Funkcialaj Ekvacioj, 28 (1985), 121-138

On Systems of Variational Inequalitieswith Mixed Boundary Conditions

By

Kazuya HAYASIDA and Haruo NAGASE

(Kanazawa University and Suzuka College of Technology, Japan)

§1. Introduction

In this paper we consider the solution $U$ of a system of variational inequalitieswith mixed boundary conditions. Our aim is to study the regularity behavior of $U$

at the $(¥mathrm{n}-2)$ -dimensional submanifold $S$, across which a boundary condition isbroken. Our main result is a $W^{2,p}$ estimate of $U$ near $S$ with $1<p<10/9$ .

Let $¥Omega$ be a smoothly bounded domain in $R^{n}$ . Let $¥partial_{1}¥Omega$ and $¥partial_{2}¥Omega$ be two disjointopen subsets of the boundary $¥partial¥Omega$ such that $¥partial¥Omega=¥overline{¥partial_{1}¥Omega}¥cup¥overline{¥partial_{2}¥Omega}$ and $S=¥overline{¥partial_{1}¥Omega}¥cap¥overline{¥partial_{2}¥Omega.}$ Wefinrst consider a closed convex set of $H^{1}(¥Omega)$ :

$K=$ {$u¥in H^{1}(¥Omega);u=0$ on $¥partial_{1}¥Omega$ and $u¥geqq 0$ on $¥partial_{2}¥Omega$ }

and a solution $u¥in ¥mathrm{K}$ of the following single variational inequality

(1.1) $(¥nabla u, ¥nabla(v-u))¥geqq$ ($f$,$ $v?u) for all $v¥in ¥mathrm{K}^{1)}$ ,

where $(, )$ is the inner product of $L^{2}(¥Omega)$ and $f$ is a given continuous function in $¥overline{¥Omega}$ .

It was proved by H. Beirao Da Veiga [2] and by M.K.V. Murthy-G. Stampacchia[12] that such $u$ is Holder continuous up to the boundary. Further the latter studiedthe regularity for $u$ in a Sobolev space $W^{m,p}(¥Omega)$ . Their method is to reduce theproblem with obstacle on $¥partial_{2}¥Omega$ to a suitable one with obstacle throughout the wholeof $¥Omega$ .

When $¥partial_{1}¥Omega=¥phi$ in particular, the variational inequality (1. 1) was previously treatedby J.L. Lions [11] and H. Brezis [3]. In this case, $¥mathrm{M}.¥mathrm{G}$ . Garroni [7] has recentlygiven a regularity property in $W^{m,p}(¥Omega)$ for nonlinear operators. When $¥partial_{1}¥Omega¥neq¥phi$ , thereis also a work of G.M. Troianiello [13], which gives an estimate of the form $-¥Delta u¥leqq$

$f+(-f)^{+}$ .Secondly we consider systems of variational inequalities in place of (1.1), where

$K$ is replaced by a closed convex subset Kin $[H^{1}(¥Omega)]^{N}$ . If $V$ is a closed subspace of$[H^{1}(¥Omega)]^{N}$ containing $[H_{0}^{1}(¥Omega)]^{N}$ and, in particular, if $¥mathrm{K}$ equals

{($u_{1}$ , $¥cdots$ , $u_{N})¥in V;u_{i}¥geqq¥psi_{i}$ in $¥Omega$ , $i=1$ , $¥cdots$ , $N$ },1) Here we consider only the most simple form.

122 K. HAYASIDA and H. NAGASE

the regularity results were obtained by J. Frehse [6]. More precisely, in [6] an $L^{2}$

estimate with a local Morrey condition was obtained for the second order derivativesof solutions. Recently $¥mathrm{A}.¥mathrm{A}$ . Arhipova [1] has treated diagonal systems for moreextensive $¥mathrm{K}$, his result being an extension of J. Frehse’s previous work [5] concerningsingle variational inequalities, where it is shown that their solutions are in a Sobolevspace $W^{2,¥infty}(¥Omega)$ .

On the other hand the study of variational inequalities for vector-valued func-tions is closely connected with differential geometric problems in parametric form.In this connection we note the work of S. Hildebrandt and $¥mathrm{K}.¥mathrm{O}$ . Widman [8], whereeach $¥mathrm{K}$ is given in more general form, for instance,

$¥mathrm{K}=$ {($u_{1}$,?, $u_{N})¥in[H_{0}^{1}(¥Omega)]^{N}$ ; $u_{i}(x)¥in Z¥mathrm{a}.¥mathrm{e}$ . in $¥Omega$ for $i=1$,?, $N$}

and $Z$ is a closed convex set in $R^{N}$ . For such $¥mathrm{K}$ they have obtained Holder con-tinuity of solutions under certain assumptions. For other topics in this field we refersome results of [4], [15] for systems of variational inequalities.

The aim of this paper is the study of systems of variational inequalities withmixed boundary conditions. Form the above mentioned it is a natural course ofevents to consider $¥mathrm{K}$ as follows

$¥mathrm{K}=¥{(u_{1^{ }},--, u_{N})¥in[H^{1}(¥Omega)]^{N}$ ; $u_{i}=0$ on $¥partial_{1}¥Omega$

and $u_{i}(x)¥in Z¥mathrm{a}.¥mathrm{e}$ . in $¥partial_{2}¥Omega$ for $i=1$ , $¥cdots$ , $N$}.

Indeed, $¥mathrm{N}.¥mathrm{N}$. Ural’tseva [14] studied systems of variational inequalities with such $¥mathrm{K}$

to derive regularity of solutions except at $S$. Her technique is the penalty methodand her result is that each solution is in $H^{2}(¥Omega^{¥prime})$ for every compact subset $¥Omega^{¥prime}$ of $¥Omega-S$.

However we are interested in the regularity behavior of solutions near $S$ .

We will prove regularity results near $S$ for any vector-valued solution of varia-tional inequalities (see Theorem 2 in Section 2). For this a weaker regularity theoremis prepared, which is valid $¥mathrm{f}¥mathrm{o}¥mathrm{r}¥overline{¥partial_{1}^{¥prime}¥mathrm{J}I}$ irredgular at $S$ (see Theorem 1 in Section 2). In[10] this result has already been given for single linear elliptic equations under theassumption that $u=0$ throughout $¥partial¥Omega$ , that is, the Dirichlet data of $u$ is zero.

In the proof of Theorem 1, our variational method yields $H^{1}(¥Omega)$ estimates withan adequate weight function. In order to prove Theorem 2, we shall use a paralleltranslation of solutions with a weight, where we use the same method as that inTheorem 8.6 of [11, Chapter 2] for the case of $¥partial_{1}¥Omega=¥phi$ . In this paper we consideronly cases in the upper-half plane for the sake of brevity.

We express our hearty thanks to the referee for many valuable comments.

§2. Preliminaries and Results

Let $¥beta$ be any fixed positive number less than 1/2. In Section 5 we choose $¥beta$ sothat it is close to 1/2. Let $¥gamma$ be a positive number such that

On Systems of Variational Inequalities 123

(2. 1) $¥mathcal{T}+¥beta¥pi<¥pi/¥mathit{2}$ .

