数理解析研究所講究録
1181 巻 2001 年 182-193

182

Regularity of Solutions to Some
Variational Inequalities for the Stokes
Equations
Norikazu SAITO(齊藤宣–)
International Institute for Advanced Studies(国際高等研究所)

1

Introduction

The main purpose of the present paper is to give a regularity result of solutions to
the following problem:

PROBLEM (F). Find
(1.1)

$u\in K_{\sigma}^{1}(\Omega)$and$p\in L^{2}(\Omega)$satisfying$a(u, v-u)-(p, \mathrm{d}\mathrm{i}\mathrm{v}(v-u))+j(v)-j(u)\geq(f, v-u)$ ,$(\forall v\in K^{1}(\Omega))$.

Here and hereafter the following notation is employed:
is a bounded domain in$m=2$ or 3. The boundary
is composed of two connected components
and which are assumed to be suitably smooth. For the sake of simplicity, we. We introduce
assume that$\Omega$$\partial\Omega$$\mathbb{R}^{m},$$\Gamma_{0}$$\Gamma$$\overline{\Gamma}_{0}\cap\overline{\Gamma}=\emptyset${ $v\in H^{1}(\Omega)^{m}|v=0$ on$K^{1}(\Omega)=$$\Gamma_{0}$},subspace ofthendenotes the solenoidal.
denotes theinner product inoraccording as scalar-valued functions or vectorfunctions.
setvaluedWe$K_{\sigma}^{1}(\Omega)$$(\mathrm{d}\mathrm{i}\mathrm{v}v=0)$$L^{2}(\Omega)$$K^{1}(\Omega)$$(\cdot, \cdot)$$L^{2}(\Omega)^{m}$$a(u, v)= \frac{1}{2}\int_{\Omega}\sum_{1\leq i,j\leq m}ei,j(u)e_{i},j(v)dx$

for $u=(u_{1}, \cdots, u_{m})$ and $v=(v_{1}, \cdots, v_{m})$ , where$e_{i,j}(v)= \frac{\partial v_{i}}{\partial x_{j}}+\frac{\partial v_{j}}{\partial x_{i}}$denotes an element of the defomation tensor $E(v)=[e_{i,j}(v)]$ . Finally(1.2)$j(v)= \int_{\Gamma}g|v|ds$,( $ds=$ the surface element of ),$\Gamma$183
where is a given scalar function defined on F.
As was described in Fujita and Kawarada [7], the variational inequiality (1.1)
arises in the study of the steady motions of the viscous imcompressible fluid under
the frictinal boundary condition, where denotes the flow velocity, the pressure,the external forces acting on the fluid, and is called the modulus function of
be thefriction. We now review the boundary condition of this type. Let$\sigma(u,p)=S(u,p)n$, where $S(u,p)=-pI+E(v)$stress vector to F. That is, we letstands for the stress tensor and the unit outer normal to . Then we pose on
that
$g$

$u$

$p$

$f$

$g$

$\sigma(u,p)$$\Gamma$$n$$\sigma(u,p)$(1.3)$|\sigma(u,p)|\leq g$

and
(1.4)

$|\sigma(u,p)|<g$$\Rightarrow$$|\sigma(u,p)|=g$$\Rightarrow$$u=0$ ,$\{$

almost everywhere on . The classical form of the firictional boundary value problem
for the Stokes equations dealt with in [7] consists of$\Gamma$

(1.5)

in

$-\triangle u+\nabla p=f$$\Omega$,$\mathrm{d}\mathrm{i}\mathrm{v}u=0$in$\Omega$$u=0$ on,$\Gamma_{0}$together with (1.3) and (1.4). (F) is a weak form of this problem.
The existence theorem was established in [7]. Assume that(H)

$f\in L^{2}(\Omega)^{m}$,$g\in L^{\infty}(\Gamma)$

,

on F.

