}\int_{0}^{\infty}(\int_{-\infty}^{\infty}\frac{\xi_{N}^{2}(e^{ly_{N}\xi_{N}y_{N}\xi_{N}}+e^{-l})}{\lambda/\gamma_{1,+}+|\xi’|^{2}+\xi_{N}^{2}}d\xi_{N})\hat{f}_{+N}(\xi’,y_{N})dy_{N}$$= \sum_{j=1}^{N-1}\frac{-1}{\gamma_{1,+}}\int_{0}^{\infty}[\frac{\xi_{j}(i\omega_{+})e^{-y_{N}\omega_{+}}}{2i\omega+}+\frac{\xi_{j}(-i\omega_{+})e^{-y_{N}\omega+}}{-2i\omega+})]\hat{f}_{+j}(\xi’, y_{N})dy_{N}$$- \frac{1}{2\pi\gamma_{1,+}}\int_{-\infty}^{\infty}\int_{0}^{\infty}(e^{iy_{N}\xi_{N}}+e^{-\iota y_{N}\xi_{N}})\hat{f}_{+N}(\xi’, y_{N})dy_{N}d\xi_{N}$$+ \frac{1}{\gamma_{1,+}}\int_{0}^{\infty}(\lambda/\gamma_{1,+}+|\xi’|^{2})(\frac{e^{-y_{N}\omega_{+}}}{2i\omega+}-\frac{e^{-y_{N}\omega_{+}}}{-2i\omega+})\hat{f}_{+N}(\xi’,y_{N})dy_{N}.$Thus, using$\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{0}^{\infty}(e^{\iota y_{N}\xi_{N}}+e^{-\iota y_{N}\xi_{N}})\hat{f}_{+N}(\xi’, y_{N})dy_{N}d\xi_{N}=\int_{-\infty}^{\infty}\mathcal{F}[f_{+N}^{e}](\xi)d\xi_{N}=\hat{f_{+N}^{e}}(\xi’, 0)=0,$

we have (2.5). Similarly, we have$\hat{u}_{-1}(\xi’,0)=0,$$( \partial_{N}\hat{u}_{-1})(\xi’,0)=\sum_{j=1}^{N-1}\frac{\xi_{j}}{\gamma_{1,-}}\int_{-\infty}^{0}e^{y_{N}\omega-j_{-j}(\xi,y_{N})dy_{N}}+\frac{\omega_{-}}{i\gamma_{1,-}}\int_{-\infty}^{0}e^{y_{N}\omega-}\hat{f}_{-N}(\xi’, y_{N})dy_{N}$.

In fact,$\hat{u}_{-1}(\xi’, 0)=\sum_{j=1}^{N-1}\frac{1}{2\pi\gamma_{1,-}}\int_{-\infty}^{0}(\int_{-\infty}^{\infty}\frac{i\xi_{j}(-e^{-iy_{N}\xi_{N}}+e^{iy_{N}\xi_{N}})}{\lambda/\gamma_{1,-}+|\xi’|^{2}+\xi_{N}^{2}}d\xi_{N})\hat{f}_{-j}(\xi’, y_{N})dy_{N}$$+ \frac{1}{2\pi\gamma_{1,-}}\int_{-\infty}^{0}(\int_{-\infty}^{\infty}\frac{i\xi_{N}(e^{-l}y_{N}\xi_{N}+e^{iy_{N}\xi_{N}})}{\lambda/\gamma_{1,-}+|\xi’|^{2}+\xi_{N}^{2}}d\xi_{N})\hat{f}_{-N}(\xi’, y_{N})dy_{N}$$= \sum_{j=1}^{N-1}\frac{1}{\gamma_{1,-}}\int_{-\infty}^{0}(-\frac{i\xi_{j}e^{y_{N}\omega-}}{2i\omega_{-}}-\frac{i\xi_{j}e^{y_{N}\omega-}}{-2i\omega_{-}})j_{-g}(\xi’,y_{N})dy_{N}$$+ \frac{1}{\gamma_{1,-}}\int_{-\infty}^{0}(\frac{i(i\omega_{-})e^{y_{N}\omega-}}{2i\omega_{-}}-\frac{i(-i\omega_{-})e^{y_{N}\omega-}}{-2i\omega_{-}})\hat{f}_{-N}(\xi’,y_{N})dy_{N}$$=0$ ;$( \partial_{N}\hat{u}_{-1})(\xi’, 0)=\sum_{j=1}^{N-1}\frac{1}{2\pi\gamma_{1,-}}\int_{-\infty}^{0}(\int_{-\infty}^{\infty}\frac{-\xi_{j}\xi_{N}(-e^{-iy_{N}\xi_{N}}+e^{\iota y_{N}\xi_{N}})}{\lambda/\gamma_{1,-}+|\xi|^{2}+\xi_{N}^{2}}d\xi_{N})\hat{f}_{-j}(\xi’,y_{N})dy_{N}$$+ \frac{1}{2\pi\gamma_{1,-}}\int_{-\infty}^{0}(\int_{-\infty}^{\infty}\frac{-\xi_{N}^{2}(e^{-iy_{N}\xi_{N}}+e^{\iota y_{N}\xi_{N}})}{\lambda/\gamma_{1,-}+|\xi’|^{2}+\xi_{N}^{2}}d\xi_{N})\hat{f}_{-N}(\xi’, y_{N})dy_{N}$

(2.6)

123
$= \sum_{j=1}^{N-1}\frac{1}{\gamma_{1,-}}\int_{0}^{\infty}(\frac{\xi_{j}(i\omega_{-})e^{y_{N}\omega_{+}}}{2i\omega_{-}}+\frac{\xi_{j}(-i\omega_{-})e^{y_{N}\omega-}}{-2i\omega+})\hat{f}_{-j}(\xi’, y_{N})dy_{N}$$- \frac{1}{2\pi\gamma_{1,-}}\int_{-\infty}^{\infty}\int_{0}^{\infty}(e^{-iy_{N}\xi_{N}}+e^{iy_{N}\xi_{N}})\hat{f}_{-N}(\xi’, y_{N})dy_{N}d\xi_{N}$$+ \frac{1}{\gamma_{1,-}}\int_{-\infty}^{0}(\lambda/\gamma_{1,-}+|\xi’|^{2})(\frac{e^{y_{N}\omega+}}{2i\omega_{-}}-\frac{e^{y_{N}\omega_{+}}}{-2i\omega_{-}})\hat{f}_{-N}(\xi’, y_{N})dy_{N}$$= \sum_{j=1}^{N-1}\frac{\xi_{j}}{\gamma_{1,-}}\int_{-\infty}^{0}e^{y_{N}\omega-}\hat{f}_{-j}(\xi’, y_{N})dy_{N}$$- \frac{1}{\gamma_{1,-}}\int_{-\infty}^{\infty}\mathcal{F}[f_{-N}^{e}](\xi)d\xi+\frac{\omega_{-}}{i\gamma_{1,-}}\int_{-\infty}^{0}e^{y_{N}\omega-}\hat{f}_{-N}(\xi’, y_{N})dy_{N},$and therefore we have (2.6).
Let

for $0<\epsilon<\pi/2.$$\Sigma_{\epsilon}=\{\lambda\in \mathbb{C}\backslash \{0\}||\arg\lambda|\leq\pi-\epsilon\}$

Let $1<q<\infty$ , and then by the Fourier multiplier theorem,(2.7)$|\lambda|^{1/2}\Vert u\pm 1\Vert_{L_{q}(\mathbb{R}^{N})}+\Vert\nabla u\pm 1\Vert_{L_{Q}(\mathbb{R}^{N})}\leq C_{q_{\rangle}\epsilon}\Vert f_{\pm}\Vert_{L_{q}(\mathbb{R}_{\pm}^{N})}$anddepending solely on
with some constantfor any. In view of (2.5) and (2.6),
Next, we construct the compensating function$\lambda\in\Sigma_{\epsilon}$$C_{q_{\rangle}\epsilon}$$\epsilon,$$q$$\gamma_{1_{\}}}\pm\cdot$$u\pm 2$

$u\pm 2$should satisfy theequations:$\{\begin{array}{l}(\lambda-\gamma_{1,\pm}\Delta)u\pm 2=0 in\mathbb{R}_{\pm}^{N},\gamma_{1,+}\partial_{N}u_{+2}|_{x_{N}=0+}-\gamma_{1,-}\partial_{N}u_{-2}|_{x_{N}=0-}=h,u_{+2}|_{xN}=0+=u_{-2}|_{xN^{=0-}},\end{array}$where $h=g-(\partial_{N}u+(\cdot, +0)-\partial_{N}u_{-}(\cdot, -0))$ . We findObviously,$\hat{u}\pm 2(\xi’, x_{N})$of the forms:(2.8)$\hat{u}\pm 2(\xi’, x_{N})=\alpha\pm e^{\mp\omega\pm xN}.$$(\lambda+\gamma_{1,\pm}|\xi’|^{2})\hat{u}\pm 2-\gamma_{1,\pm}\partial_{N}^{2}\hat{u}_{\pm 2}=0.$

Since$\gamma_{1,\pm}\partial_{N}\hat{u}\pm 2(\xi’, 0)=\mp\gamma_{1,\pm}\alpha\pm\omega\pm and\hat{u}_{\pm 2}(\xi’, 0)=\alpha\pm$, from the interface condition of (2.8) and(2.5) and (2.6), we have$-\gamma_{1},+\alpha+\omega_{+}-\gamma_{1,-}\alpha_{-}\omega_{-}=\hat{h}_{+}(\xi’, 0)-\hat{h}_{-}(\xi’, 0) , \alpha_{+}=\alpha_{-},$

which, combined with (2.5) and (2.6), we have$\alpha+=\alpha_{-}=-\frac{\hat{h}_{+}(\xi’,0)-\hat{h}_{-}(\xi’,0)}{\gamma_{1},+\omega_{+}+\gamma_{1,-}\omega_{-}}=-\frac{\hat{g}(\xi’,0)}{\gamma_{1},+\omega_{+}+\gamma_{1,-}\omega_{-}}$$+ \frac{1}{\gamma_{1,+}\omega_{+}+\gamma_{1,-}\omega_{-}}\{\sum_{j=1}^{N-1}\xi_{j}\int_{0}^{\infty}e^{-y_{N}\omega+}\hat{f}_{+j}(\xi’, y_{N})dy_{N}+i\omega_{+}\int_{0}^{\infty}e^{-y_{N}\omega+}\hat{f}_{+N}(\xi’, y_{N})dy_{N}$$+ \sum_{j=1}^{N-1}\xi_{j}\int_{-\infty}^{0}e^{y_{N}\omega-\hat{f}_{-j}(\xi’,y_{N})dy_{N}-i\omega_{-}\int_{-\infty}^{0}e^{y_{N}\omega-}\hat{f}_{-N}(\xi’,y_{N})dy_{N}\}},$

so that$\hat{u}\pm 2(\xi’, x_{N})=-\frac{e^{\mp xN}\hat{g}(\xi’,0)}{\gamma_{1,+}\omega_{+}+\gamma_{1,-}\omega_{-}}$$+ \frac{e^{\mp\omega\pm x_{N}}}{\gamma_{1},+\omega_{+}+\gamma_{1,-}\omega_{-}}\{\sum_{j=1}^{N-1}\xi_{j}\int_{0}^{\infty}e^{-y_{N}\omega+}\hat{f}_{+j}(\xi’, y_{N})dy_{N}+i\omega+\int_{0}^{\infty}e^{-y_{N}\omega+}\hat{f}_{+N}(\xi’, y_{N})dy_{N}$124$+ \sum_{j=1}^{N-1}\xi_{j}\int_{-\infty}^{0}e^{y_{N}\omega_{+}}\hat{f}_{-j}(\xi’, y_{N})dy_{N}-i\omega_{-}\int_{-\infty}^{0}e^{y_{N}\omega-}\hat{f}_{-N}(\xi’, y_{N})dy_{N}\}.$