Hereafter we consider the case in the upper half space $¥{x_{n}¥geqq 0¥}$ of $R^{n}$ with co-

ordinates $(x_{1^{ }},¥cdots, x_{n})$ . Let $r=(x_{1}^{2}+ --+x_{n}^{2})^{1/2}$ , $¥rho=(x_{n-1}^{2}+x_{n}^{2})^{1/2}$ and $¥theta=¥mathrm{A}¥mathrm{r}¥mathrm{g}$ $(x_{n1}¥_$

$+ix_{n})$ . We set

$¥chi(x_{n-1}, x_{n})=¥rho^{-¥beta}¥sin(¥beta¥theta+¥gamma)$.

By (2. 1) there is a positive constant $c¥leqq 1$ such that

(2.2) $¥chi¥geqq c¥rho^{-¥beta}$ for $x_{n}¥geqq 0$ .

Figure 1

For any fixed $R$ with $0<R<1$ we define

$¥sum=¥{(¥mathrm{x}_{1^{ }},¥cdots, x_{n});r<R, x_{n}>0¥}$

and

$¥Gamma=¥{x_{n}=0¥}¥cap¥{x_{n-1}<0, x_{1}^{2}+-+x_{n-1}^{2}<R^{2}¥}$ .

Let $G$ be a subdomain of $¥Sigma$ such that the boundary $¥partial G$ contains $¥Gamma$ (see Figure 1).Thus $¥partial G$ may be non-smooth except for $¥Gamma$ . We denote $¥partial G-¥Gamma$ by $¥partial^{¥prime}G$ . In whatfollows, we consider in $G$ instead of $¥Omega$ .

Now we set

$C_{(0)}^{1}(¥overline{G})=$ {$u¥in C^{1}(¥overline{G});u=0$ in a neighborhood of $¥partial^{¥prime}G$ }.

If $ G=¥Sigma$ particularly, $u¥in C_{(0)}^{1}(¥overline{G})$ means that $u¥in C^{1}(¥overline{¥Sigma})$ and $u=0$ near $r=R$ and$¥{x_{n}=0, x_{n-1}¥geqq 0¥}$ . The norm and the inner product in $L^{2}(G)$ are simply denoted by$||||$ and $(, )$ respectively. We define $||u||_{1}=||¥nabla u||+||u||$ for $u¥in C_{(0)}^{1}(¥overline{G})$, which is equiv-alent to $||¥nabla u||$ by Poincare’s inequality. The completion of $C_{(0)}^{1}(¥overline{G})$ with respect tothe norm $||u||_{1}$ is denoted by $H_{(0)}^{1}(G)$ .

For any $u$, $v$ $¥in C_{(0)}^{1}(¥overline{G})$ we define the inner product and the norm as follows:

$((u, v))=(¥chi^{¥delta}¥nabla u, ¥nabla v)+(¥chi^{¥delta-2}¥rho^{-2(¥beta+1)}u, v)$, $|||u|||=((u, u))^{1/2}$ .

124 K. HAYASIDA and H. NAGASE

where $¥delta$ is any fixed positive number less than 1. In Section 5, $¥delta$ may be chosen asclose to 1 as desired. By virtue of (2.2), $||||||$ satisfies the condition of the norm.We denote by $¥tilde{H}_{(0)}^{1}(G)$ the completion of $C_{(0)}^{1}(¥overline{G})$ with respect to the norm $||||||$ .

When $X$ is a Sobolev space of real valued functions defined on $G$, $[X]^{N}$ denotesthe space of $¥mathrm{N}$-vector valued functions with components in $X$. Let $U=(u_{1^{ }},--, u_{N})$

and $V=(u_{1}, --, v_{N})$ . Then we define

$||U||^{2}=¥sum_{i=1}^{N}||u_{i}||^{2}$, $(U, V)=¥sum_{i=1}^{N}(u_{i}, v_{i})$ ,

$|||U|||^{2}=¥sum_{i=1}^{N}|||u_{i}|||^{2}$, $((U, V))=¥sum_{i=1}^{N}((u_{i}, v_{i}))$ .

For any set $¥Omega¥subset R^{n}$ and for any $p>1$ , $L^{p}(¥Omega)$ is the class of all measurable functions$u$ defined in $¥Omega$ , for which $¥int_{¥Omega}|u(x)|^{p}dx<¥infty$ . We write

$||u||_{L(¥Omega)}p=(¥int_{¥Omega}|u|^{p}dx)^{1/p}$ and $||U||_{L(¥Omega)}p=¥sum_{i=1}^{N}||u_{i}||_{L^{p}(¥Omega)}$ .

We consider the following bilinear form

$B[U, V]=¥sum_{i=1}^{N}$ $(¥nabla u_{i}, ¥nabla v_{i})+¥sum_{i=1}^{N}(c_{ij}u_{j}, v_{i})$,

where $c_{ij}¥in C^{1}(¥overline{G})(1¥leqq i,j¥leqq N)$ and we assume

(2. 3) $¥mathrm{m}i,j^{¥frac{¥mathrm{x}}{G}}¥mathrm{a},|c_{ij}|¥leqq 9c^{¥delta}/(3¥mathit{2}NR^{2})$ ,

where $c$ is the positive constant in (2.2). Obviously $B[U, V]$ is continuous in$[H_{(0)}^{1}(G)]^{N}¥times[H_{(0)}^{1}(G)]^{N}$ . Since $¥mathrm{m}i,j¥mathrm{a}_{¥frac{¥mathrm{x}}{G}},|c_{ij}|¥leqq(¥mathit{2}NR^{2})^{-1}$ from (2.3), we have

$|¥sum_{i,j}(c_{ij}u_{i}, u_{j})|¥leqq(¥mathrm{m}i,j,¥mathrm{a}_{¥frac{¥mathrm{x}}{G}}|c_{ij}|)¥sum_{i,j}||u_{i}||¥cdot||u_{j}||¥leqq N(¥mathrm{m}i,j,¥mathrm{a}_{¥frac{¥mathrm{x}}{G}}|c_{ij}|)(¥sum_{i}||u_{i}||^{2})$.

By Poincare’ $¥mathrm{s}$ inequality,

$||u_{i}||^{2}¥leqq(¥mathit{2}R)^{2}(¥mathit{2}n)^{-1}||¥nabla u_{i}||^{2}$ .

More precisely, this follows from

$||u_{i}||^{2}¥leqq¥frac{1}{¥mathit{2}}(¥mathit{2}R)^{2}||¥partial_{x_{j}}u_{i}||^{22)}$ .

Hence we get

$|¥sum_{i,j}(c_{ij}u_{i}, u_{j})|¥leqq¥frac{1}{¥mathit{2}}¥sum_{i}||¥nabla u_{i}||^{2}$ .

2) See e.g., page 158 in the book of R. A. Adams “Sobolev Spaces”.

On Systems of Variational Inequalities 125

This implies that there is a positive constant $c$ such that

$B[U, U]¥geqq c||¥nabla U||^{2}$ for all $U¥in[H_{(0)}^{1}(G)]^{N}$ .

Accordingly $B[U, V]$ satisfies Lax-Milgram’s condition on $[H_{(0)}^{1}(G)]^{N}$ .Let $Z$ be a closed convex set in $R^{N}$ which contains the origin.Then we set

$H=$ { $V¥in[C_{(0)}^{1}(¥overline{G})]^{N}$ ; $u(x)¥in Z$ for all $ x¥in¥Gamma$}.