$g>0$ a. .
$\mathrm{e}$Then (F) admits of a solution $\{u,p\}$ . The velocity part is unique. However theuniqueness of the pressure part depends on cases. That is, in general, is unique
up to an additive constant and the constant is restricted via (1.3).
$u$

$p$

$p$

Theorem 1.1. Assume that $(H)$ and moreover that $g\in H^{1}(\Gamma)$ . Let $\{u,p\}$ be asolution of $(F)$ . Then $u\in H^{2}(\Omega)^{m}$ and $p\in H^{1}(\Omega)$ with$||u||_{2}+||p||_{1}\leq C(||f||+||g||_{1,\mathrm{r}})$,where $C=C(\Omega)$ is a positive constant. Moreover $\{u,p\}$ satisfies (1.5) almost
everywhere in . Furthermore we have $\sigma(u,p)\in H^{1/2}(\Gamma)^{m}$ and$\Omega$$-\sigma(u,p)\in g\partial|u|$. on .
$\Gamma$$\mathrm{a}.\mathrm{e}$instead ofandIn the above and in what follows, we write
respectively. For vector-valued functions, as longandas there is no possibility of confusion, we use the same symbols. Furthermore,denotes the subdifferential of the function $|z|=(z_{1}^{2}+\cdots+Z_{2}^{2})1/2$ . Namely,$||\cdot||,$$||\cdot||_{L^{2}}(\Omega),$$||\cdot||_{H^{S}}(\Omega)$$||\cdot||_{S}$$||\cdot||_{s,\Gamma}$$||\cdot||_{H^{S}}(\Gamma)$$\partial|\cdot|$,$z/|z|$$(z\neq 0)$$\{\zeta\in \mathbb{R}^{m}; |\zeta|\leq 1\}$$(z=0)$ .

$\partial|z|=\{$

184
In order to prove Theorem 1.1, we follow the method of Br\’ezis [2]. Namely, weof equations
approximate a solution $\{u,p\}$ of the inequality (1.1) by solutionsin (1.1). Then thewhich are obtained by replacing by a regular functionalis studied.
regularity of$\{u_{\epsilon},p_{\in}\}$$j$$j_{\epsilon}$$\{u_{\mathcal{E}},p\in\}$

The organization of the present paper is as follows. In \S 2, we describe a specific.
definition of the above mentioned regularized problem, which we will refer asisthere.
approximation
resultalso
discussed
are
The well-posedness and the\S 3. In \S 4, we have the proof of Theorem 1.1.
devoted to a regularity result for
From the view point of physics, some modififcations of (F) are much more interesting.
With this connection, in the final section (\S 5), we state leak or slip boundary value
problems of friction type and give regularity results for these problems without theproofs.
Before concluding Introduction, we shall mention a few remarks.
$(\mathrm{F}_{\epsilon})$$(\mathrm{F}_{\xi})$

Remark 1.1. To be rigorously, should be understood as the functional on$j$$j( \eta)=\int_{\Gamma}g|\eta|ds$,$(\eta\in H^{1/2}(\Gamma)^{m})$However, for the sake of simplicity, we will regard
through$j(v|_{\mathrm{r})=} \int_{\Gamma}g|v|\Gamma|ds$$j$$H^{1/2}(\Gamma)m$.

as the functional on$H^{1}(\Omega)$

,

where
$v|_{\Gamma}=$the trace of

$v$

on ,
$\Gamma$and we write as (1.2).
Remark 1.2. It is well-known that ( , for example, Duvaut and Lions [4]) there
are positive constants and such that$\mathrm{e}\mathrm{g}.$$\delta_{0}$$a(u, v)\leq\delta_{0}||u||_{1}||v||_{1}$$\delta_{1}$$(\forall u, v\in H^{1}(\Omega))$,$a(v, v)\geq\delta_{1}||v||_{1}^{2}$$(\forall v\in K^{1}(\Omega))$.

Remark 1.3. Suppose that $\{u,p\}$ is suitably regular and satisfies (1.5) in the clasthensical sense. Multiplying the both sides of $-\triangle u+\nabla p=f$ byintegrating over , we have$\psi\in K^{1}(\Omega)$$\Omega$,$a(u, \psi)-\int_{\Omega}p\mathrm{d}\mathrm{i}_{\mathrm{V}}\psi_{d}x=\int_{\Gamma}\sigma(u,p)\cdot\psi dS+\int_{\Omega}f\cdot\psi_{d}x$$(\forall\psi\in K^{1}(\Omega))$.