Therefore, we have$\hat{u}_{+2}(\xi’,x_{N})=\int_{0}^{\infty}\frac{e^{-\omega+(x_{N}+y_{N})}\partial_{N}\hat{g}+(\xi’,y_{N})}{\gamma_{1},+\omega_{+}+\gamma_{1,-}\omega_{-}}dy_{N}-\int_{0}^{\infty}\frac{e^{-\omega_{+}(xN+y_{N})}\omega+\hat{9}+(\xi’,y_{N})}{\gamma_{1},+\omega_{+}+\gamma_{1,-}\omega-}dy_{N}$$+j=1 \sum_{\overline{\gamma_{1},+\omega\gamma_{1,-}\omega_{-}}}^{N-1}\int_{0}^{\infty}e^{-\omega_{+}(x+y_{N})}N\hat{f}_{+J},(\xi’, y_{N})dy_{N}$$+ \frac{+}{\gamma_{1,+}\omega\gamma_{1},-\omega_{-}}\int_{0}^{\infty}e^{-\omega_{+}(xN+y_{N})}\hat{f}_{+N}(\xi’,y_{N})dy_{N}$$+ \sum_{j=1}^{N-1}\frac{\xi_{j}}{\gamma_{1,+}\omega_{+}+\gamma_{1,-}\omega_{-}}\int_{0}^{\infty}e^{-(\omega_{+}xN+\omega-y_{N})}\hat{f}_{-j}(\xi’, -y_{N})dy_{N}$$- \overline{\gamma_{1,+}\omega\gamma_{1,-}\omega_{-}}-\int_{0}^{\infty}e^{-((v-y_{N})}\omega_{+xN+}\hat{f}_{-N}(\xi’, -y_{N})dy_{N}$;$\hat{u}_{-2}(\xi’,x_{N})=-\int_{-\infty}^{0}\frac{e^{\omega-(x_{N}+y_{N})}\partial_{N}\hat{g}_{-}(\xi’,y_{N})}{\gamma_{1,+}\omega_{+}+\gamma_{1},-\omega-}dy_{N}-\int_{-\infty}^{0}\frac{e^{\omega-(x+y_{N})}N\omega_{-}\hat{g}-(\xi’,y_{N})}{\gamma_{1,+}\omega_{+}+\gamma_{1},-\omega_{-}}dy_{N}$$+ \sum_{J=1}^{N-1}\frac{\xi_{j}}{\gamma_{1,+}\omega_{+}+\gamma_{1,-}\omega_{-}}\int_{-\infty}^{0}e^{(\omega-xN+\omega+y_{N})}\hat{f}_{+j}(\xi’, -y_{N})dy_{N}$$+ \frac{i\omega_{-}}{\gamma_{1,+}\omega_{+}+\gamma_{1},-\omega_{-}}\int_{-\infty}^{0}e^{(\omega_{+}y)}\omega-xN+N\hat{f}_{+N}(\xi’, -y_{N})dy_{N}$$+ \sum_{j=1}^{N-1}\frac{\xi_{j}}{\gamma_{1,+}\omega_{+}+\gamma_{1},-\omega_{-}}\int_{-\infty}^{0}e^{\omega-(xN+y_{N})}\hat{f}_{-j}(\xi’,y_{N})dy_{N}$$- \overline{\gamma_{1,+}\omega\gamma_{1,-}\omega_{-}}-\int_{-\infty}^{0}e^{\omega-(xN+y_{N})}\hat{f}_{-N}(\xi’,y_{N})dy_{N}$To estimate$u\pm 2$$(\xi’, \lambda)\in(\mathbb{R}^{N-1}\backslash \{0\})\cross$$\xi\in \mathbb{R}^{N-1}\backslash \{O\}$(1)$m(\xi’, \lambda)$(2.9), we introduce some symbol classes.

Definition 2.1. Letto.

be a domain in
Assume thatfor each$\mathbb{C}$and let $m(\xi’, \lambda)(\lambda=\gamma+i\tau\in$
be a function defined foris an infinitely many diﬀerentiable function with respect$m(\xi, \lambda)$$\lambda\in$is called a multiplier of order$s$with type 1 on

if the estimates:(2.10)$|\partial_{\xi}^{\kappa’},m(\xi’, \lambda)|\leq C_{\alpha’}(|\lambda|^{1/2}+|\xi’|)^{s-|\kappa’|}$hold for any multi-index
depending solely on and$\kappa’\in N_{O}^{N-1}$

and

and

$(\xi’, \lambda)\in$$(\xi’, \lambda)\in$

with some constant$C_{\kappa’}$$\kappa’$(2)$m(\xi’, \lambda)$

is called a multiplier of order$s$with type 2 onif the estimates:(2.11)$|\partial_{\xi}^{\kappa’},m(\xi’, \lambda)|\leq C_{\kappa’}(|\lambda|^{1/2}+|\xi’|)^{s}|\xi’|^{-|\kappa’|}$hold for any multi-index
and$\kappa’\in N_{0}^{N-1}$

and$(\xi’, \lambda)\in$

with some constants$C_{\kappa’}$depending solely on$\kappa’$Let$M_{s,\iota}(---)$

be the set ofan multipliers of order with type on
$s$

$i$

$(i=1,2)$ .

Obviously, $M_{s,i}(---)$ are vector spaces on . Moreover, the following lemma follows from the fact:
and the Leibniz rule immediately.
$\mathbb{C}$$(|\lambda|^{1/2}+|\xi’|)^{-|\alpha’|}\leq|\xi’|^{-|\alpha’|}$125
Lemma 2.2. Let$s_{1},$$s_{2}$be two real numbers. Then, the following three assertions hold.

(1)

Given $m_{i}\in M_{s_{\rangle}1}(---)(i=1,2)$ , we have $m_{1}m_{2}\in M_{s_{1}+s_{2},1}(---)$ .

(2)

Given $\ell_{l}\in M_{s.,i}(---)(i=1,2)$ , we have(3)

Given $n_{i}\in M_{s.,2}(---)(i=1,2)$ , we have $m_{1}m_{2}\in M_{s_{1}+s_{2},2}(---)$ .

$\ell_{1}\ell_{2}\in M_{s_{1}+s_{2},2}(---)$.

In what follows, we use the following lemma due to Shibata and Shimizu [3, Lemma 5.4].
Lemma 2.3. Let $0<\theta<\pi/2$ and $1<q<\infty$ . Given
we define the operators $L_{j}(\lambda)(j=1,2,3,4)$ by$\ell_{0}(\xi’, \lambda)\in \mathbb{M}_{0,1}(\Sigma_{\theta})$and$\ell_{1}(\xi’, \lambda)\in \mathbb{M}_{0,2}(\Sigma_{\theta})$,$[L_{1}( \lambda)h](x)=\int_{0}^{\infty}\mathcal{F}^{;-1}[\ell_{0}(\xi’, \lambda)\lambda^{1/2}e^{-A_{k}(\xi’,\lambda)(x_{N}+y_{N})}\mathcal{F}[h](\xi’, y_{N})](x’)dy_{N},$$[L_{2}( \lambda)h](x)=\int_{0}^{\infty}\mathcal{F}^{J-1}[\ell_{1}(\xi, \lambda)Ae^{-A_{k}(\xi’,\lambda)(x_{N}+y_{N})}\mathcal{F}’[h](\xi’, y_{N})|(x’)dy_{N},$$[L_{3}( \lambda)h](x)=\int_{0}^{\infty}\mathcal{F}^{;-1}[\ell_{1}(\xi’, \lambda)Ae^{-A(x_{N}+y_{N})}\mathcal{F}[h](\xi’, y_{N})|(x’)dy_{N},$$[L_{4}( \lambda)h](x)=\int_{0}^{\infty}\mathcal{F}_{\xi’}^{;-1}[\ell_{1}(\xi’, \lambda)A^{2}\mathcal{M}_{k}(\xi’, x_{N}+y_{N}, \lambda)\mathcal{F}’[h](\xi’, y_{N})](x’)dy_{N}.$Then,$L_{\iota}$

is a bounded linear operator on$L_{q}(\mathbb{R}_{+}^{N})$and$\Vert L_{\iota}(\lambda)h\Vert_{L_{q}(\mathbb{R}_{+}^{N})}\leq C\Vert h\Vert_{L_{q}(\mathbb{R}_{+}^{N})}.$

Using the identities:$\omega\pm=\underline{\lambda\gamma|\xi’|^{2}}, 1=\underline{\lambda\gamma|\xi’|^{2}},$and applying Lemma 2.3, we have$\Vert\lambda^{1/2}u\pm 2, \nabla u\pm 2)\Vert_{L_{q}(\mathbb{R}_{\pm}^{N})}\leqC\{\Vert f_{+}|_{L_{q}(\mathbb{R}_{+}^{N})}+\Vert f_{-}|_{L_{q}(\mathbb{R}_{-}^{N})}+\Vert(\lambda^{1/2}g\pm, \nabla_{9}\pm)\Vert_{L_{q}(R_{\pm}^{N})}\}$

Setting $u\pm=u\pm 1+u\pm 2$ and combining (2.7) and (2.12) yield that$u\pm$.

(2.12)

satisfy the estimate:$\Vert(\lambda^{1/2}u\pm, \nabla u\pm)\Vert_{L_{q}(\mathbb{R}_{\pm}^{N})}\leq C\{\Vert f+\Vert_{L_{q}(\mathbb{R}_{+}^{N})}+\Vert f_{-}\Vert_{L_{q}(\mathbb{R}_{-}^{N})}+\Vert(\lambda^{1/2}g\pm, \nabla g\pm)\Vert_{L_{q}(\mathbb{R}_{\pm}^{N})}\}$.

(2.13)

satisfies
andMoreover, by the Fourier multiplier theorem and Lemma 2.3, we see that
(2.2). Since $f_{\pm}|_{xN^{=\pm 0}}=0$ , assuming that $g=0$ in (2.2), using the integration by parts and defining, we haveby $u(x)=u\pm(x)$ for$u\pm\in W_{q}^{2}(\mathbb{R}_{\pm}^{N})$$u\pm$$u$$x\in \mathbb{R}_{\pm}^{N}$

Theorem 2.4. Let $1<q<\infty$ and $0<\theta<\pi/2$ . Set$\Sigma_{\theta}=\{\lambda\in \mathbb{C}\backslash \{0\}||\arg\lambda|\leq\pi-\theta\}.$Then, for any
andsatisfying the estimate:$f\in L_{q}(\dot{\mathbb{R}}^{N})$$\lambda\in\Sigma_{\theta}$, the variational problem (2.1) admits a unique solution $u\in$$W_{q}^{1}(\mathbb{R}^{N})$$\Vert(\lambda^{1/2}u, \nabla u)\Vert_{L_{q}(R^{N})}\leq C\Vert f\Vert_{L_{q}(\dot{\mathbb{R}}^{N})}$

with some constant $C>0.$.

(2.14)

126
2.2

Bent half-space problem

Let :be its inverse map. Writing $\nabla\Phi(x)=\mathcal{A}+B(x)$be a bijection ofclass and let
are orthonormal matrices with constantand $\nabla\Phi^{-1}(y)=\mathcal{A}_{-1}+B_{-1}(y)$ , we assume thatand
coeﬃcients and $B(x)$ and $B_{-1}(y)$ are matrices of functions inwith $N<r<\infty$ such that$\Phi$$\mathbb{R}^{N}arrow \mathbb{R}^{N}$$\Phi^{-1}$$C^{1}$$\mathcal{A}$$\mathcal{A}_{-1}$$W_{r}^{1}(\mathbb{R}^{N})$$\Vert(B, B_{-1})\Vert_{L_{\infty}(\pi)}N\leq M_{1}, \Vert\nabla(B, B_{-1})\Vert_{L,(\mathbb{R}^{N})}\leq M_{2}$We will choose
following. Set.