The completion of $H$ with respect to the norm $||||_{1}$ (resp. $||||||$) is denoted by $¥mathrm{K}$

(resp. $¥tilde{¥mathrm{K}}$). Clearly $¥mathrm{K}(¥tilde{¥mathrm{K}})$ is a closed convex set in $[H_{(0)}^{1}(G)]^{N}([¥tilde{H}_{(0)}^{1}(G)]^{N})$ contaningthe zero vector of $R^{N}$ , respectively. It is well-known that for any given $F¥in[L^{2}(G)]^{N}$

there is a unique solution $U¥in ¥mathrm{K}$ of

(2.4) $B[U, V-U]¥geqq(F, V-U)$ for all $V¥in ¥mathrm{K}$

(see e.g., [9]).For $¥epsilon>0$ we denote by $¥Sigma_{¥epsilon}$ the set $¥{r<R/¥mathit{2}¥}¥cap¥{x_{n}>¥epsilon|x_{n-1}|¥}$ . And we denote

simply by $¥partial^{k}$ any $¥mathrm{k}$-th order derivative with respect to the variables $x_{j}(1¥leqq j¥leqq n)$ .

Under the assumptions as above we shall prove the following theorem:

Theorem 1. Let $U$ be the solution of (2.4). Then $U$ is in $¥tilde{¥mathrm{K}}$ and it satisfies$|||U|||¥leqq C||¥rho^{1-¥beta¥delta/2}F||$ ,

where $¥beta$ , $¥delta$ are the numbers appearing in the definition of the norm $||||||$ and $C$ is inde-pendent of $F$ and $U$.

Theorem 2. Suppose that $ G=¥sum$ in particufar. Let $U$ be the solution of (2.4).Then for any $¥epsilon_{1}$ , $¥epsilon_{2}>0$ it holds that

$¥partial^{2}U¥in[L^{10/9-¥epsilon_{1}}(¥sum_{¥mathrm{S}2})]^{N}$

and

$||¥partial^{2}U||_{L^{10/9}}-¥epsilon_{1}(¥Sigma_{¥epsilon_{2}})¥leqq C||¥rho^{3/4}F||$ ,

where $C$ is independent of $F$ and $U$, but depending on $¥epsilon_{1}$ and $¥epsilon_{2}$ .

§3. Lemmas

In what follows, let $ 0<¥beta$ , $¥gamma<1/¥mathit{2}$ and let us assume (2. 1). Further let $0<¥delta<1$ .

First we have

Lemma 1. Let $¥chi$ be as in the previous section. Then it holds that

126 K. HAYASIDA and H. NAGASE

$¥Delta x^{¥delta}=¥beta^{2}¥delta(¥delta-1)¥chi^{¥delta-2}¥rho^{-2(¥beta+1)}$ , $|¥nabla¥chi^{¥delta}|=¥beta¥delta¥chi_{¥rho^{-(¥beta+1)}}^{¥delta-1}$

and

$¥partial_{x_{n}}¥chi^{¥delta}|_{x_{n}=0,x_{n-1}<0}<0$ .

Proof. We can rewrite

$¥chi=-¥mathrm{I}¥mathrm{m}e^{-¥gamma i}(x_{n-1}+ix_{n})^{-¥beta}$ ,

which means that $¥chi$ is a harmonic function with respect to the variables $(x_{n-1}, x_{n})$ .On the other hand we see

$¥partial_{x_{n-1}}¥chi=-¥beta¥rho^{-(¥beta+1)}¥sin((¥beta+1)¥theta+¥mathcal{T})$

and

$¥partial_{x_{n}}¥chi=¥beta¥rho^{-(¥beta+1)}¥cos((¥beta+1)¥theta+¥mathcal{T})$,

from which

$|¥nabla¥chi|=¥beta¥rho^{-(¥beta+1)}$ .

Hence the first and second equalities follow. The last inequality is valid from (2. 1).Q.E.D.

Lemma 2. Let $F¥in[L^{2}(G)]^{N}$ and $V¥in[¥tilde{H}_{(0)}^{1}(G)]^{N}$ . Then

$|(¥chi^{¥delta}F, V)|¥leqq C||¥rho^{1-¥beta¥delta/2}F||¥cdot|||V|||$ .

Proof. By Schwarz’ inequality we have

$|(¥chi^{¥delta}F, V)|¥leqq C||¥rho^{1-¥beta¥delta/2}F||¥cdot||¥rho^{-1-¥beta¥delta/2}V||¥leqq C||¥rho^{1-¥beta¥delta/2}F||¥cdot|||V|||$ . Q.E.D.

In succession we consider cases in the upper half space $¥{x_{n}¥geqq 0¥}$ . Let $¥alpha$ be anyfixed real number with $4/5<¥alpha<1$ , which may be chosen so slose to 4/5. Let $¥tilde{¥rho}=$

$(x_{n-1}^{2}+x_{n}^{2a})^{1/2}$. Since $x_{n}¥leqq¥tilde{¥rho}^{1¥prime a}$ , we have

(3. 1) $|¥partial_{x_{n-1}}¥tilde{¥rho}|¥leqq C$, $|¥partial_{x_{n}}¥tilde{¥rho}|¥leqq C¥tilde{¥rho}^{1-1/a}$.

Hereafter let $h$ be positive and sufficiently small. We define the followingmapping from $R^{n}$ into itself:

$¥Phi_{h}$ : $¥left¥{¥begin{array}{l}y_{j}=x_{j}(j¥neq n-1)¥¥y_{n-1}=x_{n-1}+h¥tilde{¥rho}.¥end{array}¥right.$

Then we see that

On Systems of Variational Inequalities 127

(3.2)

$[|¥mathrm{r}_{¥partial_{x_{n}}y_{1}¥partial_{x_{n}}y_{n}}^{¥partial_{x_{1}}y_{1}¥partial_{x_{1}}y_{n}}¥ldots¥cdot.¥cdot¥cdots.¥cdot¥ldots]=¥left¥{¥begin{array}{lll}1 & & 0¥¥ & 1 & ¥¥0 & 1+h¥partial_{x_{n- 1}}¥tilde{¥rho} & 0¥¥ & fr¥partial_{x_{n}}¥tilde{¥rho} & 1¥end{array}¥right¥}$

$¥left¥{¥begin{array}{lll}¥partial_{y1}x_{1} & ¥cdots & ¥partial_{y1}x_{n}¥¥¥cdots & ¥cdots & ¥cdots¥¥¥partial_{y_{n}}x_{1} & ¥cdots & ¥partial_{?}/nx_{n}¥end{array}¥right¥}=[^{1}0$ $10*]$

where

$*=¥frac{1}{1+h¥partial_{x_{n-1}}¥tilde{¥rho}}$ $¥left(¥begin{array}{lll}1 & 0 & ¥¥-h¥partial_{x_{n}}¥tilde{¥rho} & 1+h¥partial_{x_{n}} & 1¥tilde{¥rho}¥end{array}¥right)$ .

Let $J_{h}$ be the Jacobian of $¥Phi_{h}$ , which is equal to $1+h¥partial_{x_{n-1}}¥tilde{¥rho}$. From (3. 1) there is apositive constant $c$ such that $c¥leqq J_{h}¥leqq c^{-1}$ in $R^{n}$ . Hence $¥Phi_{h}$ is one-to-one mappingfrom $¥{x_{n}¥geqq 0¥}$ onto itself. Combining (3. 1) and (3.2) one finds

$(3.1)^{¥prime}$ $|¥partial_{y_{n-1}}¥tilde{¥rho}¥circ¥Phi_{h}^{-1}|¥leqq C$ and $|¥partial_{yn}¥tilde{¥rho}¥circ¥Phi_{h}^{-1}|¥leqq C(¥tilde{¥rho}¥circ¥Phi_{h}^{-1})^{1-1/¥alpha}$ .