According to this identity, we can say that$\sigma(u,p)=\omega$on$\Gamma$as theis the Neumann or natural boundary condition corresponding toellipticity form. Concerning such boundary conditions, we refer to Ladyzhenskaya[10] or Saito [13].
$a(\cdot, \cdot)$$H^{1}-$

Remark 1.4. In the subsequent sections, denotes various generic constant. If it
which may not be numbers, we shall indicate it
depends on parameters
by $C=C(q_{1}, \cdots, q_{M})$ .
$C$$q_{1},$$\cdots,$ $q_{M}$

;

185

2
Let

Regularized Problem$\epsilon>0$$(\mathrm{F}_{\epsilon})$. We introduce(2.1)$j_{\epsilon j}(v)= \int_{\Gamma}g\rho_{\epsilon}(v)ds$,$(v\in H^{1}(\Omega)^{m})$

,

where
(2.2)

$|v|-\in/2$$(|v|>\mathit{6})$,$|v|^{2}/(2\epsilon)$$(|v|\leq\epsilon)$.

$\rho_{\epsilon}(v)=\{$

Then we consider

PROBLEM$(\mathrm{F}_{\Xi})$. Find$u_{\epsilon}\in K_{\sigma}^{1}(\Omega)$andsatisfying$p_{\mathcal{E}}\in L^{2}(\Omega)$$a(u_{\epsilon}, v-u_{\epsilon})-(p_{\epsilon}, \mathrm{d}\mathrm{i}_{\mathrm{V}}(v-u_{\Xi}))+j_{\xi}(v)-j_{\mathcal{E}}(u\mathcal{E})$(2.3)$\geq(f, v-u_{\epsilon})$

Theorem 2.1. Assume that $(H)$ and let
with$\epsilon>0$. Then$(F_{\xi})$,$(\forall v\in K^{1}(\Omega))$.

admits a unique solution$\{u_{\epsilon},p\mathcal{E}\}$$||u_{\epsilon}||_{1}+||p_{\in}||\leq C(\Omega)(||f||+||g||_{L(}2\Gamma))$Furthermore,$\{u_{\mathit{6}},p\xi\}$

is a weak solution of (1.5) together with$-\sigma(u_{\epsilon},p\xi)=g\alpha_{\epsilon}(u_{\mathcal{E}})$

Namely,(2.4)$\{u_{\epsilon},p\mathit{6}\}$.

$\mathrm{a}.\mathrm{e}$. on , (In particular$\Gamma$$\sigma(u_{\epsilon},p\epsilon)\in L^{2}(\Gamma)^{m}$).

satisfies$a(u, \psi)-(p, \mathrm{d}\mathrm{i}\mathrm{v}\psi)+\int_{\Gamma}g\alpha_{\overline{\mathrm{c}}}(u)\cdot\psi ds=(f, \psi)$$(\forall\psi\in K^{1}(\Omega))$,

where we have put(2.5)$v/|v|$$(|v|>\in)$$v/\epsilon$$(|v|\leq\epsilon)$$\alpha_{\epsilon}(v)=\{$

Remark 2.1. In Theorem 2.1,
defined by$\sigma(u,p)$

is understood as a functional on$\langle\sigma, \eta\rangle=a(u_{\epsilon}, \psi_{\eta})-(p_{\mathit{6}}, \mathrm{d}\mathrm{i}\mathrm{V}\psi\eta)-(f, \psi_{\eta})$where$\psi_{\eta}\in K^{1}(\Omega)$.

,$(\forall\eta\in H^{1/2}(\Gamma)m)$$H^{1/2}(\Gamma)^{m}$,is any extension of .
$\eta$

Proof of Theorem 2.1. From standard theory of convex analysis (e.g., Ekeland andTemam [5], or Glowinski [8] , the minimization problem: Find$)$$u\in K_{\sigma}^{1}(\Omega)$$J_{\epsilon}(u)=v \in K1(\inf_{\Omega)\sigma}J_{\epsilon}(v), J_{\mathit{6}}(v)=\frac{1}{2}a(v, v)-(f, v)+j\epsilon(v)$satisfying

186
has a unique solution$u$which is characteraized by

$r$

(2.6)

$a(u, v-u)+j_{\epsilon}(v)-j\in(u)\geq(f, v-u)$ ,.

$(\forall v\in K_{\sigma}^{1}(\Omega))$

We are going to show that a scalar function can be taken as $\{u,p\}$ sloves (2.3). Le, we have
and $t>0$ . Substituting into (2.6) $v=u+t\phi$ and letting$p$$\phi\in K_{\sigma}^{1}(\Omega)$$tarrow \mathrm{O}$$a(u, \phi)+\int_{\Gamma}g\alpha_{\epsilon}(u)\mathrm{g}\emptyset ds=(f, \phi)$,$(\forall\phi\in K^{1}\sigma(\Omega))$.