(2.15)

small enough eventually, so that we may assume that $0<M_{1}\leq 1\leq M_{2}$ in the,be the unit outer normal to . Setting $\Phi^{-1}=$and let, we see thatis represented by $x_{N}=\Phi_{-1,N}(y)=0$ , which furnishes that$M_{1}$$D\pm=\Phi(\mathbb{R}_{\pm}^{N})$$(\Phi_{-1,1}, \ldots, \Phi_{-1,N})$$\Gamma_{0}=\Phi(\mathbb{R}_{C}^{N})$$\Gamma_{0}$$n_{0}$$\Gamma_{0}$$n_{0}=\frac{\nabla\Phi_{-1,N}}{|\nabla\Phi_{-1,N}|}=\frac{(\mathcal{A}_{N1}+B_{N1},\ldots,\mathcal{A}_{NN}+B_{NN})}{(\sum_{\iota=1}^{N}(\mathcal{A}_{Ni}+B_{N_{l}})^{2})^{1/2}}$

where we have set$\mathcal{A}_{-1}=(\mathcal{A}_{ij})$

and $B_{-1}=(B_{\iota j})$ . In particular,, by (2.15)$\sum_{\iota=1}^{N}(\mathcal{A}_{Ni}+B_{N\iota})^{2}=1+\sum_{i=1}^{N}(2\mathcal{A}_{N\iota}B_{N\iota}+B_{N\iota}^{2})$$n_{0}$,(2.16)

is defined on the whole$\mathbb{R}^{N}$. Since$\Vert\nabla n_{0}\Vert_{L_{r}(R^{N})}\leq C_{N}M_{2}.$Moreover, we have$\frac{\partial}{\partial y_{j}}=\sum_{k=1}^{N}\frac{\partial x_{k}}{\partial y_{j}}\frac{\partial}{\partial x_{k}}=\sum_{k=1}^{N}(\mathcal{A}_{\iota j}+B_{ij}(\Phi(x)))\frac{\partial}{\partial x_{k}}$.

(2.17)By (2.15),$\Vert B_{jk}o\Phi\Vert_{L_{\infty}(\mathbb{R}^{N})}\leq CM_{1}, \Vert\nabla(B_{jk}o\Phi)\Vert_{L_{r}(R^{N})}\leq CM_{2}$Let$\ddot{\mathbb{R}}^{N}=D+\cup D_{-}$.

(2.18), and$(u, v)_{\ddot{\mathbb{R}}^{N}}= \int_{D_{+}}u(x)\overline{v(x)}dx+\int_{D-}u(x)\overline{v(x)}dx..$

In this subsection, we consider the variational equations:$\lambda(u, \varphi)_{\ddot{R}^{N}}+(\gamma_{1}\nabla u, \nabla\varphi)_{\ddot{R}^{N}}=(f, \nabla\varphi)_{R^{N}}$for any(2.19)$\varphi\in W_{q}^{1}(\mathbb{R}^{N})$with $f=(f_{1}, \ldots, f_{N})\in L_{q}(\mathbb{R}^{N})$ , andis a piecewise smooth ﬃnction defined bywith
By the transformation: $y=\Phi(x)$ , the equation (2.19) is transformed to theequation:$\gamma_{1}|_{D_{\pm}}=\gamma_{1,\pm}$$\gamma_{1}$some positive constants$\gamma\pm\cdot$$\lambda(vJ, \varphi)_{\dot{R}^{N}}+((\gamma_{1}\circ\Phi)J\sum_{j,k,\ell=1}^{N}(\mathcal{A}_{jk}+B_{jk}o\Phi)(\mathcal{A}_{j\ell}+B_{jl}o\Phi)\partial_{k}v, \partial_{\ell}\varphi)_{\dot{R}^{N}}=(F, \nabla\varphi)_{\dot{R}^{N}}$for any$\varphi\in W_{q}^{1},$$(\mathbb{R}^{N})$(2.20), where $J=\det\Phi$ and $F=(F_{1}, \ldots, F_{N})$ with $F_{k}= \sum_{j=1}^{N}(\mathcal{A}_{jk}+B_{jk}o\Phi)f_{j}$ . By

(2.15),$\Vert J-1\Vert_{L_{\infty}(R^{N})}\leq CM_{1}, \Vert\nabla J\Vert_{L_{r}(R^{N})}\leq CM_{2}$.

(2.21)

Let $Jv=w$ , and then$J \sum_{j,k,l=1}^{N}(\mathcal{A}_{jk}+B_{jk}\circ\Phi)(\mathcal{A}_{j\ell}+B_{j\ell}o\Phi)\partial_{k}v$$= \sum_{j,k,\ell=1}^{N}(\mathcal{A}_{jk}+B_{jk}o\Phi)(\mathcal{A}_{j\ell}+B_{j\ell}o\Phi)\partial_{k}w-\{\sum_{j,k,\ell=1}^{N}(\mathcal{A}_{jk}+B_{jk}o\Phi)(\mathcal{A}_{j\ell}+B_{jl}\circ\Phi)\partial_{k}J\}J^{-2}w.$

Noting$\sum_{j=1}^{N}\mathcal{A}_{jk}\mathcal{A}_{jl}=\delta_{k\ell}=\{\begin{array}{ll}1 for j=k,0 for j\neq k,\end{array}$127
and letting $P=(P_{k\ell}(x))$ and $Q(x)=(Q_{1}(x), \ldots, QN(x))$ with
$P_{k\ell}= \sum_{j=1}^{N}\{\mathcal{A}_{jk}(B_{j\ell}o\Phi)+\mathcal{A}_{j\ell}(B_{jk}o\Phi)+(B_{jk}\circ\Phi)(B_{jl}o\Phi$$Q_{f}=- \sum_{j,k=1}^{N}(\mathcal{A}_{jk}+B_{jk}\circ)(\mathcal{A}_{j}\ell+B_{j\ell}o)(\partial_{k}J)J^{-2},$

we have(2.22)$(\lambda w, \varphi)_{\dot{\pi}}N+((\gamma_{1}0\Phi)\nabla w, \nabla\varphi)_{\dot{\mathbb{R}}^{N}}+((\gamma_{1}0\Phi)(P\nabla w+Qw), \nabla\varphi)_{\dot{\mathbb{R}}^{N}}=(F, \nabla\varphi)_{\mathbb{R}^{N}}$for any$\varphi\in W_{q}^{1},(\mathbb{R}^{N})$. By Sobolev’s imbedding theorem,(2.23)$\Vert ab\Vert_{L_{q}(\dot{R}^{N})}\leq C\Vert a\Vert\Vert b\Vert_{L_{q}^{N}}^{1-\frac{N}{(\dot{\mathbb{R}}r}}\Vert\nabla b\Vert_{q}^{\frac{N}{Lr}})(\dot{\pi})$

andfor any
and using (2.18) and (2.21), we have$a\in L_{r}(\dot{\mathbb{R}}^{N})$$b\in W_{q}^{1}(\dot{\mathbb{R}}^{N})$provided $N<r<\infty$ (cf. [2, Lemma 2.4]). So, applying (2.23)(2.24)$\Vert P\nabla w+Qw\Vert_{L_{q}(\dot{\mathbb{R}}^{N})}\leq C(M_{1}+\sigma)\Vert\nabla w\Vert_{L_{q}(\dot{\mathbb{R}}^{N})}+C_{\sigma}M_{2}\Vert w\Vert_{L_{q}(\dot{\mathbb{R}}^{N})}$, whereis a constant such thatfor any small $\sigma>0$ with some constants and, letbe a solution to the variational equation:
Given
$C$

$C_{\sigma}$$C_{\sigma}$$C_{\sigma}arrow\infty$as(2.25)$(\lambda w, \varphi)_{\dot{\pi}^{N}}+((\gamma_{1}0\Phi)\nabla w, \nabla\varphi)_{\dot{\mathbb{R}}^{N}}=(F-(\gamma_{1}0\Phi)(P\nabla z+Qz), \nabla\varphi)_{\mathbb{R}^{N}}$for any$\varphi\in W_{q}^{1},$ $(\mathbb{R}^{N})$$\sigmaarrow 0.$$w\in W_{q}^{1}(\mathbb{R}^{N})$$z\in W_{q}^{1}(\mathbb{R}^{N})$. By Theorem 2.4 and (2.24), such$w$

uniquely exists, which satisfies the estimate:$\Vert(\lambda^{1/2}w, \nabla w)\Vert_{L_{q}(\mathbb{R}^{N})}\leq C(M_{1}+\sigma)\Vert\nabla z\Vert_{L_{q}(\pi^{N})}+C_{\sigma}M_{2}\Vert z\Vert_{L_{q}(R^{N})}+C\Vert f\Vert_{L_{q}(R^{N})}.$

Choosing $\sigma>0$ and $M_{1}>0$ small enough andhave$|\lambda|$

large enough, by the Banach fixed point theorem weTheorem 2.5. Let $1<q<\infty$ and $0<\theta<\pi/2$ . For$\lambda_{0}>0$, we set$\Sigma_{\theta,\lambda_{0}}=\{\lambda\in\Sigma_{\theta}||\lambda|\geq\lambda_{0}\}.$

Then, there exists asolutionsuch that for any
satisfying the estimate:$\lambda_{0}>0$$u\in W_{q}^{1}(\mathbb{R}^{N})$$\lambda\in\Sigma_{\theta,\lambda_{0}}$and$f\in L_{q}(\mathbb{R}^{N})$problem (2.19) admits a unique$\Vert(\lambda^{1/2}u, \nabla u)\Vert_{L_{q}(\pi)}N\leq C\Vert f\Vert_{L_{q}(\mathbb{R}^{N})}.$

Next, for the later use we consider two more variational problems. The first one is the variational:
problem in(2.26),
for any$\mathbb{R}^{N}$$\varphi\in W_{q}^{1},$$\lambda(u, \varphi)_{R^{N}}+(\gamma\nabla u, \nabla\varphi)_{\mathbb{R}^{N}}=(f, \nabla\varphi)_{\mathbb{R}^{N}}$where$\gamma$$(\mathbb{R}^{N})$is a positive constant. Then, we have

Theorem 2.6. Let $1<q<\infty$ and $0<\theta<\pi/2$ . Then, for any
satisfying the estimate:admits a unique solution$\lambda\in\Sigma_{\theta}$

and$f\in L_{q}(\mathbb{R}^{N})$problem (2.27)$u\in W_{q}^{1}(\mathbb{R}^{N})$$\Vert(\lambda^{1/2}u, \nabla u)\Vert_{L_{q}(R^{N})}\leq C\Vert f\Vert_{L_{q}(R^{N})}.$The second one is the variational problem in$D_{+}$:$\lambda(u, \varphi)_{D_{+}}+(\gamma\nabla u, \nabla\varphi)_{D_{+}}=(f, \nabla\varphi)_{D_{+}}$

for any$\varphi\in W_{q,0}^{1}(D_{+})$.

(2.27)

Employing the similar argumentation to the proof of Theorem 2.5, we havesuch that for any
Theorem 2.7. Let $1<q<\infty$ and $0<\theta<\pi/2$ . Then, there exists aand $f\in L_{q}(D_{+})$ problem (2.19) admits a unique solution $u\in W_{q,0}^{1}(D_{+})$ satisfying the estimate:$\lambda_{0}>0$$\Vert(\lambda^{1/2}u, \nabla u)\Vert_{L_{q}(D_{+})}\leq C\Vert f\Vert_{L_{Q}(D_{+})}.$$\lambda\in\Sigma_{\theta,\lambda_{0}}$128
A proof of Theorem 1.5

2.3

To prove Theorem 1.5, first we consider the variational problem:

for any$\lambda(u, \varphi)_{\dot{\Omega}}+(\gamma^{1}\nabla u, \nabla\varphi)_{\dot{\Omega}}=(f, \nabla\varphi)_{\dot{\Omega}}$$\varphi\in W_{q,0}^{1}(\Omega)$.