In what follows, the $¥mathrm{y}$-variable is always connected with the $¥mathrm{x}$-variable by $y=¥Phi_{h}(x)$ .For the notational convenience, we usually write $¥tilde{¥rho}$ in place of $¥tilde{¥rho}¥circ¥Phi_{h}^{-1}$ , because thedependence of $¥tilde{¥rho}$ on $x$ or $y$ will be understood from the context.

We put $x^{¥prime¥prime}=(x_{1^{ }},--, x_{n-2})$ and $y^{/¥prime}=(y_{1}, --, y_{n-2})$ for $x=(x_{1^{ }},¥cdots, x_{n})$ and $y=$

$(y_{1}, --, y_{n})$ . The following operators are defined:

$(S_{h}u)(x)=u(x^{¥prime¥prime}, x_{n-1}+h¥tilde{¥rho}, x_{n})$, $(T_{h}u)(y)=u(y^{¥prime¥prime}, y_{n-1}-h¥tilde{¥rho}, y_{n})$,

$(P_{h}u)(x)=h^{-1}((S_{h}u)(x)-u(x))$ ,

$(Q_{h}u)(y)=h^{-1}((T_{h}u)(y)-u(y))$.

If we set

$¥nabla_{x}(S_{h}u)=S_{h}¥nabla_{x}u+hF_{h}(¥nabla_{x}u)$

and

$¥nabla_{y}(T_{h}u)=T_{h}¥nabla_{y}u+hG_{h}(¥nabla_{y}u)$,

it is easily seen that

$F_{h}(¥nabla_{x}u)=(0, --, 0, ¥partial_{x_{n-1}}¥tilde{¥rho}¥cdot S_{h}¥partial_{x_{n-1}}u, ¥partial_{x_{n}}¥tilde{¥rho}¥cdot S_{h}¥partial_{x_{n-1}}u)$,(3.2)

$G_{h}(¥nabla_{y}u)=(¥mathrm{o}, -, 0, -¥partial_{y_{n-1}}¥tilde{¥rho}¥cdot T_{h}¥partial_{y_{n-1}}u, -¥partial_{y_{n}}¥tilde{¥rho}¥cdot T_{h}¥partial_{y_{n-1}}r/)$,

128 K. HAYASIDA and H. NAGASE

The following lemma follows immediately from (3.1), $(3. 1^{¥prime})$ and (3.3).

Lemma 3. For all $u¥in C_{(0)}^{1}(¥overline{G})$

$|F_{h}(¥nabla_{x}u)|¥leqq C¥tilde{¥rho}^{1-1/a}|S_{h}¥partial_{x_{n-1}}u|$

and

$|G_{h}(¥nabla_{y}u)|¥leqq C¥tilde{¥rho}^{1-1/¥alpha}|T_{h}¥partial_{y_{n-1}}u|$ ,

where $C$ is independent of $h$ and $u$ .

We note that any function $u¥in H_{(0)}^{1}(G)$ can be extended over $¥{x_{n}¥geqq 0¥}$ in such away that $u=0$ in $¥{x_{n}¥geqq 0¥}¥cap G^{c}$ .

Lemma 4. Suppose that $ G=¥sum$ in particular. Let $¥beta¥delta/¥mathit{2}>1/¥alpha-1$ and $fet$ $¥zeta$ be $a$

function in $C_{0}^{¥infty}(¥{r<R¥})^{3)}$ such that $0¥leqq¥zeta¥leqq 1$ . Then $¥zeta S_{h}U$ and $¥zeta T_{h}U¥in Kfor$ any $U¥in¥tilde{¥mathrm{K}}$.

Proof. Recalling the definition of $¥tilde{¥mathrm{K}}$, one takes a sequence $¥{U_{j}¥}¥in H$ such that

$||¥rho^{-¥beta¥delta/2}¥nabla(U_{j}-U)||+||¥rho^{-¥beta¥delta/2-1}(U_{j}-U)||¥rightarrow 0$ as $ j¥rightarrow¥infty$ .

Since $¥tilde{¥rho}¥geqq¥rho$ and $¥beta¥delta/2>1/¥alpha-1$ , it follows that

(3.4) $||U_{j}-U||+||¥tilde{¥rho}^{1-1/¥alpha}¥nabla(U_{j}-U)||¥rightarrow 0$ as $ j¥rightarrow¥infty$ .

Clearly $¥zeta S_{h}$ $U_{j}¥in H$ and, by (3.4),

$¥lim_{j-¥infty}||¥zeta S_{h}(U_{j}-U)||=¥lim_{j¥rightarrow¥infty}||¥zeta S_{h}¥nabla(U_{j}-U)||=0$ ;

so that it suffices to show

$¥lim_{j-¥infty}||F_{h}(¥nabla_{x}(U_{j}-U))||=0$

to complete the proof. Note that Lemma 3 implies

$|F_{h}(¥nabla_{x}(U_{j}-U))|¥leqq C¥tilde{¥rho}^{1-1/a}|S_{h}¥partial_{x_{n-1}}(U_{j}-U)|$.

Since $¥tilde{¥rho}^{1-1/a}$ is bounded from above by $C(¥tilde{¥rho}¥circ¥Phi_{h})^{1-1/a}$ , the proof will be completedwith the aid of (3.4). Q. $¥mathrm{E}.¥mathrm{D}$ .

We denote by $(, )_{x}$ and $||||_{x}$ ((, ) and $||||_{y}$) the $¥mathrm{L}^{2}$-inner product and the$¥mathrm{L}^{2}$-norm of functions on $¥{x_{n}¥geqq 0¥}(¥{y_{n}¥geqq 0¥})$ , respectively.

Lemma 5. For u $¥in H_{(0)}^{1}(¥Sigma)$

3) Namely, $¥zeta$ is in $C^{¥infty}$ with compact support in $¥{r<R¥}$ .

On Systems of Variational Inequalities 129

$||¥tilde{¥rho}^{-1}P_{h}u||_{x}¥leqq C||¥partial_{x_{n-1}}u||_{x}$

and

$||¥tilde{¥rho}^{-1}Q_{h}u||_{y}¥leqq C||¥partial_{y_{n-1}}u||_{y}$ ,

where $C$ is idependent of $h$ and $u$ .

Proof. When $u¥in C_{(0)}^{1}(¥overline{¥Sigma})$ in particular, we see

(3.5) $(P_{h}u)(x)=¥tilde{¥rho}¥int_{0}^{1}(¥partial_{x_{n-1}}u)(x^{¥prime¥prime}, x_{n-1}+¥theta h¥tilde{¥rho}, x_{n})d¥theta$ .

By Schwarz’ inequality,

$||¥tilde{¥rho}^{-1}P_{h}u||_{x}^{2}¥leqq¥int_{0}^{1}¥int_{x_{n}>0}[(¥partial_{x_{n-1}}u)(x^{¥prime¥prime}, x_{n-1}+¥theta h¥tilde{¥rho}, x_{n})]^{¥mathit{2}}dxd¥theta$

$¥leqq¥int_{0}^{1}||(J_{¥theta h}^{-1})^{1/2}¥partial_{y_{n-1}}u||_{y}^{2}d¥theta¥leqq C||¥partial_{x_{n-1}}u||_{x}^{2}$.

Hence the first half has been proved for $u¥in C_{(0)}^{1}(¥overline{¥Sigma})$ . When $u¥in H_{(0)}^{1}(¥Sigma)$ , it is suffi-cient to take an approximating sequence from $C_{(0)}^{1}(¥overline{¥Sigma})$ .