By using this, in the same line as Solonnikov and \v{S}\v{c}adilov [16], we can take a uniquesatisfying (2.4).
Thanks to the convexity of ,$p\in L^{2}(\Omega)$$j_{\epsilon}$

(2.7)

,

$\int_{\Gamma}g\alpha_{\mathcal{E}}(v)\cdot(w-v)ds\leq j_{\epsilon}(w)-j\epsilon(v)$$(\forall v, w\in H^{1}(\Omega)^{m})$In view of (2.4) and (2.7), we can easily verify that $\{u,p\}$ solves
hand, (2.4) yields$(\forall\eta\in H^{1/}2(\Gamma)^{m})$$\langle\sigma, \psi_{\eta}\rangle+\int_{\Gamma}g\alpha_{\mathit{6}}(u)\cdot\psi\eta d_{S}=0$

where$\psi_{\eta}\in K^{1}(\Omega)$$L^{2}(\Gamma)^{m}$that$(\mathrm{F}_{\epsilon})$.

. On the other,is any extension of . Consequently, it follows fromand$\eta$$g\alpha_{\epsilon}(u)\in$$\sigma(u,p)\in L^{2}(\Gamma)^{m}$$-\sigma(u,p)=g\alpha_{\in}(u)$$\mathrm{a}.\mathrm{e}$. on$\Gamma$,which completes the proof.

$\square$Remark 2.2. As mentioned above, in order to derive (2.4), we follow the methodof [16], in which the following facts are applied. Through Riesz’s representation
theorem, we define an operator from
by
to
$B$

$L^{2}(\Omega)$$(Bq, v)H^{1}(\Omega)^{m}=(p, \mathrm{d}\mathrm{i}_{\mathrm{V}v})$,$K^{1}(\Omega)$$(\forall q\in L^{2}(\Omega);\forall v\in K^{1}(\Omega))$

The range $R(B)$ of $B$ forms a closed subspace of
decomposition$K^{1}(\Omega)$.

. Moreover, the orthogonal$K^{1}(\Omega)=R(B)\oplus K_{\sigma}^{1}(\Omega)$

holds. For the proof, we refer to Saito et al. [14].

Theorem 2.2. Assume that $(H)$ holds. Let $\{u,p\}$ and, respectively. Then we have:and$\{u_{\epsilon},p\mathcal{E}\}$be solutions of $(F)$$(F_{\epsilon})$

(2.8)

where

,

$||u_{\epsilon}-u||_{1}+||\tilde{p}\mathcal{E}-\tilde{p}||\leq C(\Omega, g)\sqrt{\epsilon}$$\tilde{p}$stands for the normalization of$\tilde{p}=p-\frac{1}{|\Omega|}\int_{\Omega}pdx$

and where the meaning of$\tilde{p}_{\mathit{6}}$is same.

,

$p$

(

subject to$|\Omega|$: the measure of ),$\Omega$

187
is essentially same as Kikuchi
and Oden [9], we omit to mention it and proceed to the estimate of the pressure, we havepart. Putting

Proof. Since the derivation of$||u-u_{\epsilon}||_{1}\leq C(\Omega, g)\sqrt{\epsilon}$$q_{\mathcal{E}}=\tilde{p}_{\epsilon}-\tilde{p}$(2.9)$(\forall\phi\in H_{0}^{1}(\Omega)^{m})$$a(u-u_{\epsilon}, \emptyset)=(q_{\xi}, \mathrm{d}\mathrm{i}\mathrm{V}\emptyset)$

In view of Babu\v{s}ca-Aziz’s lemma ([2]), we can take
. Now substituting
in with$\Omega$$q_{\epsilon}$$||w_{\epsilon}||_{1}\leq C(\Omega)||q\epsilon||$.

subject tointo (2.9), we deduce$w_{\epsilon}\in H_{0}^{1}(\Omega)^{m}$$\phi=w_{\epsilon}$$\mathrm{d}\mathrm{i}\mathrm{v}w_{\mathit{6}}=$$||q_{\epsilon}||^{2}=a(u-u_{\epsilon}, w_{\epsilon})\leq\delta_{0}||u-u_{\epsilon}||_{1}||w\mathcal{E}||1\leq\delta_{0}C(\Omega)||u-u\mathcal{E}||_{1}||q_{\epsilon}||$.

Combining this with the estimate of the velocity part, we arrive at (2.8).