(2.28)

And then, we have
Theorem2.8. Leet $1<q<\infty,$ $N<r<\infty$ and $0<\theta<\pi/2$ . Assume that $\max(q, q’)\leq r$ and that
. Then, there exists asuch that for any
andare compact hypersurfaces of classsatisfying the estimate:, problem (2.28) admits a unique solution
and$W_{r}^{2-1/r}$$\Gamma_{-}$$\Gamma$$f\in L_{q}(\Omega)^{N}$$\lambda_{1}>0$$u\in W_{q,0}^{1}(\Omega)$$\lambda\in\Sigma_{\theta,\lambda_{1}}$$\Vert(\lambda^{1/2}u, \nabla u)\Vert_{L_{q}(\Omega)}\leq C\Vert f\Vert_{L_{q}(\Omega)}.$To prove Theorem 2.1, we start with. Set
be compact hyper-surfaces of
Proposition 2.9. Let $N<r<\infty$ and let and
. Let
and
be any positive number $\in(O, 1)$ . Then, there exist constants $M_{2}>0,$ $0<d<1,$
open sets, finitely many $N$ -vector of functions $\Phi_{j}^{l}\in W_{r}^{2}(\mathbb{R}^{N})^{N}(i=0,1,j=1, \ldots, K_{i})$ , and$x_{j}^{i}\in\Gamma_{\iota}(i=0,1,j=1,\ldots\rangle K_{l})$ such that the following assertions hold:points$\Gamma_{1}=\Gamma_{-}$$W_{r}^{2-1/r}$$\Gamma_{-}$$\Gamma$$\Gamma_{0}=\Gamma$$M_{1}$$U\pm\subset\Omega\pm$

(i)

(ii)

The maps:$\mathbb{R}^{N}\ni x\mapsto\Phi_{j}^{l}(x)\in \mathbb{R}^{N}(i=0,1)$$\Omega=(\bigcup_{j=1}^{K_{0}}\Phi_{j}^{0}(\mathbb{R}^{N})\cap B_{d}(x_{j}^{0}))\cup(\bigcup_{j=1}^{K_{1}}\Phi_{j}^{I}(\mathbb{R}_{+}^{N})\cap B_{d}(x_{j}^{1}))\cup U+\cup U_{-},$$\Phi_{j}^{0}(\mathbb{R}_{0}^{N})\cap B_{d}(x_{j}^{0})=\Gamma\cap B_{d}(x_{J}^{0})$,$\Phi_{j}^{1}(\mathbb{R}_{0}^{N})\cap B_{d}(x_{j}^{1})=\Gamma_{-}\cap B_{d}(x_{j}^{1})$(i\"u)

There exist$C^{\infty}$$0\leq\zeta_{j}^{i},$functions$\tilde{\zeta}_{j}^{l}\leq 1,$$\zeta_{j}^{l},$$0\leq\zeta_{\pm}^{2},$$\Vert(\zeta_{j}^{i},\tilde{\zeta}_{j}^{l})\Vert_{W_{\infty}^{2}(R^{N})},$

Here,$c_{O}$$\Phi_{j}^{0}(\mathbb{R}^{N})\cap B_{d}(x_{j}^{0})=\Omega\cap B_{d}(x_{j}^{0})$,$\tilde{\zeta}_{j}^{l}(i=0,1, j=1, \ldots, K_{i})$$\tilde{\zeta}_{\pm}^{2}\leq 1$

$on$

supp

supp

and
mal matrices, andwhich satisfy the conditions:$B_{J}^{l}$$B_{j,-}^{l}$$\zeta_{\pm}^{2}$, and$\tilde{\zeta}_{j}^{l}\subset B_{d}.(x_{j}^{i})$on supp$\tilde{\zeta}_{j}^{l}=1$.

$M_{2},$

are

$N,$

and

$q$

and
where
matrices of$\mathcal{A}_{j}^{l}$for $i=0$ , 1 and $j=1$ , . . .

,$supp\zeta_{\pm}^{2},$ $supp\tilde{\zeta}_{\pm}^{2}\subset U\pm,$$\tilde{\zeta}_{\pm}^{2}=1$$\zeta_{\pm}^{2},$.

$\mathcal{A}_{j}^{i}$,-are. Here,constant orthonorfunctions defined onand$N\cross N$$W_{r}^{1+i}(\mathbb{R}^{N})$$\Vert B_{j,-}^{i}\Vert_{L_{\infty}(R^{N})}\leq M_{1},$$K_{\iota}$on supp$r.$$N\cross N$$\Vert B_{j}^{i}\Vert_{L_{\infty}(R^{N})}\leq M_{1},$such that$\tilde{\zeta}_{\pm}^{2}$$\zeta_{j}^{i},$$\sum_{j=1}^{\infty}\zeta_{j}^{l}=1on\Gamma^{\iota}(i=0,1)$$\nabla(\Phi_{j}^{i})^{-1}=\mathcal{A}_{j,-}^{i}+B_{j}^{l}$$\Vert\nabla B_{j,-}^{l}\Vert_{W_{\dot{r}}(R^{N})}\leq M_{2}$$\zeta_{j}^{l},$,$\overline{\Omega},$

is a constant which depends on$\nabla\Phi_{j}^{l}=\mathcal{A}_{j}^{i}+B_{j}^{i},$,$\Phi_{j}^{1}(\mathbb{R}_{+}^{N})\cap B_{d}(x_{j}^{1})=\Omega_{-}\cap B_{d}(x_{j}^{1})$$\Vert(\zeta_{\pm}^{2},\tilde{\zeta}_{\pm}^{2})\Vert_{W_{\infty}^{2}(R^{N})}\leqc_{0},$$\sum_{i=0}^{1}\sum_{j=1}^{K}\zeta_{j}^{l}+\zeta_{+}^{2}+\zeta_{-}^{2}=1$(iv)

are bijective.

$\mathbb{R}^{N}$$\Vert\nabla B_{j}^{l}\Vert_{W_{\dot{r}}(R^{N})}\leq M_{2}$$W_{r}^{0}(\mathbb{R}^{N})=L_{r}(\mathbb{R}^{N})$.

class, employing the argumentations due toandare compact hyper-surfaces ofEnomoto and Shibata [1, Proposition 6.1], we can prove Proposition 2.9, so that we may omit its proof.
, let
, and
. Given,Let
be solutions to the following variational problems:
and

Since

$\Gamma$$\ddot{\mathbb{R}}_{j}^{N}=\Phi_{j}^{0}(\mathbb{R}_{+}^{N})\cup\Phi_{j}^{0}(\mathbb{R}_{-}^{N})$$u_{j}^{1}$$W_{r}^{2-1/r}$$\Gamma_{-}$$D_{j}^{1}=\Phi_{j}^{1}(\mathbb{R}_{+}^{N})$$\Gamma_{j}^{1}=\partial D_{j}^{1}=\Phi_{j}^{1}(\mathbb{R}_{0}^{N})$$\lambda(u_{j}^{0}, \varphi)_{\ddot{R}_{g}^{N}}+(\gamma_{j}^{0}\nabla u_{j}^{0},\nabla\varphi)_{\ddot{R}_{J}^{N}}=(\tilde{\zeta}_{j}^{0}f, \nabla\varphi)_{R^{N}}$$\lambda(u_{j}^{1}, \varphi)_{D_{J}^{1}}+(\gamma_{1,-}\nabla u_{j}^{1}, \nabla\varphi)_{D_{J}^{1}}=(\tilde{\zeta}_{j}^{1}f, \nabla\varphi)_{D_{g}^{1}}$$\lambda(u_{\pm}^{2}, \varphi)_{R^{N}}+(\gamma_{1,\pm}\nabla u_{\pm}^{2}, \nabla\varphi)_{R^{N}}=(\tilde{\zeta}_{\pm}^{2}f, \nabla\varphi)_{R^{N}}$for any$\varphi\in W_{q}^{1},(\mathbb{R}^{N})$for any$\varphi\in W_{q,0}^{1}(D_{j}^{1})$for any$\varphi\in W_{q}^{1},(\mathbb{R}^{N})$are piece-wise constant functions defined bysuch that for any
and Theorem 2.7, there exists a constantHere,$f\in L_{q}(\Omega)^{N}$$u_{j}^{0},$$u_{\pm}^{2}$$\gamma_{j}^{0}$$\gamma_{j}^{0}|_{\Phi_{J}^{0}(R_{\pm}^{N})}=\gamma_{1,\pm}$$\lambda_{1}\geq 1$,.

(2.29),(2.30)(2.31). By Theorem 2.5, Theorem 2.6$\lambda\in\Sigma_{\theta,\lambda_{1}}$problems (2.29), (2.30) and129(2.31) admit unique solutions$u_{j}^{0}\in W_{q}^{1}(\mathbb{R}^{N})$,$u_{j}^{1}\in W_{q}^{1}(D_{j}^{1})$and$u_{\pm}^{2}\in W_{q}^{1}(\mathbb{R}^{N})$satisfying the estimates:$\Vert(\lambda^{1/2}u_{j}^{0}, \nabla u_{j}^{0})\Vert_{L_{q}(\mathbb{R}^{N})}\leq C\Vert\tilde{\zeta}_{j}^{0}f\Vert_{L_{q}(R^{N})},$$\Vert(\lambda^{1/2}u_{j}^{1}, \nabla u_{j}^{1})\Vert_{L_{q}(D_{g}^{1})}\leq C\Vert\tilde{\zeta}_{j}^{1}f\Vert_{L_{Q}(D_{J}^{1})}$,(2.32)$\Vert(\lambda^{1/2}u_{\pm}^{2}, \nabla u_{j}^{0})\Vert_{L_{q}(R^{N})}\leq C\Vert\tilde{\zeta}_{\pm}^{2}f\Vert_{L_{q}(\mathbb{R}^{N})}.$

Let$\mathcal{A}(\lambda)$

be an operator defined by$\mathcal{A}(\lambda)f=\sum_{i=0}^{1}\sum_{j=1}^{K}\zeta_{j}^{l}u_{j}^{l}+\zeta_{+}^{2}u_{+}^{2}+\zeta_{-}^{2}u_{-}^{2},$and then noting that$( \sum_{l=0}^{1}\sum_{j=1}^{K}\nabla\zeta_{j}^{l}+\nabla\zeta_{+}^{2}+\nabla\zeta^{\underline{2}}=0, by (2.29)$, (2.30) and (2.31) we have

(2.33)$\lambda(\mathcal{A}(\lambda)f, \varphi)_{\dot{\Omega}}+(\gamma^{1}\nabla \mathcal{A}(\lambda)f, \nabla\varphi)_{\dot{\Omega}}=(f+\mathcal{R}_{1}(\lambda)f, \nabla\varphi)_{\dot{\Omega}}+(\mathcal{R}_{2}(\lambda)f, \varphi)_{\dot{\Omega}}+(\mathcal{R}_{3}(\lambda)f, \varphi)_{\Gamma}$

for any

with$\varphi\in W_{q,0}^{1}(\Omega)$$\mathcal{R}_{1}(\lambda)f=2\sum_{i=0}^{1}\sum_{j=1}^{K}\gamma_{j}^{l}(\nabla\zeta_{j}^{i})u_{j}^{i}+\gamma_{1,+}(\nabla\zeta_{+}^{2})u_{+}^{2}+\gamma_{1,-}(\nabla\zeta_{-}^{2})u_{-}^{2}$$\mathcal{R}_{2}(\lambda)f=-\{\sum_{\iota=0}^{1}\sum_{j=1}^{K}\gamma_{j}^{l}(\triangle\zeta_{j}^{l})u_{j}^{i}+\gamma_{1,+}(\Delta\zeta_{+}^{2})u_{+}^{2}+\gamma_{1,-}(\Delta\zeta_{-}^{2})u_{-}^{2}\},$$\mathcal{R}_{3}(\lambda)f=-\sum_{j=1}^{K_{0}}(\gamma_{1,+}-\gamma_{1,-})(u_{j}^{0}(\nabla\zeta_{j}^{0})\cdot n)|r.$

By Poincar\’es’ inequality,(2.34)$\Vert\varphi\Vert_{L_{q}(\Omega)}\leq C\Vert\nabla\varphi\Vert_{L_{q}(\Omega)}$

for any$\varphi\in W_{q,0}^{1}(\Omega)$, so that by (2.32)$|(\mathcal{R}_{2}(\lambda)f, \varphi)_{\dot{\Omega}}|\leq C|\lambda|^{-1/2}\Vert f\Vert_{L_{q}(\Omega)}\Vert\nabla\varphi\Vert_{L_{q}(\Omega)}$

for any$\varphi\in W_{q,0}^{1}(\Omega)$.