The proof of the later half is completely analogous. Q.E.D.

Lemma $¥epsilon$ . Let $u¥in H_{(0)}^{1}(¥Sigma)$ . Thena) $||P_{h}u-¥tilde{¥rho}¥partial_{x_{n-1}}u||_{x}¥rightarrow 0$ ,

b) $||Q_{h}u+¥tilde{¥rho}¥partial_{y_{n-1}}u||_{y}¥rightarrow 0$ as $h¥rightarrow 0$ .

Proof. If $u¥in C_{(0)}^{1}(¥overline{¥Sigma})$ , we have from (3.5)

$||P_{h}u-¥tilde{¥rho}¥partial_{x_{n-1}}u||_{x}^{2}$

$¥leqq¥int_{0}^{1}¥int_{x_{n}>0}¥tilde{¥rho}^{2}[(¥partial_{x_{n-1}}u)(x^{¥prime¥prime}, x_{n-1}+¥theta h¥tilde{¥rho}, x_{n})-¥partial_{x_{n-1}}u(x)]^{2}dxd¥theta$ ,

which implies a) for $u¥in C_{(0)}^{1}(¥overline{¥Sigma})$ .

If $u¥in H_{(0)}^{1}(¥Sigma)$ , there is a sequence $¥{u_{j}¥}¥subset C_{(0)}^{1}(¥overline{¥Sigma})$ such that $u_{j}¥rightarrow u$ in $H_{(0)}^{1}(¥Sigma)$ .

Using Lemma 5, we see that

$||P_{h}u-¥tilde{¥rho}¥partial_{x_{n-1}}u||_{x}¥leqq||P_{h}(u-u_{j})||_{x}+||P_{h}u_{j}-¥tilde{¥rho}¥partial_{x_{n-1}}u_{j}||_{x}+||¥tilde{¥rho}(¥partial_{x_{n-1}}u_{j}-¥partial_{x_{n-1}}u)||_{x}$

$¥leqq C(||¥nabla(u_{j}-u)||_{x}+||P_{h}u_{j}-¥tilde{¥rho}¥partial_{x_{n-1}}u_{j}||_{x})$ .

Thus we have proved $¥mathrm{a}$). Similarly b) can be proved. Q.E.D.

§4. Proof of Theorem 1

Let us define a new bilinear form:

130 K. HAYASIDA and H. NAGASE

$¥tilde{B}[U, V]=¥sum_{i=1}^{N}$ $(¥chi^{¥delta}¥nabla u_{j}, ¥nabla v_{i})$

$+¥sum_{i=1}^{N}(v_{i}¥nabla¥chi^{¥delta}, ¥nabla u_{i})+¥sum_{i,j=1}^{N}(c_{ij}¥chi^{¥delta}u_{j}, v_{i})$.

In view of Lemma 1, it holds that for any $U$, $V¥in[¥tilde{H}_{(0)}^{1}(G)]^{N}$

$|¥tilde{B}[U, V]|¥leqq C|||U|||¥cdot|||V|||$ .

For $v¥in C_{(0)}^{1}(¥overline{G})$ we have

$(¥partial_{x_{n}}¥chi^{¥delta}, ¥partial_{x_{n}}v^{2})=¥lim_{¥epsilon-+0}¥int_{x_{n}>¥mathrm{e}}¥partial_{x_{n}}¥chi^{¥delta}.¥partial_{x_{n}}v^{2}dx$

and

$¥int_{x_{n}>¥mathrm{e}}¥partial_{x_{n}}¥chi^{¥delta}.¥partial_{x_{n}}v^{2}dx=-¥int_{x_{n}>¥epsilon}¥partial_{x_{n}}^{2}¥chi^{¥delta}.v^{2}dx-¥int_{x_{n}=¥mathrm{e}}¥partial_{x_{n}}¥chi^{¥delta}.v^{2}dx_{1}¥cdots dx_{n-1}$.

Therefore from Lemma 1 we obtain

$(¥partial_{x_{n}}¥chi^{¥delta}, ¥partial_{x_{n}}v^{2})¥geqq-(¥partial_{x_{n}}^{2}¥chi_{U^{2}}^{¥delta},)$,

which implies that for any $V¥in[C_{(0)}^{1}(¥overline{¥Sigma})]^{N}$

(4. 1) $¥tilde{B}[V, V]¥geqq¥sum_{i}(¥chi^{¥delta}¥nabla v_{i}, ¥nabla v_{i})-¥frac{1}{¥mathit{2}}(¥Delta¥chi^{¥delta}, ¥sum_{i}v_{i}^{2})+¥sum_{i,j}(c_{ij}¥chi^{¥delta}v_{j}, v_{i})$.

Let $w(¥rho)$ be a $¥mathrm{C}^{1}$ -function in $R^{1}$ with $w(0)=w(R)=0$. Then for any $k>1$ wehave

$¥int_{0}^{R}¥rho^{k-1}w^{2}d¥rho¥leqq 4k^{-2}¥int_{0}^{R}¥rho^{k+1}(w^{¥prime})^{2}d¥rho$,

which is obtained from

$ k¥int_{0}^{R}¥rho^{k-1}w^{2}d¥rho=-¥mathit{2}¥int_{0}^{R}¥rho^{k}ww^{¥prime}d¥rho$

$¥leqq¥frac{k}{¥mathit{2}}¥int_{0}^{R}¥rho^{k-1}w^{2}d¥rho+¥frac{¥mathit{2}}{k}¥int_{0}^{R}¥rho^{k+1}(w^{¥prime})^{2}d¥rho$.

Thus putting $ k=¥mathit{2}-¥beta¥delta$ , we see that

$¥int_{0}^{R}¥rho^{1-¥beta¥delta}v_{i}^{2}d¥rho¥leqq 4R^{2}(¥mathit{2}-¥beta¥delta)^{-2}¥int_{0}^{R}¥rho^{1-¥beta¥delta}(¥partial_{¥rho}v_{i})^{2}d¥rho$.

Noting tiie well-known inequality $|¥nabla v_{i}|^{2}¥geqq(¥partial_{¥rho}v_{i})^{2}$ , we arrive at

On Systems of Variational Inequalities 131

$(¥chi^{¥delta}U_{i}, U_{i})¥leqq ¥mathit{4}R^{2}(¥mathit{2}-¥beta¥delta)^{-2}c^{-¥delta}(¥chi^{¥delta}¥nabla v_{i}, ¥nabla v_{i})$

$¥leqq¥frac{16}{9}R^{2}c^{-¥delta}(¥chi^{¥delta}¥nabla v_{i}, ¥nabla v_{i})$ .

Therefore it follows that

$|¥sum_{i,j}$ $(c_{ij}¥chi^{¥delta}v_{j}v_{i})|¥leqq N(¥mathrm{m}i,j^{¥frac{¥mathrm{x}}{G}}¥mathrm{a},|c_{ij}|)¥sum_{i}$$(¥chi^{¥delta}v_{i}, v_{i})$

$¥leqq¥frac{16}{9}c^{-¥delta}NR^{2}(¥mathrm{m}i,j¥mathrm{a}_{¥frac{¥mathrm{x}}{G}},|c_{ij}|)¥sum_{i}$ $(¥chi^{¥delta}¥nabla v_{i}, ¥nabla v_{i})$.