Regularity Results for

3

Concering a regularity of a solution

(3.1)

of

$(\mathrm{F}_{\Xi})$

and, we havehold. For any
with$g\in H^{1}(\Gamma)$$u_{\epsilon}\in H^{2}(\Omega)^{m}$$(F_{\epsilon})$$(\mathrm{F}_{\mathit{6}})$$\{u_{\mathcal{E}},p\in\}$. Theorem 3.1. Assume that $(H)$ and. Thenbe a solution of$\square$$\epsilon>0$, let$\{u_{\epsilon},p\mathcal{E}\}$$p_{\epsilon}\in H^{1}(\Omega)$$||u_{\epsilon}||_{2}+||p_{\epsilon}||_{1}\leq C(\Omega)(||f||+||g||_{1,\mathrm{r}})$.

We firstly review a regularity result for the Stokes equations under the Neumannboundary condition.
andLemma 3.1. Let
is a weak solution of (1.5) with$\omega\in H^{1/2}(\Gamma)^{m}$$f\in L^{2}(\Omega)^{m}$. Suppose that $\{u,p\}\in H^{1}(\Omega)^{m}\cross$$L^{2}(\Omega)$$\sigma(u,p)=\omega$on .
$\Gamma$Namely, $\{u,p\}$ satisfies$a(u, \psi)-(p, \mathrm{d}\mathrm{i}\mathrm{v}\psi)=\int_{\Gamma}\omega\cdot\psi ds-(f, \psi)$

Then$u\in H^{2}(\Omega)^{m}$and$p\in H^{1}(\Omega)$,$(\forall\psi\in K^{1}(\Omega))$

.

with

$||u||_{2}+||p||_{1}\leq C(\Omega)(||f||+||\omega||1/2,\mathrm{r})$.

was described in Solonnikov [15] with a mention
Lemma 3.1 in the case of
on Solonnikov and \v{S}\v{c}adilov [16] concerning the method of the proof. However it
is not explicitly stated in these
seems that the complete proof for the case of
papers; In this connection, we refer to a forthcoming paper Saito [13].
$\omega\equiv 0$$\omega\not\equiv 0$

be a solution ofLemma 3.2. Let
Thus,
.
have
from Lemma 3.1,$\{u_{\mathcal{E}},p\epsilon\}$$\omega_{\epsilon}\in H^{1/2}(\Gamma)^{m}$$(F_{\epsilon})$, and put$\omega_{\epsilon}=g\alpha_{\mathcal{E}}(u_{\mathcal{E}})|_{\Gamma}$$u_{\mathcal{E}}\in H^{2}(\Omega)^{m}$and. Then we$p_{\epsilon}\in H^{1}(\Omega)$.

188
Proof. Firstly we have
(3.2)

with

$\alpha_{\epsilon}(u_{\mathit{6}})\in H^{1/2}(\Gamma)^{m}$$||\alpha_{\mathcal{E}}(u\epsilon)||_{1/2,\mathrm{r}}\leq C(\Omega, \epsilon)||u_{\mathcal{E}}|\mathrm{r}||_{1}/2,\mathrm{r}$.

This is essentially due to Br\’ezis [2], where he dealt with the scalar case. It is possible
to extend his result into our vector-values case; See [14] or [12]. Let us denote by
the weak harmonic extension of $g\in H^{1/2}(\Gamma)$ :$\tilde{g}\in H^{1}(\Omega)$$\triangle\tilde{g}\backslash =0$in$\Omega$,on$\tilde{g}=0$$\Gamma_{0}$,$\tilde{g}=g$

on .
$\Gamma$

It follows from the maximum principle that
. On the other
hand, we take the weak harmonic extensionof. That is, we extend
each component of
into by the harmonic function. By the definition ofand again using the maximum principle,
have.
Therefore, since, the.
$||\tilde{g}||_{L(\Omega)}\infty\leq||g||_{L(\Gamma)}\infty$$\tilde{\alpha}_{\epsilon}\in H^{1}(\Omega)$$\alpha_{\epsilon}(u_{\epsilon})$$\Omega$$\alpha_{\epsilon}(u_{\xi})$$\alpha_{\epsilon}$$||\tilde{\alpha}_{\epsilon}||_{L^{\infty}(\Omega)}\leq||\alpha_{\epsilon}(v)||_{L^{\infty}(\Gamma)}\leq C(m)\coprod$$\mathrm{c}\mathrm{w}\mathrm{e}$$\tilde{g}\tilde{\alpha}_{\epsilon}\in H^{1}(\Omega)^{m}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\omega_{\epsilon}\in H^{1/2}(\Gamma)^{m}$Remark 3.1. Our chooice of a regularized functional is based on the Yosida regularization. Namely, putting $\rho(z)=|z|$ for, then we have$z\in \mathbb{R}^{m}$(3.3)“the Yosida regularization of$\partial\rho$”$=\nabla\rho_{\epsilon}=\alpha_{\epsilon}$.