(2.35). By the interpolation inequality for the trace operator and (2.32) we have$\Vert \mathcal{R}_{3}(\lambda)f\Vert_{L_{q}(\Gamma)}\leq(\sum_{j=1}^{K_{0}}||(\nabla\zeta_{j}^{0})u_{j}^{0}\Vert_{L_{q}(\Omega)}^{1-1/q}\Vert\nabla((\nabla\zeta_{j}^{0})u_{j}^{0})\Vert_{L_{q}(\Omega)}^{1/q})\leq C|\lambda|^{-4-}\overline{2}q’\Vert f\Vert_{L_{q}(\Omega)},$which, combined with (2.34), furnishes that$|(\mathcal{R}_{3}(\lambda)f, \varphi)_{\Gamma}|\leq C|\lambda|^{-}2q’\Vert f\Vert_{L_{q}(\Omega)}\Vert\nabla\varphi\Vert_{L_{q},(\Omega)}$

for any
that$\varphi\in W_{q,0}^{1}(\Omega)$.

(2.36). By the Hahn-Banach theorem, there exists an operator$\mathcal{R}_{4}(\lambda)\in \mathcal{L}(L_{q}(\Omega)^{N})$such$(\mathcal{R}_{2}(\lambda)f, \varphi)_{\dot{\Omega}}+(\mathcal{R}_{3}(\lambda)f,\varphi)_{\Gamma}=(\mathcal{R}_{4}(\lambda)f, \nabla\varphi)_{\dot{\Omega}}$

for any$\varphi\in W_{q,0}^{1}(\Omega)$, and moreover by (2.35) and (2.36)(2.37)$\Vert \mathcal{R}_{4}(\lambda)f\Vert_{L_{q}(\Omega)}\leq C|\lambda|^{-\lrcorner}2q^{7}\Vert f\Vert_{L_{q}(\Omega)}$

for any
By (2.33) we have$\lambda\in\Sigma_{\theta,\lambda_{1}}.$$\lambda(\mathcal{A}(\lambda)f, \varphi)_{\dot{\Omega}}+(\gamma^{1}\nabla \mathcal{A}(\lambda)f, \nabla\varphi)_{\dot{\Omega}}=((I+(\mathcal{R}_{1}(\lambda)+\mathcal{R}_{4}(\lambda))f, \nabla\varphi)_{\Omega}$

for any$\varphi\in W_{q,0}^{1}(\Omega)$

,

130
where

$I$

denotes the identity operator from

$L_{q}(\Omega)$

onto itself. By (2.32), we have$\Vert \mathcal{R}_{1}(\lambda)f\Vert_{L_{q}(\Omega)}\leq C|\lambda|^{1/2}\Vert f\Vert_{L_{q}(\Omega)},$which, combined with (2.37), furnishes that$\Vert \mathcal{R}_{1}(\lambda)+\mathcal{R}_{4}(\lambda)\Vert_{\mathcal{L}(L_{q}(\Omega)^{N})}\leq 1/2$

. Thus,with some large constantfor any
problem (2.28) uniquely, which satisfies the estimate:$\lambda_{2}\geq\lambda_{1}$$\lambda\in\Sigma_{\theta,\lambda_{2}}$$v=\mathcal{A}(\lambda)(I+\mathcal{R}_{1}(\lambda)+\mathcal{R}_{4}(\lambda))^{-1}f$solves$\Vert(\lambda^{1/2}v, \nabla v)\Vert_{L_{q}(\Omega)}\leq C\Vert f\Vert_{L_{q}(\Omega)}.$This completes the proof of Theorem 2.8.

$\square$Finally, we give afor any, we have
Proof of Theorem 1.5. For any and$div$(f–g) $=0$ in , so that we consider the quotient spaceprovided that
. By the Helmholtz decomposition,, where $PL_{q}(\Omega)=\{f\in L_{q}(\Omega)^{N}|$ divf $=0$ in
uniquely such that $f=g+\nabla\psi$ andand, there exist
for any
is a unique solution to the weak Neumann problem:
divg $=0$ in . Here,$g\in L_{q}(\Omega)^{N}$$f$$(f, \nabla\varphi)_{\Omega}=(g, \nabla\varphi)_{\Omega}$$\dot{L}_{q}(\Omega)=$$\Omega$$\varphi\in W_{q,0}^{1}(\Omega)$$\Omega\}$$L_{q}(\Omega)^{N}\backslash PL_{q}(\Omega)$$g\in L_{q}(\Omega)^{N}$$f\in L_{q}(\Omega)^{N}$$\Omega$$\psi\in W_{q}^{1}(\Omega)$$\psi\in W_{q}^{1}(\Omega)$

for any$(\nabla\psi, \nabla\varphi)_{\Omega}=(f, \nabla\varphi)_{\Omega}$

In other words,$\psi$( ).
$\Omega$$\varphi\in W_{q}^{1},$is a weak solution to the Neumann problem:$\triangle\psi=divf in\Omega, n\cdot\nabla\psi=n_{-}\cdot f on\Gamma_{-}.$, and then $[f]=\nabla\psi$ . If $divf=0$ , then $[f]=0.$, let $[f]$ be the representation of inprovidedMoreover, by the regularity theorem of the solutions to the Neumann problem,
For

$f$

$f\in L_{q}(\Omega)^{N}$$\dot{L}_{q}(\Omega)$$[f]\in W_{q}^{1}(\Omega)^{N}$that
Under these preparation, we prove Theorem 1.5. Let be a large positive number such that, problemis the number given in Theorem 2.8, and then by Theorem 2.8 for any
where
.
Let
be
satisfying the estimate:(2.28) admits a unique solution. We look for a solution (1.9) of the form:
defined by
an operator, and thenwith$f\in W_{q}^{1}(\Omega)^{N}.$$\lambda>\lambda_{1},$$\lambda$$f\in\dot{L}_{q}(\Omega)^{N}$$\lambda_{1}$$\mathcal{R}$$u\in W_{q,0}^{1}(\Omega)$$\Vert u\Vert_{W_{q}^{1}(\Omega)}\leq C\Vert f\Vert_{L_{q}(\Omega)}$$u=\mathcal{R}g$$\mathcal{R}f=u$$\in \mathcal{L}(\dot{L}_{q}(\Omega), W_{q,0}^{1}(\Omega))$$g\in\dot{L}_{q}(\Omega)$

for any$(\gamma_{1}\nabla u, \nabla\varphi)_{\dot{\Omega}}=(g, \nabla\varphi)_{\dot{\Omega}}+(\lambda u, \varphi)_{\dot{\Omega}}$

Since is a bounded domain whose boundary is a hyper-surface ofsolving the Dirichlet problem:$\Omega$$\Delta h=-\lambda u$in$\Omega,$$\varphi\in W_{q,0}^{1}(\Omega)$$W_{r}^{2-1/r}$.

(2.38)

class, there exists a$h\in W_{q}^{2}(\Omega)$$h|r_{-}=0$

uniquely and satisfying the estimate:$\Vert h\Vert_{W_{q}^{2}(\Omega)}\leq C\Vert\lambda u\Vert_{L_{q}(\Omega)}\leq C\Vert g\Vert_{L_{q}(\Omega)}.$

Let

$S$

be an operator defined by $Sg=[\nabla h]$ , and then$(\lambda u, \varphi)_{\dot{\Omega}}=-(\Delta h, \varphi)_{\dot{\Omega}}=(\nablaSg, \nabla\varphi)_{\dot{\Omega}}$

for any$\varphi\in W_{q,0}^{1}(\Omega)$,

and therefore the equation (2.38) is transformed to$(\gamma_{1}\nabla \mathcal{R}g, \nabla\varphi)_{\dot{\Omega}}=((I+S)g, \nabla\varphi)_{\dot{\Omega}}$

for any. Sinceis the identity operator inbounded, so that the Rellich compactness theorem yields that

where

$I$

$\mathcal{L}(\dot{L}_{q}(\Omega))$$\varphi\in W_{q,0}^{1}(\Omega)$$\nabla h\in W_{q}^{1}(\Omega)^{N},$$S$(2.39),$Sg\in W_{q}^{1}(\Omega)^{N}$. Moreover,is a compact operator from$\dot{L}_{q}(\Omega)$

is
into

$\Omega$

131
itself. Thus, in view of the Riesz-Schauder theorem, especially the Fredholm alternative principle, inorder to. prove that the $I+S$ is invertible, it sufices to prove that the kernel of $I+S$ is trivial. Thus,
satisfy $(I+S)g=0$ , and then by (2.39)
let$g\in L_{q}(\Omega)$

for any$(\gamma_{1}\nabla \mathcal{R}g, \nabla\varphi)_{\dot{\Omega}}=0$$\varphi\in W_{q,0}^{1}(\Omega)$.

(2.40), that is, divg $=0$ . First, we consider the case $2\leq q<\infty$ . Since, that is,, settingin (2.40), we haveThus, $\mathcal{R}g(x)=c\pm forx\in\Omega\pm$ with somein , which yields thatis constant in. Thus, $c+=c_{-}=0,$, which furnishes thatandBut,
constants. So, we have $g=0$ in. In fact, we have
which implies that
The task is to prove $g=0$ in$\dot{L}_{q}(\Omega)$$\mathcal{R}g\in W_{q,0}^{1}(\Omega)\subset W_{2,0}^{1}(\Omega)\subset W_{q,0}^{1}(\Omega)$$\varphi=\mathcal{R}g$$\Vert\gamma_{1}\nabla \mathcal{R}g\Vert_{L_{2}(\dot{\Omega})}=0$$\dot{\Omega}$$\gamma_{1}\nabla \mathcal{R}g=0$$\Omega\pm\cdot$$\mathcal{R}g$$[[\mathcal{R}g]]=0$$\mathcal{R}g\in W_{q,0}^{1}(\Omega)$$c\pm\cdot$$\mathcal{R}g|_{\Gamma-}=0$$\dot{L}_{q}(\Omega)$$\mathcal{R}g=0$

for any$0=\lambda(\mathcal{R}g, \varphi)_{\dot{\Omega}}+(\gamma_{1}\nabla \mathcal{R}g,\nabla\varphi)_{\dot{\Omega}}=(g, \nabla\varphi)_{\dot{\Omega}}$$\varphi\in W_{q,0}^{1}(\Omega)$,. In viewwhich furnishes that divg $=0$ . Thus, the inverse operator $(I+S)^{-1}$ of $I+S$ exists in, $u=\mathcal{R}(I+S)^{-1}f\in W_{q,0}^{1}(\Omega)$ is a solution of (1.9) satisfying theof (2.39), we see that for any
estimate:$\mathcal{L}(\dot{L}_{q}(\Omega))$$f\in\dot{L}_{q}(\Omega)$$\Vert u\Vert_{W_{q}^{1}(\Omega)}\leq C\Vert f\Vert_{\dot{L}_{q}(\Omega)}.$Next, we consider the case where $1<q<2$ . In this case, $2<q’<\infty$ , so that for any
exists awhich solves the variational equation:$h\in L_{q’}(\Omega)^{N}$

there$\varphi\in W_{q,0}^{1}(\Omega)$$(\gamma_{1}\nabla\varphi, \nabla\psi)_{\dot{\Omega}}=(h, \nabla\psi)_{\dot{\Omega}}$

Setting$\psi=\mathcal{R}g$

for any$\psi\in W_{q,0}^{1}(\Omega)$.