By virtue of (2.3), (4. 1) becomes

$¥tilde{B}[V, V]¥geqq c|||V|||^{2}$ for all $V¥in[¥tilde{H}_{(0)}^{1}(G)]^{N}$ ,

where $c$ is a positive constant. Accordingly $¥tilde{B}[U, V]$ satisfies Lax-Milgram’ $¥mathrm{s}$

condition.By Lemma 2 and a well-known fact (see, e.g., page 24 in [9]) there is a unique

solution $¥tilde{U}¥in¥tilde{¥mathrm{K}}$ of

(4.2) $¥tilde{B}[¥tilde{U}, V-¥tilde{U}]¥geqq(¥chi^{¥delta}F, V-¥tilde{U})$ for all $V¥in¥tilde{¥mathrm{K}}$.

Let us put $V^{;}=(1-t¥chi^{-¥delta})¥tilde{U}+t¥chi^{-¥delta}W$ for any given $W¥in H$ and for sufficientlysmall $t>0$ . Evidently $V^{¥prime}¥in¥tilde{¥mathrm{K}}$ . Setting $V=V^{¥prime}$ in (4.2) and dividing by $t$ , we have

$B[¥tilde{U}, W-¥tilde{U}]¥geqq(F, W-¥tilde{U})$ .

which is valid also for any $ W¥in$ K. The unique solvability of (2.4) implies that $¥tilde{U}$

coincides with the solution $U$ of (2.4). Putting $V=(0, --, 0)$ in (4.2) and usingLemma 2, we arrive at the conclusion of Thdorem 1. Q.E.D,

§5. Proof of Theorem 2

Let us take a function $¥zeta(x)¥in C_{0}^{¥infty}(¥{r<R¥})$ such that $0¥leqq¥zeta¥leqq 1$ and $¥zeta=1$ in aneighborhood of the origin. Let $V^{¥prime}=(1-¥zeta^{2})U+¥zeta^{2}V$ for any fixed $ V¥in$ K. Since$V^{¥prime}¥in ¥mathrm{K}$, we have from (2.4)

$B[U, ¥zeta^{2}(V-U)]¥geqq(F, ¥zeta^{2}(V-U))$.

Let us define

$H[U, V]=B[U, ¥zeta^{2}(V-U)]-B[¥zeta U, ¥zeta(V-U)]$ .

Then

$H[U, V]=¥sum_{i}[(¥nabla u_{i}, ¥zeta(v_{i}-u_{i})¥nabla¥zeta)-(u_{t}¥nabla¥zeta, ¥nabla(¥zeta(v_{i}-u_{i})))]$,

and also,

132 K. HAYASIDA and H. NAGASE

(5.1) $B[¥zeta U, ¥zeta(V-U)]¥geqq(F, ¥zeta^{2}(V-U))-H[U, V]$ for all $ V¥in$ K.

Here we note that $U¥in¥tilde{¥mathrm{K}}$ in view of Theorem 1. Let $J_{h}^{-1}$ be the Jacobian of theinverse mapping $¥Phi_{h}^{-1}$, where $¥Phi_{h}$ is the mapping defined in Section 3. We write $J_{h}=$

$1+hJ_{1}$ and $J_{h}^{-1}=1+hJ_{2}$ . More precisely, $dy=J_{h}(x)dx$, $dx=J_{h}^{-l}(y)dy$, $J_{h}(x)=1+$

$hJ_{1}(x)$ , $J_{h}^{-1}(y)=1+hJ_{2}(y)$ and $J_{h}(x)J_{h}^{-1}(y)=1$ . It is obvious that $J_{1}$ , $J_{2}$ are boundedin $¥{x_{n}¥geqq 0¥}$ and

(5.2) $J_{1}+J_{2}=O(h)$ with $y=¥Phi_{h}(x)$ $(h¥rightarrow 0)$ .

From now on we put $w_{i}=¥zeta u_{i}$ for the sake of brevity. We observe that

$(¥nabla w_{i}, ¥nabla S_{h}w_{i})_{x}=(¥nabla w_{i}, S_{h}¥nabla w_{i})_{x}+h(¥nabla w_{i}, F_{h}(¥nabla w_{i}))_{x}$

and

$(¥nabla w_{i}, S_{h}¥nabla w_{i})_{x}=(J_{h}^{-1}T_{h}¥nabla w_{i}, ¥nabla w_{i})_{y}$.

Hence

$(¥nabla T_{h}w_{i}, ¥nabla w_{i})_{y}=(¥nabla w_{i}, ¥nabla S_{h}w_{i})_{x}-h(J_{2}T_{h}¥nabla w_{i}, ¥nabla w_{i})_{y}$

$-h(¥nabla w_{i}, F_{h}(¥nabla w_{i}))_{x}+h(G_{h}(¥nabla w_{i}), ¥nabla w_{i})_{y}$.

Moreover we see that

$(¥nabla T_{h}w_{i}, ¥nabla T_{h}w_{i})_{y}=(T_{h}¥nabla w_{i}, T_{h}¥nabla w_{i})_{y}+¥mathit{2}h(T_{h}¥nabla w_{i}, G_{h}(¥nabla w_{i}))_{y}$

$+h^{2}(G_{h}(¥nabla w_{i}), G_{h}(¥nabla w_{i}))_{y}$ ,$(T_{h}¥nabla w_{i}, T_{h}¥nabla w_{i})_{y}=(¥nabla w_{i}, ¥nabla w_{i})_{x}+h(J_{1}¥nabla w_{i}, ¥nabla w_{i})_{x}$ .

Consequently the following equality holds:

$(¥nabla Q_{h}w_{i}, ¥nabla Q_{h}w_{i})_{y}=h^{-2}[(¥nabla T_{h}w_{i}, ¥nabla T_{h}w_{i})_{y}-(¥nabla T_{h}w_{i}, ¥nabla w_{i})_{y}$

$+(¥nabla w_{i}, ¥nabla(w_{i}-T_{h}w_{i}))_{y}]$

$=h^{-2}[(¥nabla w_{i}, ¥nabla(w_{i}-S_{h}w_{i}))_{x}+(¥nabla w_{i}, ¥nabla(w_{i}-T_{h}w_{i}))_{y}]$

$+h^{-1}[¥mathit{2}(T_{h}¥nabla w_{i}, G_{h}(¥nabla w_{i}))_{y}+(J_{1}¥nabla w_{i}, ¥nabla w_{i})_{x}$

$+(J_{2}T_{h}¥nabla w_{i}, ¥nabla w_{i})_{y}+(¥nabla w_{i}, F_{h}(¥nabla w_{i}))_{x}$

$-(G_{h}(¥nabla w_{i}), ¥nabla w_{i})_{y}]+(G_{h}(¥nabla w_{i}), G_{h}(¥nabla w_{i}))_{y}$ .

This is rewritten in the form

(5.3) $(¥nabla Q_{h}w_{i}, ¥nabla Q_{h}w_{i})_{y}=h^{-2}[I_{1}+I_{2}]+h^{-1}[¥mathit{2}I_{3}+I_{4}+I_{5}+I_{6}-I_{7}]+I_{8}$.

Now we shall prove the following inequality

(5.4) $(¥nabla Q_{h}w_{i}, ¥nabla Q_{h}w_{i})_{y}¥leqq C[||¥tilde{¥rho}^{1-1/a}¥nabla U||(||¥tilde{¥rho}F||+||¥tilde{¥rho}^{1-1/a}¥nabla U||+||¥nabla Q_{h}(¥zeta U||_{y})$

$+||¥tilde{¥rho}F||¥cdot||¥nabla Q_{h}(¥zeta U)||_{y}]$ .