A property of the Yosida regularization (or a direct calculation) gives$| \alpha_{\epsilon}(z)-\alpha\epsilon(w)|\leq\frac{1}{\epsilon}|z-w|$,$(z, w\in \mathbb{R}^{m})$which is needed to prove (3.2). On the other hand, in view of (3.3) and Proposition
3 (Appendice I, Br\’ezis [2]), we also have“the Yosida regularization of$\partial j$

”

$=”$

the G\^ateaux defivative of”$j_{\epsilon}$

We proceed to the derivation of (3.1); We need another device.
Lemma 3.3. Let$\beta_{\epsilon}=u_{\epsilon}|_{\Gamma}$(3.4). Under the same assumptions of Theorem 3.1, we have$||\beta \mathit{6}||_{3/}2,\mathrm{r}\leq o(\Omega)(||f||+||g||_{1,\mathrm{r}})$.

Because of the limitation of the page number, we cannnot state the complete
proof of Lemma 3.3; Below we shall describe a sketch of the proof under a simple
situation. Namely, we assume that$\Omega=\mathbb{R}_{+}^{2}\equiv\{x=(x_{1}, x_{2});x_{2}>0\}$,$\Gamma=\{x=(X_{1}, X_{2});x_{2}=0\}$

and, for $R>0$ , put$\mathcal{O}_{R}=\{X=(_{X_{1}}, X_{2});|X|>R\}\cap\Omega$

Moreover we assume thatPut

$u_{\epsilon}\equiv 0$in$\mathcal{O}_{R/2}$$\varphi=-\frac{\partial v}{\partial x_{1}}$,.

. We simply write as$v= \frac{\partial u}{\partial x_{1}}$.

$u=u_{\epsilon}$

and$p=p_{\epsilon}$

.

189
by

$\mathrm{M}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{y}\mathrm{i}\mathrm{n}\mathrm{g}-\triangle u+\nabla p=f$

(3.5)$\varphi$then integrating over$\Omega$, we have.

$a(u, \varphi)-\int_{\Omega}p\mathrm{d}\mathrm{i}\mathrm{v}\varphi dx=\int_{-R}^{R}\sigma(u,p)\cdot\varphi dX_{1}+\int_{\Omega}f\cdot\varphi dx$

We obtain$a(u, \varphi)=a(v, v)\geq\delta_{1}||v||_{1}^{2}$

,

since
,

$\int_{\Omega}\frac{\partial u_{i}}{\partial x_{j}}\frac{\partial\varphi_{l}}{\partial x_{k}}dx=\int_{\Omega}\frac{\partial v_{i}}{\partial x_{j}}\frac{\partial v_{l}}{\partial x_{k}}dx$

By virtue of$\nabla_{z}\alpha_{\epsilon}(z)w\cdot w\geq 0$$\int_{-R}^{R}\sigma(u,p)\cdot\varphi dX_{1}$

for

$z,$

$w\in \mathbb{R}^{m}$$(i,j, k, l=1,2)$ .

, we get$=$$- \int_{-R}^{R}\frac{\partial}{\partial x_{1}}(g\alpha\in(u))\cdot vdx_{1}$$=$$- \int_{-R}^{R}g’\alpha_{\mathcal{E}}(u)\cdot vdx1^{-}\int_{-R}^{R}g(\nabla u\alpha\epsilon(u)v\cdot v)dx_{1}$$\leq$$\int_{-R}^{R}|g’|\cdot|\alpha(\epsilon u)|\cdot|v|dx_{1}$$\leq$$C||g’||_{L^{2}}(\Gamma)||v||_{1}$.

Moreover we can easily check that$\int_{\Omega}p\mathrm{d}\mathrm{i}\mathrm{v}\varphi dx=0$,$\int_{\Omega}f\cdot\varphi dx\leq C||f||||v||_{1}$.