, by (2.40) we have
$0=(\gamma_{1}\nabla \mathcal{R}g, \nabla\varphi)_{\dot{\Omega}}=(h, \nabla \mathcal{R}g)_{\dot{\Omega}}.$in St, so thatwith some constants
The arbitrary choice of implies thatBut,
, so that. Therefore, employing the same argumentation as above, we see, which furnishes the existence of a solutionof problem
that $(I+S)^{-1}$ exists in(1.9) for any
satisfying the estimate:
The uniqueness of solutions follows from the existence of solutions of the dual problem. In fact, letsatisfy the homogeneous equation:$f$$\mathcal{R}g=c\pm in\Omega\pm$$\nabla \mathcal{R}g=0$$c\pm\cdot$$\mathcal{R}g=0$$\mathcal{R}g\in W_{q,0}^{1}(\Omega)$$u\in W_{q,0}^{1}(\Omega)$$\mathcal{L}(\dot{L}_{q}(\Omega))$$f\in L_{q}(\Omega)^{N}$$\Vert u\Vert_{W_{q}^{1}(\Omega)}\leq C\Vert f\Vert_{L_{q}(\Omega)}.$$u\in W_{q,0}^{1}(\Omega)$$(\gamma_{1}\nabla u, \nabla\varphi)_{\Omega}=0$

For any$f\in L_{q’}(\Omega)$, let$\varphi\in W_{q,0}^{1}(\Omega)$

for any$\varphi\in W_{q,0}^{1}(\Omega)$.

be a solution of the variational problem:$(\gamma_{1}\nabla\varphi, \nabla\psi)_{\dot{\Omega}}=(f, \nabla\psi)_{\dot{\Omega}}$

for any$\psi\in W_{q,0}^{1}(\Omega)$.

Setting $\psi=u$ , we have
$0=(\gamma_{1}\nabla\varphi, \nabla u)_{\dot{\Omega}}=(f, \nabla u)_{\dot{\Omega}}.$, so
The arbitrary choice of implies that $\nabla u=0$ in , so that is a constant in . But,
that $u=0$ . This proves the uniqueness of solutions, so that we have completed the proof of Theorem 1.5.
$f$$\Omega$$u$$\Omega$$u\in W_{q,0}^{1}(\Omega)$

Stokes equations and reduced Stokes equations

3

Let be a complex number. In this section, we consider the following generalized resolvent problem of
the Stokes equations:$\lambda$$\{\begin{array}{ll}divv =f in \dot{\Omega},\lambda v-\gamma_{1}(DivS(v)-\nabla \mathfrak{p})=g in \dot{\Omega},{[}[(S(v)-\mathfrak{p}I)n]]=[[h]], [[v]]=0 on \Gamma,(S_{-}(v_{-})-\mathfrak{p}_{-}I)n_{-}|_{\Gamma-}=h_{-}|r_{-} on \Gamma_{-}.\end{array}$

(3.1)

132
Let $K(u)$ be a solution to the variational problem:
for any$(\gamma_{1}\nabla K(u), \nabla\varphi)_{\dot{\Omega}}=(\gamma_{1}DivS(u)-\nabladivu, \nabla\varphi)_{\dot{\Omega}}$$\varphi\in W_{q,0}^{1}(\Omega)$

(3.2)

,

subject to$[[K(u)]]=[[<S(u)n, n>+divu K(u)|r_{-}=(<S_{-}(u_{-})n_{-}, n_{-}>+divu_{-})|\Gamma-\cdot$

In view of Theorem 1.5, for any
possessing the estimate:$u\in W_{q}^{2}(\dot{\Omega})^{N}\cap W_{q}^{1}(\Omega)^{N}$

problem (3.2) admits a unique solution $K(u)\in$$W_{q}^{1}(\dot{\Omega})$$\Vert K(u)\Vert_{W_{q}^{1}(\dot{\Omega})}\leq C\Vert u\Vert_{W_{q}^{2}(\dot{\Omega})}$

.

(3.3)

We also consider the following equations:(3.4)$\{\begin{array}{ll}\lambda u-\gamma_{1}(DivS(u)-\nabla K(u))=g in\dot{\Omega},{[}[(S(u)-K(u)I)n]]=[[h]], [[u]]=0 on \Gamma,(S_{-}(u_{-})-K(u)I)n_{-}|_{\Gamma_{-}}=h_{-}|_{\Gamma-} on \Gamma_{-}.\end{array}$

Problem (3.1) is called the Stokes equations, while problem (3.4) is called the reduced Stokes equations.
We discuss the equivalence of both problems (3.1) and (3.4).
be the set of all $N$-vector of functionsFirst, we assume that problem (3.1) is solvable. Let
satisfying the condition:(3.5).
for any$g$$J_{q}(\Omega)$$\varphi\in W_{q,0}^{1}(\Omega)$$(g, \nabla\varphi)_{\dot{\Omega}}=0$. Let and be solutions of problem (3.1) with $f=0$ . Then, noting that divv $=0$ in St
Let
and $[[v]]=0$ , by (3.1), (3.2) and (3.5) we have$g\in J_{q}(\Omega)$$v$$\mathfrak{p}$$0=(\lambda v-\gamma_{1}(DivS(u)-\nabla \mathfrak{p}), \nabla\varphi)_{\dot{\Omega}}$$=-\lambda(divv, \varphi)_{\dot{\Omega}}-(\gamma_{1}DivS(u)-\nabla divv, \nabla\varphi)_{\dot{\Omega}}+(\gamma_{1}\nabla \mathfrak{p}, \nabla\varphi)_{\dot{\Omega}}$$=(\gamma_{1}\nabla(\mathfrak{p}-K(u)), \nabla\varphi)_{\dot{\Omega}}$

for any$\varphi\in W_{q,0}^{1}(\Omega)$.

Moreover,$[[\mathfrak{p}]]=[[<S(v)n, n>]]=[[K(v)]], \mathfrak{p}|_{\Gamma_{-}}=<\mu-S_{-}(u_{-})n_{-}, n_{-}>|r_{-}=K(v)|_{\Gamma-},$, which shows that is a solution
because $divv=0$ in . Thus, the uniqueness implies thatproblem(3.4).
ofConversely, we assume that problem (3.4) is uniquely solvable. Moreover, we assume that it does
hold the uniqueness of the variational problem:$\mathfrak{p}=K(v)$$\dot{\Omega}$$(\lambda w, \varphi)_{\dot{\Omega}}+(\nabla w, \nabla\varphi)_{\dot{\Omega}}=(f, \nabla\varphi)_{\dot{\Omega}}$

for any$\varphi\in W_{q,0}^{1}(\Omega)$,
subject to $[[w]]=g$ and $w|_{\Gamma-}=g|r_{-}$ . Let
be a solution to the variational problem:
be given. Let$f\in W_{q}^{1}$ $(\dot{\Omega})$$v$$g\in L_{q}(\Omega)^{N},$

(3.6)

,

$h\in W_{q}^{1}(\dot{\Omega})^{N}$and$h_{-}\in L_{q}(\Omega_{-})^{N}$$\theta\in W_{q}^{1}(\dot{\Omega})$$(\gamma_{1}\nabla\theta, \nabla\varphi)_{\dot{\Omega}}=-(g, \nabla\varphi)_{\dot{\Omega}}$

for any$\varphi\in W_{q,0}^{1}(\Omega)$

,

(3.7)

subject to $[[\theta]]=<[[h]],$ $n>and\theta|r_{-}=<h_{-}|r_{-},$ $n_{-}>$ . Using this , we rewrite (3.1) of the form:$\theta$$\{\begin{array}{ll}divv=f\pm in \dot{\Omega},\lambda v-\gamma_{1}(DivS(v)-\nabla(\mathfrak{p}+\theta))=g+\gamma_{1}\nabla\theta in \dot{\Omega},{[}[(S(v)-(\mathfrak{p}+\theta)I)n]]=[[h]]-[[\theta]]n, [[v]]=0 on \Gamma,(S_{-}(v_{-})-(\mathfrak{p}+\theta)I)n_{-}|_{\Gamma-}=h_{-}|_{\Gamma-}-\theta|r_{-}n_{-} on \Gamma_{-}.\end{array}$$<[[h]],$ $n>-[[\theta]]=0$ on, and
By (3.7),
in (3.4) we may assume that$g+\gamma_{1}\nabla\theta\in J_{q}(\Omega),$$\Gamma$$<h_{-}|_{\Gamma-},$ $n_{-}>-\theta|_{\Gamma_{-}}=0$$g\in J_{q}(\Omega) , <[[h]], n>=0, <h_{-}|r_{-}, n_{-}>=0$.

on$\Gamma_{-}$. Thus,
(3.8)

133
Let

$\psi\in W_{q}^{1}(\dot{\Omega})$

be a solution to the variational equation:$(\gamma_{1}\nabla\psi, \nabla\varphi)_{\dot{\Omega}}=-\{\lambda(f, \varphi)_{\dot{\Omega}}+(\nabla f, \nabla\varphi)_{\dot{\Omega}}\}$

for any$\varphi\in W_{q,0}^{1}(\Omega)$

(3.9)

,

. In fact, since
subject to $[[\psi]]=[[f]]$ andfor any
Poincar\’es’
frominequality, by the Harn-Banach theorem there exists a
as follows
such thatfor any$\psi|r_{-}=f|_{\Gamma_{-}}$$|\lambda(f,\varphi)_{\dot{\Omega}}|\leq|\lambda|\Vert f\Vert_{L_{q}(\Omega)}\Vert\nabla\varphi\Vert_{L_{q},(\Omega)}$$\varphi\in W_{q,0}^{1}(\Omega)$$\lambda(f, \varphi)_{\dot{\Omega}}=(G, \nabla\varphi)_{\dot{\Omega}}$with$\varphi\in$$G\in L_{q}(\Omega)^{N}$$W_{q,0}^{1}(\Omega)$$\Vert G\Vert_{L_{q}(\Omega)}\leq C|\lambda|\Vert f\Vert_{L_{q}(\Omega)}$. Let$u\in W_{q}^{2}(\dot{\Omega})\cap W_{q}^{1}(\Omega)$be a solution to the reduced Stokes equations:(3.10)$\{\begin{array}{ll}\lambda u-\gamma_{1}(DivS(u)-\nabla K(u))=g+\gamma_{1}\nabla\psi in \dot{\Omega},{[}[(S(u)-K(u)I)n]]=[[h]]-[[f]]n, [[u]]=0 on \Gamma,(S_{-}(u_{-})-K(u)I)n_{-}|r_{-}=h_{-}|r_{-}-f|n on \Gamma_{-}.\end{array}$

Note that $[[u]]=0$ in (3.10). For any$\varphi\in W_{q,0}^{1}(\Omega)$, by (3.8) and (3.10) we have$(\gamma_{1}\nabla\psi, \nabla\varphi)_{\dot{\Omega}}=(\lambda u-\gamma_{1}(DivS(u)-\nabla K(u)), \nabla\varphi)_{\dot{\Omega}}$$=-\{\lambda(divu, \varphi)_{\dot{\Omega}}+(\nabla divu, \nabla\varphi)_{\dot{\Omega}}\},$

which, combined with (3.9), furnishes that$\lambda(f-divu, \varphi)_{\dot{\Omega}}+(\nabla(f-divu), \nabla\varphi)_{\dot{\Omega}}=0$

for any$\varphi\in W_{q,0}^{1}(\Omega)$.

Moreover, by (3.2), (3.8) and (3.10),[[div u]] $=-[[<S(u)n, n>-K(u)]]=[[f]],$$divu|r_{-}=-(<S(u)n_{-}, n_{-}>-K(u))|r_{-}=f|r_{-},$and therefore by the uniqueness we have $divu=f$ in St. Since $[[\psi]]=[[f]]$ andsolve problem (3.1).