On Systems of Variational Inequalities 133

To do this, we shall first prove

(5.5) $h^{-2}[I_{1}+I_{2}]¥leqq ¥mathrm{t}¥mathrm{h}¥mathrm{e}$ right-hand side of (5.4).

Let $¥tilde{¥zeta}¥in C_{0}^{¥infty}(¥{r<R¥})$ be such that $0¥leqq¥tilde{¥zeta}¥leqq 1$ and $¥tilde{¥zeta}=1$ on the support of $¥zeta$ . ByLemma 4, $¥tilde{¥zeta}S_{h}U¥in$ K. (5. 1) yields

$B[¥zeta U, ¥zeta(S_{h}U-U)]_{x}¥geqq(F, ¥zeta^{2}(S_{h}U-U))_{x}-H[U,¥tilde{¥zeta}S_{h}U]_{x}^{4)}$.

Hence

$I_{1}¥leqq(f_{i}, ¥zeta(w_{i}-¥zeta S_{h}u_{i}))_{x}+H[U,¥tilde{¥zeta}S_{h}U]_{x}$

(5.6)$-(¥nabla w_{i}, ¥nabla(S_{h}w_{i}-¥zeta S_{h}u_{i}))_{x}+(c_{ij}w_{j}, ¥zeta S_{h}u_{i}-w_{i})_{x}$ ,

where $F=(f, --,f_{N})$ . Amost similarly we have

$I_{2}¥leqq(f_{i}, ¥zeta(w_{i}-¥zeta T_{h}u_{i}))_{y}+H[U,¥tilde{¥zeta}T_{h}U]_{y}$

(5.7)$-(¥nabla w_{i}, ¥nabla(T_{h}w_{i}-¥zeta T_{h}u_{i}))_{y}+(c_{ij}w_{j}, ¥zeta T_{h}u_{i}-w_{i})_{y}$.

We calculate the sum of each first term on the right-hand sides of (5.6) and (5.7).We see that

$(f_{i}. ¥zeta(w_{i}-¥zeta S_{h}u_{i}))_{x}+(f_{i}, ¥zeta(w_{i}-¥zeta T_{h}u_{i}))_{y}$

$=(f_{i}, ¥zeta(w_{i}-S_{h}w_{i}))_{x}+(f_{i}, ¥zeta(S_{h}¥zeta-¥zeta)S_{h}u_{i})_{x}$

$+(f_{i}, ¥zeta(w_{i}-T_{h}w_{i}))_{y}+(f_{i}, ¥zeta(T_{h}¥zeta-¥zeta)T_{h}u_{i})_{y}$

$=(f_{i}, ¥zeta(w_{i}-S_{h}w_{i}))_{x}+(f_{i}, ¥zeta(S_{h}¥zeta-¥zeta)S_{h}u_{i})_{x}$

$+(S_{h}f_{i}, S_{h}¥zeta¥cdot(S_{h}w_{i}-w_{i}))_{x}+(S_{h}f_{i}, S_{h}¥zeta¥cdot(¥zeta-S_{h}¥zeta)u_{i})_{x}$

$+h[(J_{1}S_{h}f_{i}, S_{h}¥zeta¥cdot(S_{h}w_{i}-w_{i}))_{x}+(J_{1}S_{h}f_{i}, S_{h}¥zeta¥cdot(¥zeta-S_{h}¥zeta)u_{i})_{x}]$,

$ f_{i}¥zeta-S_{h}f_{i}¥cdot S_{h}¥zeta=f_{i}(¥zeta-S_{h}¥zeta)+(f_{i}-S_{h}f_{i})S_{h}¥zeta$ ,

and

$f_{i}¥zeta S_{h}u_{i}-S_{h}f_{i}¥cdot S_{h}¥zeta¥cdot u_{i}$

$=(f_{i}¥zeta-S_{h}f_{i}¥cdot S_{h}¥zeta)S_{h}u_{i}+S_{h}f_{i}¥cdot S_{h}¥zeta¥cdot(S_{h}u_{i}-u_{i})$ .

Thus

$h^{-2}|(f_{i}, ¥zeta(w_{i}-¥zeta S_{h}u_{i}))_{x}+(f_{i}, ¥zeta(w_{i}-¥zeta T_{h}u_{i}))_{y}|$

$¥leqq|(f_{i}, P_{h}¥zeta¥cdot P_{h}w_{i})_{x}|+|(P_{h}f_{i}, S_{h}¥zeta¥cdot P_{h}w_{i})_{x}|+|(f_{i}, (P_{h}¥zeta)^{2}S_{h}u_{i})_{x}|$

(5.8)$+|(P_{h}f_{i}, S_{h}¥zeta¥cdot P_{h}¥zeta¥cdot S_{h}u_{i})_{x}|+|(S_{h}f_{i}, S_{h}¥zeta¥cdot P_{h}¥zeta¥cdot P_{h}u_{i})_{x}|$

$+|(J_{1}S_{h}f_{i}, S_{h}¥zeta¥cdot P_{h}w_{i})_{x}|+|(J_{l}S_{h}f_{i}, S_{h}¥zeta¥cdot P_{h}¥zeta¥cdot u_{i})_{x}|$ .

4) The notation $B[, ]_{x}$ , $H[, ]_{x}$ means the forms with respect to variables $x$ . The replace-ment of $x$ with $y$ is similar.

134 K. HAYASIDA and H. NAGASE

We estimate the second term on the right-hand side of (5.8). In general the follow-ing equality holds:

(5.9) $(P_{h}f, g)_{x}=-(S_{h}f, P_{h}g)_{x}+(J_{2}f, g)_{y}$ .

Hence from (3.4) and Lemma 5

$|(P_{h}f_{i}, S_{h}¥zeta¥cdot P_{h}w_{t})_{x}|¥leqq C||¥tilde{¥rho}f_{i}||(||¥tilde{¥rho}^{-1}P_{h}(S_{h}¥zeta¥cdot P_{h}w_{i})||_{x}+||¥tilde{¥rho}^{-1}P_{h}w_{i}||_{x})$

$¥leqq C||¥tilde{¥rho}f_{i}||(||¥partial_{x_{n-1}}(S_{h}¥zeta¥cdot P_{h}w_{i})||_{x}+||¥nabla w_{i}||)$.

On the other hand we have from (3.2)

$|¥partial_{x_{n-1}}(S_{h}¥zeta¥cdot P_{h}w_{i})|¥leqq C|¥nabla_{y}(¥zeta Q_{h}w_{i})|$

and by Poincare’ $¥mathrm{s}$ inequality

$||Q_{h}w_{i}||¥leqq C||¥nabla(Q_{h}w_{i})||_{y}$ .

Consequently we get

$|(P_{h}f_{i}, S_{h}¥zeta¥cdot P_{h}w_{i})|¥leqq the$ right-hand side of (5.4).

We choose the preceding $¥zeta$ in such a for $¥mathrm{m}$ that $¥zeta(x)=¥zeta_{1}(¥chi_{n-1}, ¥chi_{n})¥cdot¥zeta_{2}(x^{¥prime/})$. Then$P_{h}¥zeta=¥zeta_{2}P_{h}¥zeta_{1}$ , which implies that $P_{h}¥zeta=0$ in a neighborhood of the origin. Fromthis the other terms on the right-hand side of (5.8) can be estimated more easily.Therefore it follows that

$h^{-2}|(f_{i}, ¥zeta(w_{i}-¥zeta S_{h}u_{i}))_{x}+(f_{i}, ¥zeta(w_{i}-¥zeta T_{h}u_{i}))_{y}|$

(5. 10)$¥leqq ¥mathrm{t}¥mathrm{h}¥mathrm{e}$ right-hand side of (5.4).