Substituing these results of calculations into (3.5), we have$||v||_{1}\leq C(||f||+||g||_{1,\mathrm{r}})$which implies that$\beta_{\epsilon}\in H^{3/2}(\Gamma)$

and$||\beta_{\mathcal{E}}||_{3/}2,\mathrm{r}\leq c(||f||+||g||_{1,\mathrm{r}})$

Proof of Theorem 1.1

4
Let

.

$\epsilon>0$$||u_{\epsilon}||_{2}$$\{u_{\epsilon’}\}$. By virtue of Theorem 3.1, sequences
be a solution of, respectively. Hence, there are subsequences
are bounded assuch that, and letandand$||p_{\Xi}||_{1}$$\{p_{\mathit{6}};\}$$\{u_{\mathcal{E}},p\in\}$$u_{\epsilon’}arrow u^{*}$$(\mathrm{F}_{\epsilon})$$\mathit{6}\downarrow 0$

weakly in$H^{2}(\Omega)^{m}$,$p_{\epsilon’}arrow p^{*}$

weakly in$H^{1}(\Omega)$

and$||u|*|_{2}+||p|*|_{1}\leq C(\Omega)(||f||+||g||_{1,\mathrm{r}})$.

is a solution of (F). Next let $\{u,p\}$ be any
According to Theorem 2.2,
solution of (F). By the uniqueness of the velocity part, we have $u=u^{*}$ . On the$\{u^{*},p^{*}\}$

190
other hand, $p-p^{*}=k$ and a constant
and we deduce$k$is restricted via (1.3). Therefore $p\in H^{1}(\Omega)$ ,$\sigma(u,p)-\sigma(u,p*)=kn$. on .
$\Gamma$$\mathrm{a}.\mathrm{e}$This, together with (1.3), implies that $|k|\leq 2g$ holds almost everywhere on . Hencedenotes the measure of . By making use of this, whereestimate, we have$\Gamma$$|k|\leq 2|\Gamma|^{-}1/2||g||_{L()}2\Gamma$$\Gamma$$|\Gamma|$$||u||_{2}+||p||_{1}$$||u||_{2}+||p^{*}||1+|k|\sqrt{|\Omega|}$$\leq$$C(\Omega)(||f||+||g||_{1,\Gamma})$$\leq$,

which completes the proof.

Other Problems of Friction Type5andIn gerenal, for a vector-valued function , let
nent and the tangential components of , respectively;$v$$v_{N}$$v_{T}$

denote the normal compo-$v$$v_{N}=v\cdot n$

5.1,$v_{T}=v-v_{N}n$.

Leak Problem of Friction TypeWe consider the Stokes flow $\{u,p\}$ satisfying (1.5) together with
(5.1)$|\sigma_{N}(u,p)|\leq g_{N}$

and
(5.2)

$|\sigma_{N}(u,p)|<g_{N}$$\Rightarrow$$|\sigma_{N}(u,p)|=gN$$\Rightarrow$$u_{N}=0$ ,$\{$almost everywhere on$\Gamma$

, and

(5.3)

$u_{T}=0$

on .
$\Gamma$

The above problem was introduced H. Fujita ([6]) and is called the leak boundary
value problem of friction type. As was described in [6], this can be reduced to

PROBLEM$(\mathrm{L}\mathrm{F})$. Find$u\in K_{L,\sigma}^{1}(\Omega)$and$p\in L^{2}(\Omega)$satisfying(5.4) $a(u, v-u)-(p, \mathrm{d}\mathrm{i}\mathrm{v}(v-u))+j_{N}(v)-jN(u)\geq(f, v-u)$ ,$(\forall v\in K_{L}^{1}(\Omega))$

where$K_{L}^{1}(\Omega)=${ $v\in K^{1}(\Omega);v_{T}=0$ on },$\Gamma$$K_{L,\sigma}^{1}(\Omega)=K_{L}^{1}(\Omega)\cap K_{\sigma}^{1}(\Omega)$and$j_{N}(v)= \int_{\Gamma}g_{N}|v_{N}|ds$

.

,

,

191
Concerning the existence and theAssume that
and$\{u,p\}$ ofpart. The velocity
is unique andconstant and the constant is restricted via (5.1).
The following theorem is proved in Saito [12].