$\psi|_{\Gamma_{1}}=f|_{\Gamma_{1}},$

$u$

and

$\mathfrak{p}=K(u)-\psi$4

A proof of Theorem 1.4

Concerning the reduced Stokes equations (3.4), we have
Theorem 4.1. Let $1<q<\infty,$ $N<r<\infty$ and $0<\epsilon<\pi/2$ . Let$\Sigma_{\epsilon,\lambda_{0}}=\{\lambda\in \mathbb{C}||\arg\lambda|\leq\pi-\epsilon, |\lambda|\geq\lambda_{0}\}$O. Assume that $r \geq\max(q, q’)$ with $q’=q/(q-1)$ and thatare both compact
andhyper-surfaces ofclass. Then, there exists a constantsuch that for any
and$g\in L_{q}(\Omega)^{N},$, and, problem (3.4) admits a unique solutionfor$\lambda_{0}>$$\Gamma$$W_{r}^{2-1/r}$$h\in W_{q}^{1}(\dot{\Omega})^{N}$$\lambda_{0}>0$$\Gamma_{-}$$\lambda\in\Sigma_{\epsilon,\lambda_{0}}$$u\in W_{q}^{2}(\dot{\Omega})^{N}$$h_{-}\in W_{q}^{1}(\Omega_{-})$

possessing the estimate:$\Vert(\lambda u, \lambda^{1/2}\nabla u_{\rangle}\nabla^{2}u)\Vert_{L_{q}(\dot{\Omega})}\leq C\{\Vert g\Vert_{L_{q}(\dot{\Omega})}+\Vert(\lambda^{1/2}h, \nabla h)\Vert_{L_{q}(\dot{\Omega})}+\Vert(\lambda^{1/2}h,\nabla h_{-})\Vert_{L_{q}(\Omega-}\}$

.

(4.1)

Theorem 4.1 was proved by Saito and Sri [5]. In fact, the interface problem in the whole space withinterface $x_{N}=0$ and the one phase problem in the half-space were treated in Shibata and Shimizu [3, 4].
And then, according the the method due to Shibata [2], we can construct a parametrix by using thepartition of unity, so that arguing similarly to Shibata [2] we can prove Theorem 4.1.
Moreover, if $g,$ and h– in Theorem 4.1 satisfy the orthogonal condition (1.6), then satisfies theorthogonal condition (1.7), that is
$h$

$u$

$(\gamma_{1}^{-1}u, p_{\ell})_{\dot{\Omega}}=0$

In fact, multiplyingthe first equation in (3.4) byfor all $P=1$ , . . . , M.
$\gamma_{1}^{-1}u$(4.2)

and using divergence theorem of GauB, we have$(\gamma_{1}^{-l}g, p_{l})_{\dot{\Omega}}=\lambda(\gamma_{1}^{-1}, u, p_{\ell})_{\dot{\Omega}}-(Div(S(u)-K(u)I), p\ell)_{\dot{\Omega}}$134$= \lambda(\gamma_{1}^{-1}, u, p_{\ell})_{\dot{\Omega}}-([[h]], p_{\ell})_{\Gamma}-(h_{-}, p_{\ell})_{\Gamma}+\frac{1}{2}(D(u),D(p_{\ell}))_{\dot{\Omega}}-(K(u),divp_{l})_{\dot{\Omega}}.$O. Since
Since $D(p_{\ell})=0$ and $divp_{\ell}=0$ , the condition (1.6) yields thatsatisfies the orthogonal condition (4.2).
iFYom now on, we prove Theorem 1.4. For this purpose, first we consider the reduced Soktes equations:$\lambda\neq 0,$$\lambda(\gamma_{1}^{-1}, u, p_{\ell})_{\dot{\Omega}}=$(4.3)$\{\begin{array}{ll}-\gamma_{1}(DivS(v)-\nabla K(v))=g in \dot{\Omega},{[}[(S(v)-K(v)I)n]]=[[h]], [[v]]=0 on \Gamma,(S_{-}(v_{-})-K(v)I)n_{-}|r_{-}=h_{-}|r_{-} on \Gamma_{-}.\end{array}$

By Theorem 4.1, there exists a large
possessing the estimate:solution$\lambda_{1}>0$$u$such that problem (3.4) with$\lambda=\lambda_{1}$admits a unique$u\in W_{q}^{2}(\dot{\Omega})$$\Vert u\Vert_{W_{q}^{2}(\dot{\Omega})}\leq C\{\Vert g\Vert_{L_{q}(\dot{\Omega})}+\Vert h\Vert_{W_{q}^{1}(\dot{\Omega})}+\Vert h_{-}\Vert_{W_{q}^{1}(\Omega-)}\}$.

is a fixed positive constant. We look for a solution
Here and in the following(4.3) of the form $v=u+w$ , and then $w$ satisfies the equations:$\lambda_{1}>0$

(4.4)
$v$

of the equations(4.5)$\{\begin{array}{ll}-\gamma_{1}(DivS(w)-\nabla K(w))=f in \dot{\Omega},{[}[(S(w)-K(w)I)n]]=0, [[w]]=0 on \Gamma,(S_{-}(w_{-})-K(w)I)n_{-}|_{\Gamma_{-}}=0 on \Gamma_{-}.\end{array}$witn $f=-\lambda_{1}u$ . Moreover, by (4.2),$f$satisfies the orthogonal condition:$(\gamma_{1}^{-1}f, p_{\ell})_{\dot{\Omega}}=0$

for all $\ell=1$ , . . . , M.

In view of Theorem 4.1 again, there exists a unique$z\in W_{q}^{2}(\dot{\Omega})$(4.6)

which solves the equations:(4.7)$\{\begin{array}{ll}\lambda_{1}z-\gamma_{1}(DivS(z)-\nabla K(z))=f in\dot{\Omega},{[}[(S(z)-K(z)I)n]]=0, [[z]]=0 on \Gamma,(S_{-}(z_{-})-K(z)I)n_{-}|_{\Gamma-}=0 on \Gamma_{-}.\end{array}$Moreover, satisfies the orthogonal condition (4.6), so that
. Then, we have
Let be an operator defined by
$f$

$z$

also satisfies the orthgonal condition (4.6).

$\mathcal{R}f=z$$\mathcal{R}$$\Vert \mathcal{R}f\Vert_{W_{q}^{2}(\dot{\Omega})}\leq C\Vert f\Vert_{L_{q}(\dot{\Omega})},$$(\gamma_{1}^{-1}\mathcal{R}f, p_{l})_{\dot{\Omega}}=0$

We look for a solution of (4.5) of the form
have

$z=\mathcal{R}g$

for all $\ell=1$ , . . . , M.

(4.8)

. Inserting this formula into (4.5) and using (4.7), we(4.9)$\{\begin{array}{ll}-\gamma_{1}(DivS(\mathcal{R}g)-\nabla K(\mathcal{R}g))=g-\lambda_{1}\mathcal{R}g in \dot{\Omega},{[}[(S(\mathcal{R}g)-K(\mathcal{R}g)I)n]]=0, [[\mathcal{R}g]]=0 on \Gamma,(S_{-}((\mathcal{R}g)_{-})-K(\mathcal{R}g)I)n_{-}|_{\Gamma-}=0 on r_{-}.\end{array}$. Moreover, is bounded, so that the Rellichand since $[[\mathcal{R}g]]=0,$
Sinceinto itself. Thus, to prove thatis a compact operator fromcompactness theorem yields thatis trivial, where denotes theis invertible, it is suﬃcient to prove that the kernel of. The task issuch that. Let be an element in
identity operator of, and then by (4.9),
satisfies the equations:to prove that $g=0$ . Let$\mathcal{R}g\in W_{q}^{1}(\Omega)$$\mathcal{R}g\in W_{q}^{2}(\dot{\Omega})^{N}$$\Omega$$L_{q}(\dot{\Omega})^{N}$$\mathcal{R}$$I$$I-\lambda_{1}\mathcal{R}$$(I-\lambda_{1}\mathcal{R})$$\mathcal{L}(L_{q}(\dot{\Omega})^{N})$$u=\mathcal{R}g$$g$$L_{q}(\dot{\Omega})^{N}$$(I-\lambda_{1}\mathcal{R})g=0$$u\in W_{q}^{2}(\dot{\Omega})$$\{\begin{array}{ll}-\gamma_{1}(DivS(u)-\nabla K(u))=0 in \dot{\Omega},{[}[(S(u)-K(u)I)n]]=0, [[u]]=0 on \Gamma,(S_{-}((u_{-})-K(u)I)n_{-}|_{\Gamma_{-}}=0 on \Gamma_{-}.\end{array}$(4.10)

135
As was observed in Sect. 3,$u$satisifies the divergence free condition:
divu $=0$

In fact, for any$\varphi\in W_{q,0}^{1}(\Omega)$by (4.10) with$\gamma_{1}=1$$in\dot{\Omega}$.

(4.11)and (3.2),$0=(DivS(u)-\nabla K(u)) , \nabla\varphi)_{\dot{\Omega}}=(\nabla divu, \nabla\varphi)_{\dot{\Omega}},$subject to $[[divu]]=[[<S(u)n, n>-K(u)]]=0$ and $divu|r_{-}=(<S_{-}(u_{-})n_{-}, n_{-}>-K(u))|r_{-}=0.$
Thus, the uniqueness implies (4.11).
, so thatFirst, we consider the case where $2\leq q<\infty$ . In this case,multiplying the first equation in (4.10) by and using the divergence theorem of GauB, the interface
condition and the boundary condition in (4.10) and (4.11), we have$u\in W_{q}^{2}(\dot{\Omega})^{N}\subset W_{2}^{2}(\dot{\Omega})^{N}$$u$$\frac{\mu+}{2}\Vert D(u)\Vert_{L_{2}(\Omega_{+})}^{2}+\frac{\mu_{-}}{2}\Vert D(u)\Vert_{L_{2}(\Omega_{-})}^{2}=0$

But,$[u, v]$$u$.

(4.12)

with respec to the inner-productsatisfies (2.40) andis the orthonormal basis of(4.9),, so that $u=\mathcal{R}g=0$ . Since
we have $g=0$ , which furnishes thatsatisfies. Thus,solves the equations (4.5). Moreover, by (4.8) $w$$\{p_{l}\}_{\ell=,1}^{M}$$t’$$\mathcal{R}_{d}$$:=(\gamma_{1}^{-1}u, v)_{\dot{\Omega}}$$\mathcal{R}g$$(I-\lambda_{1}\mathcal{R})^{-1}\in \mathcal{L}(L_{q}(\dot{\Omega})^{N})$$w=\mathcal{R}(I-\lambda_{1}\mathcal{R})^{-1}f$satisifes$\Vert w\Vert_{W_{q}^{2}(\dot{\Omega})}\leq C\Vert f\Vert_{L_{q}(\dot{\Omega})},$

for all $\ell=1$ , . . . , M.

$(\gamma_{1}^{-1}w, p_{\ell})_{\dot{\Omega}}=0$(4.13)

Next, we consider the case where $1<q<2$ . We consider the problem(4.14)$\{\begin{array}{ll}-(DivS(z)-\nabla K’(z))=g in \dot{\Omega},{[}[(S(z)-K’(z)I)n]]=0, [[z]]=0 on \Gamma,(S_{-}(z_{-})-K’(z)I)n_{-}|_{\Gamma-}=0 on r_{-},\end{array}$where $K’(z)$ is a solution to the variational equation:$(\nabla K’(z), \nabla\varphi)_{\dot{\Omega}}=(DivS(z)-\nabla$divz,$\nabla\varphi)_{\Omega}$

for any$\varphi\in W_{q,0}^{1}(\Omega)$,

subject to$[[K’(z)]]=[[<S(z)n, n>+divz K’(z)|r_{-}=(<S_{-}(z_{-})n_{-}, n_{-}>+divz_{-})|_{\Gamma-}.$Employing the same argument as above, we may assume that problem (4.14) is uniquely solvable for$2\leq q’<\infty$ . Thus, for any, let
be a solution of (4.14). As was seen in Sect. 3,. Noting satisfies the first equation of (4.10) withand using (4.14) and the divergence
theorem of GauB, we have$g\in J_{q’}(\Omega)$$z\in J_{q’}(\Omega)$$z\in W_{q}^{2},(\dot{\Omega})^{N}$$\gamma_{1}=1$$u$$( u,g)_{\dot{\Omega}}=-(u, DivS(z)-\nabla K’(z))_{\dot{\Omega}}=\frac{\mu+}{2}(D(u), D(z))_{\Omega_{+}}+\frac{\mu_{-}}{2}(D(u),D(z))_{\Omega_{-}},$$0=( DivS(u)-\nabla K(u), z)_{\Omega}=\frac{\mu+}{2}(D(u), D(z))_{\Omega_{+}}+\frac{\mu_{-}}{2}(D(u), D(z))_{\Omega-},$and therefore,
problem:$(u, g)_{\dot{\Omega}}=$O. For any$f\in L_{q’}(\dot{\Omega})^{N}$$(\nabla\psi, \nabla\varphi)_{\dot{\Omega}}=(f, \varphi)_{\dot{\Omega}}$Since$f-\nabla\psi\in J_{q}(\Omega)$, let$\psi\in W_{q,0}^{1}(\Omega)$

for any

be a solution to the variational$\varphi\in W_{q,0}^{1}(\Omega)$.

, we have$(u, f)_{\dot{\Omega}}=(u, f-\nabla\psi)_{\dot{\Omega}}+(u, \nabla\psi)_{\dot{\Omega}}=(u, \nabla\psi)_{\dot{\Omega}}.$, which, combined with theSince
as follows from (4.11) and $[[u]]=0$ , we havearbitrary choice of , furnishes that $u=0$ . Thus, employing the same argument as above,, so thatexists insolves the equations (4.5) and satisfies (4.13). The
uniqueness of solutions follows from the existence of solutions of the dual problem, so that we have proved$(u, f)_{\dot{\Omega}}=0$$(u, \nabla\psi)_{\dot{\Omega}}=0$$f$$\mathcal{L}(L_{q}(\dot{\Omega})^{N})$$(I-\lambda_{1}\mathcal{R})^{-1}$$w=\mathcal{R}(I-\lambda_{1}\mathcal{R})^{-1}f$

136
Theorem 4.2. Let $1<q<\infty,$ $N<r<\infty$ and $0<\epsilon<\pi/2$ . Assume that $r \geq\max(q,q’)$ withclass. Then, for any$q’=q/(q-1)$ and that and -are both compact hyper-surfaces of$g\in L_{q}(\Omega)$ ,satisfying the orthogonal condition (1.6), problem (4.3)
andsatisfying the orthogonal condition (1.4) and the estimate:admits a unique solution$W_{r}^{2-1/r}$$\Gamma$$\Gamma$$h_{-}\in W_{q}^{1}(\Omega_{-})$$h\in W_{q}^{1}(\dot{\Omega})$$v\in W_{q}^{2}(\dot{\Omega})$$\Vert v\Vert_{W_{q}^{2}(\dot{\Omega})}\leq C\{\Vert g\Vert_{L_{q}(\dot{\Omega})}+\Vert(\lambda^{1/2}h, \nabla h)\Vert_{L_{q}(\dot{\Omega})}+\Vert(\lambda^{1/2}h_{-}, \nabla h_{-})\Vert_{L_{q}(\Omega_{-}}\}$.

(4.15),
As was discussed in Sect. 3, Theorem 1.4 can be proved by using Theorem 4.2. In fact, let(1.6).
Let
orthogonaltheconditionsatisfy
be
given
and
and
be a solution to the variational equations:$f\in W_{q}^{1}(\dot{\Omega})$$g\in L_{q}(\dot{\Omega})^{N},$$h_{-}\in W_{q}^{1}(\Omega_{-})^{N}$$h\in W_{q}^{1}(\dot{\Omega})$$\theta\in W_{q}^{1}(\dot{\Omega})$

for any$(\gamma_{1}\nabla\theta, \nabla\varphi)_{\dot{\Omega}}=-(g, \nabla\varphi)_{\dot{\Omega}}$

subject to $[[\theta]]=<[[h]],$ $n>on\Gamma$ and $\theta|r_{-}=<h_{-},$
satisfies the estimate:$n_{-}>|_{\Gamma-}$$\varphi\in W_{q,0}^{1}(\Omega)$on$\Gamma_{-}$,. By Theorem 1.5, such exists and$\theta$$\Vert\theta\Vert_{W_{q}^{1}(\dot{\Omega})}\leq C(\Vert g\Vert_{L_{q}(\dot{\Omega})}+\Vert h\Vert_{W_{q}^{1}(\dot{\Omega})}+\Vert h_{-}\Vert_{W_{q}^{1}(\Omega_{-})})$

Using this , we rewrite problem (1.5) with$\theta$$\gamma_{0}=1$.

of the form:$\{\begin{array}{ll}divv =f in\dot{\Omega},-\gamma_{1}(DivS(v)-\nabla(\mathfrak{p}+\theta))=g+\gamma_{1}\nabla\theta in \dot{\Omega},{[}[(S(v)-(\mathfrak{p}+\theta)I)n]]=[[h]]-<[[h]], n>n, [[v]]=0 on \Gamma,(S_{-}(v_{-})-K(v)I)n_{-}|r_{-}=(h_{-}-<h_{-}, n_{-}>n_{-})|r_{-} on \Gamma_{-}.\end{array}$

Note that$<[[h]]-<[[h]],$ $n>n,$ $n>=0$ and $<h_{-}-<h_{-},$ $n_{-}>n_{-},$ $n_{-}>=0.$
be a solution to the variational equation:$g+\gamma_{1}\nabla\theta\in J_{q}(\Omega),$

Next, let$\psi\in W_{q}^{1}(\dot{\Omega})$$(\gamma_{1}\nabla\psi, \nabla\varphi)_{\dot{\Omega}}=-(\nabla f, \nabla\varphi)_{\dot{\Omega}}$

subject to $[[\psi]]=[[f]]$ on
Stokes equations:$\Gamma$and$\psi|_{\Gamma-}=f|r_{-}$

on$\Gamma_{-}$

for any. Let$\varphi\in W_{q,0}^{1}(\Omega)$$v\in W_{q}^{2}(\dot{\Omega})^{N}$,(4.17)

be a solution to the reduced$\{\begin{array}{ll}-\gamma_{1}(DivS(v)-\nabla K(v))=g+\gamma_{1}\nabla\theta+\gamma_{1}\nabla\psi in \dot{\Omega},{[}[(S(v)-K(v)I)n]]=[[h]]-<[[h]], n>n-[[f]]n, [[v]]=0 on \Gamma,(S_{-}(v_{-})-K(v)I)n_{-}|r_{-}=(h_{-}-<h_{-}, n_{-}>n_{-}-fn_{-})|r_{-} on \Gamma_{-}.\end{array}$Since $g,$$h$(4.16)(4.18)and h-satisfy the orthogonal condition (1.6), we have$(\gamma_{1}^{-1}(g+\gamma_{1}\nabla\theta+\gamma_{1}\nabla\psi), p_{\ell})_{\dot{\Omega}}+([[h]]-<[[h]], n>n-[[f]]n, p_{\ell})_{\Gamma}$$+(h_{--}<h_{-}, n_{-}>n_{-}-fn_{-}, p_{\ell})_{\Gamma-}$$=([[\theta]]n, p_{\ell})_{\Gamma}+(\theta n_{-}, p_{\ell})_{\Gamma_{-}}+([[\psi]]n, p_{\ell})_{\Gamma}+(\psi n_{-}, p_{\ell})r_{-}+(\theta+\psi, divp_{\ell})_{\dot{\Omega}}$$-(<[[h]], n>, p_{\ell})_{\Gamma}-(<h_{-}, n_{-}>n_{-}, p_{l})_{\Gamma_{-}}-([[f]]n, p_{\ell})_{\Gamma}-(fn_{-}, p_{\ell})r_{-}$$=0.$Thus, by Theorem 4.2, problem (4.18) admits a unique solution
condition (1.7) and the estimate:$v\in W_{q}^{2}(\dot{\Omega})^{N}$

satisfying the orthogonal$\Vert v\Vert_{W_{q}^{2}(\dot{\Omega})}\leq C\{\Vert g\Vert_{L_{q}(\dot{\Omega})}+\Vert h\Vert_{W_{q}^{1}(\dot{\Omega})}+\Vert h_{-}\Vert_{W_{q}^{1}(\Omega_{-})}+\Vert f\Vert_{W_{q}^{1}(\dot{\Omega})}\},$

where we have used the estimate:$\Vert\nabla\theta\Vert_{W_{q}^{1}(\dot{\Omega})}+\Vert\nabla\psi\Vert_{W_{q}^{1}(\dot{\Omega})}\leq C\{\Vert g\Vert_{L_{q}(\dot{\Omega})}+\Vert h\Vert_{W_{q}^{1}(\dot{\Omega})}+\Vert h_{-}\Vert_{W_{q}^{1}(\Omega_{-})}+\Vert f\Vert_{W_{q}^{1}(\dot{\Omega})}\}.$

137
Moreover, for any$\varphi\in W_{q,0}^{1}(\Omega)$, by (4.18) we have$(\gamma_{1}\nabla\psi, \nabla\varphi)_{\dot{\Omega}}=(-\gamma_{1}Div(S(v)-K(v)I)) , \nabla\varphi)_{\dot{\Omega}}$$=-(\gamma_{1}DivS(v)-\nabla divv, \nabla\varphi)_{\dot{\Omega}}+(\gamma_{1}\nabla K(v), \nabla\varphi)_{\dot{\Omega}}-(\nabla divv, \nabla\varphi)_{\dot{\Omega}},$which, combined with (3.2) and (4.17), furnishes that$(\nabla$( –divv),$f$$\nabla\varphi)_{\dot{\Omega}}=0$

for any$\varphi\in W_{q,0}^{1}(\Omega)$.

Furthermore, by (3.2), (4.18) and (4.17)[[divv]] $=-[[<S(v)n, n>-K(v)]]=[[f]]$$divv|r_{-}=-(<S_{-}(v_{-})n_{-}, n_{-}>-K(v))|r_{-}=f|r_{-}$

onon$\Gamma,$$\Gamma_{-},$

where the facts that $<[[h]]-<[[h]]n,$ $n>n,$ $n>=0$ on and $<h_{-}-<$ h-n-, $n_{-}>n_{-},$ $n_{-}>=0$
on F-have been used. Thus, the uniqueness of the variational problem implies that divv $=f$ in
Setting $\mathfrak{p}=-\theta+K(v)-\psi$ , we see that and are required solutions of problem (1.5) withwhich completes the proof of Theorem 1.4.
$\Gamma$$\dot{\Omega}.$$v$$\mathfrak{p}$$\gamma_{0}=1,$

References[1] Y. Enomoto and Y. Shibata, On the -sectoriality and its application to some mathematical study of theviscous compressible fluids, Funkcial Ekvaj. 56 (3) (2013), 441-505.
$\mathcal{R}$[2] Y. Shibata, Generalized Resolvent Estimates of the Stokes Equations with first order boundary condition in a
general domain, J. Math. Fluid Mech. 15 (2013), 1-40.
[3] Y. Shibata and S. Shimizu, On the
maximal regularety of the Stokes problem vnth first order boundary
condition, Model Problem, J. Math. Soc. Japan 64 (2), 561-626 (2012)$L_{p}-L_{q}$regulanty for the two-phase Stokes equations; Model problems, J.
[4] Y. Shibata and S. Shimizu, Maximal
Diﬀerential Equations bf 251 (2011), 373-419.
$L_{p}-L_{q}$[5] Sri Maryani and H. Saito On theequations, Preprint in Nov. 2015.

$\mathcal{R}$-boundedness of solution operator families for two-phase Stokes resolvent