-uniqueness, we know ([6]):. Then there exists a solution
is unique up to an additive$\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{S}\mathrm{S}/\mathrm{n}\mathrm{o}\mathrm{n}$$f\in L^{2}(\Omega)^{m},$$g_{N}\in L^{\infty}(\Gamma)$$g_{N}>0\mathrm{a}.\mathrm{e}$$u$$(\mathrm{L}\mathrm{F})$$p$

Theorem 5.1. In additon to the assumptions mentioned above, we assume that
. Let $\{u,p\}$ be a solution of $(LF)$ . Then $u\in H^{2}(\Omega)^{m}$ and $p\in H^{1}(\Omega)$
with$||u||_{2}+||p||_{1}\leq C(||f||+||gN||_{1,\mathrm{r})}$ .

$g_{N}\in H^{1}(\Gamma)$

Moreover we have $\sigma_{N}(u,p)\in H^{1/2}(\Gamma)$ and$-\sigma_{N}(u,p)\in g\partial|u_{N}|$$\mathrm{a}.\mathrm{e}$. on F.

Slip Problem of Friction Type

5.2

The slip boundary value problem of friction type consists of (1.5) together with
(5.5)$|\sigma_{T}(u,p)|\leq g_{T}$

and
(5.6)

$|\sigma_{T}(u,p)|<g_{T}$$\Rightarrow$$|\sigma_{T}(u,p)|=g_{T}$$\Rightarrow$$u_{T}=0$,$\{$almost everywhere on$\Gamma$

, and

(5.7)

$u_{N}=0$on$\Gamma$

The weak formulation using the variational inequality is as follows.
PROBLEM(5.8)$(\mathrm{S}\mathrm{F})$. Find$u\in K_{S,\sigma}^{1}(\Omega)$and$p\in L^{2}(\Omega)$satisfying$a(u, v-u)-(p, \mathrm{d}\mathrm{i}\mathrm{v}(v-u))+j_{T}(v)-j\tau(u)\geq(f, v-u)$,$(\forall v\in K_{S}^{1}(\Omega))$

,

where
$K_{S}^{1}(\Omega)=${ $v\in K^{1}(\Omega);v_{N}=0$ on },$\Gamma$$K_{S,\sigma}^{1}(\Omega)=K_{S}^{1}(\Omega)\cap K_{\sigma}^{1}(\Omega)$,and$j \tau(v)=\int_{\Gamma}g_{T}|v\tau|ds$.

admits a solution $\{u,p\}$ if
As was mentioned in [6],. The velocity part is unique and is unique except for an additive
andconstant. In this case, the restriction for an additive constant is absent.
$f\in L^{2}(\Omega)^{m},$ $g_{T}\in L^{\infty}(\Gamma)$$(\mathrm{S}\mathrm{F})$$g_{N}>0\mathrm{a}.\mathrm{e}$

$u$

$p$

192
Theorem 5.2. In additon to the assumptions mentioned above, we assume that
. Let $\{u,p\}$ be a solution of $(SF)$ . Then $u\in H^{2}(\Omega)^{m}$ and $p\in H^{1}(\Omega)$with$g_{T}\in H^{1}(\Gamma)$$||u||_{2}+||p||_{1}\leq C(||f||+||g\tau||_{1,\mathrm{r}})$Moreover we have$\sigma_{T}(u,p)\in H^{1/2}(\Gamma)^{m}$.

and$-\sigma_{T}(u,p)\in g\partial|u_{T}|$. on .
$\Gamma$$\mathrm{a}.\mathrm{e}$For the proof, we refer to [12].

Acknowledgment
The author wishes to express his sincere gratitude to Professors Haim Br\’ezis andHiroshi Fujita for their valuable advise and suggestions.

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[12] N. SAITO: On the regularity of the Stokes flow under leak or slip boundarytype, to appear.
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: On a boundary value problem for[16] V.A. SOLONNIKOV ANDa stationary system of Navier-Stokes equations, Proc. Steklov Inst. Math., 125(1973), pp. 186-199.
$\mathrm{V}.\mathrm{E}.\check{\mathrm{S}}\check{\mathrm{C}}\mathrm{A}\mathrm{D}\mathrm{I}\mathrm{L}\mathrm{O}$

Norikazu SAITO
International Institute for Advanced Studies
9-3 Kizugawadai, Kizu-cho, Kyoto 619-0225, JAPAN-mail: nsaito@kurims.kyoto-u.ac.jp$\mathrm{E}$(財) 、際高等研究所
619-0225 京都府相楽郡木津町木津川台 9-3
Tel: 0774-73-4000 Fax: 0774-73-4024-mail: nsaito@kurims.kyoto-u.ac.jp$\mathrm{E}$