Next we estimate the sum of each $¥mathrm{t}¥mathrm{n}^{1}$$¥mathrm{i}¥mathrm{r}¥mathrm{d}$ term on the right-hand sides of (5.6)and (5.7). Since $¥nabla_{y}=¥mathrm{A}¥nabla_{x}$ for $A=$ $(¥partial_{y_{j}}x_{i})_{i,j=1,n}¥cdots,$ , we see

$(¥nabla w_{i}, ¥nabla(T_{h}w_{i}-¥zeta T_{h}u_{i}))_{y}$

$=(J_{h}S_{h}¥nabla w_{i}, A¥nabla(w_{i}-S_{h}¥zeta¥cdot u_{i}))_{x}$

(5. 11) $=(S_{h}¥nabla w_{i}, ¥nabla(w_{i}-S_{h}¥zeta¥cdot u_{i}))_{x}$

$+h(J_{1}S_{h}¥nabla w_{i}, ¥nabla(w_{i}-S_{h}¥zeta¥cdot u_{i}))_{x}$

$+(J_{h}S_{h}¥nabla w_{i},(¥mathrm{A}-I)¥nabla(w_{i}-S_{h}¥zeta¥cdot u_{i}))_{x}$ ,

where I is the unit matrix. On the other hand

$h^{-2}|(¥nabla w_{i}, ¥nabla(S_{h}w_{i}-¥zeta S_{h}u_{i}))_{x}+(S_{h}¥nabla w_{i}, ¥nabla(w_{i}-S_{h}¥zeta¥cdot u_{i}))_{x}|$

$¥leqq|(P_{h}¥nabla w_{i}, ¥nabla(P_{h}¥zeta¥cdot S_{h}u_{i}))_{x}|+|(S_{h}¥nabla w_{i}, ¥nabla(P_{h}¥zeta¥cdot P_{h}u_{i}))_{x}|$ .

Recalling $P_{h}¥zeta=0$ in a neighborhood of the origin and using Lemmas 3, 5 and (5.9),

On Systems of Variational Inequalities 135

we see that the right-hand side of this inequality is estimated by that of (5.4). By(3.2), $¥tilde{¥rho}^{1/¥alpha-1}¥cdot(A-I)=O(h)(h¥rightarrow 0)$ . Hence the remaining terms of (5. 11) are similarlyestimated. The sum of the remaining terms on the right-hand sides of (5.6) and(5.7) can be estimated more easily. Therefore (5.5) has been proved.

Next we shall estimate $h^{-1}[¥mathit{2}I_{3}+ ---I_{7}]$ with frequent use of Lemmas 3 and 5.From (3.2), we get

$¥partial_{y_{n-1}}¥tilde{¥rho}=(1+h¥partial_{x_{n-1}}¥tilde{¥rho})^{-1}¥partial_{x_{n-1}}¥tilde{¥rho}$,$¥partial_{y_{n}}¥tilde{¥rho}=(1+h¥partial_{x_{n-1}}¥tilde{¥rho})^{-1}¥partial_{x_{n}}¥tilde{¥rho}$.

Hence we have

$¥partial_{x_{n-1}}¥tilde{¥rho}¥cdot S_{h}¥partial_{x_{n-1}}w_{i}-¥partial_{y_{n-1}}¥tilde{¥rho}¥cdot T_{h}¥partial_{y_{n-1}}w_{i}$

$=h(1+h¥partial_{x_{n-1}}¥tilde{¥rho})^{-1}(¥partial_{x_{n-1}}¥tilde{¥rho})^{2}S_{h}¥partial_{x_{n-1}}w_{i}$

$+¥partial_{y_{n-1}}¥tilde{¥rho}¥cdot(S_{h}¥partial_{x_{n-1}}w_{i}-T_{h}¥partial_{y_{n-1}}w_{i})$

and

$¥partial_{x_{n}}¥tilde{¥rho}¥cdot S_{h}¥partial_{x_{n-1}}w_{i}-¥partial_{y_{n}}¥tilde{¥rho}¥cdot T_{h}¥partial_{y_{n-1}}w_{i}$

$=h(1+h¥partial_{x_{n-1}}¥tilde{¥rho})^{-1}¥partial_{x_{n-1}}¥tilde{¥rho}¥cdot¥partial_{x_{n}}¥tilde{¥rho}¥cdot S_{h}¥partial_{x_{n-1}}w_{i}$

$+¥partial_{y_{n}}¥tilde{¥rho}¥cdot(S_{h}¥partial_{x_{n-1}}w_{i}-T_{h}¥partial_{y_{n-1}}w_{i})$ .

On the other hand from (3.3)

$F_{h}(¥nabla w_{i})+G_{h}(¥nabla w_{i})$

$=(0,$ $-$ , 0, $¥partial_{x_{n-1}}¥tilde{¥rho}¥cdot S_{h}¥partial_{x_{n-1}}w_{i}-¥partial_{y_{n-1}}¥tilde{¥rho}¥cdot T_{h}¥partial_{y_{n-1}}w_{i}$ ,$¥partial_{x_{n}}¥tilde{¥rho}¥cdot S_{h}¥partial_{x_{n-1}}w_{i}-¥partial_{y_{n}}¥tilde{¥rho}¥cdot T_{h}¥partial_{y_{n-1}}w_{i})$.

This yields

$|(¥nabla w_{i}, F_{h}(¥nabla w_{i})+(G_{h}(¥nabla w_{i}))(y))_{x}|$

$¥leqq C||¥tilde{¥rho}^{1-1/¥alpha}¥nabla w_{i}||(h||S_{h}¥nabla w_{i}||+||S_{h}¥nabla w_{i}-(T_{h}¥nabla w_{i})(y)||_{x})$ .

Since

$(T_{h}¥nabla w_{i}, G_{h}(¥nabla w_{i}))_{y}$

$=(¥nabla w_{i}, (G_{h}(¥nabla w_{i}))(y))_{x}+h(J_{1}¥nabla w_{i}, (G_{h}(¥nabla w_{i}))(y))_{x}$ ,

we get

$h^{-1}|I_{3}+I_{6}|¥leqq C||¥tilde{¥rho}^{1-1/a}¥nabla w_{i}||(||¥nabla w_{i}||+||Q_{h}¥nabla w_{i}||_{y})$.

Evidently

136 K. HAYASIDA and H. NAGASE

$||Q_{h}¥nabla w_{i}||_{y}¥leqq||¥nabla Q_{h}w_{i}||_{y}+C||¥tilde{¥rho}^{1-1/a}¥nabla w_{i}||$.

Therefore we obtain

$h^{-1}|I_{3}+I_{6}|¥leqq ¥mathrm{t}¥mathrm{h}¥mathrm{e}$ right-hand side of (5.4).

More easily we see that

$h^{-1}|I_{3}-I_{7}|$ , $h^{-1}|I_{4}+I_{5}|$ , $I_{8}$

$¥leqq ¥mathrm{t}¥mathrm{h}¥mathrm{e}$ right-hand side of (5.4),

where we have used (5.2). Combining these inequalities and (5.5), we conclude theproof of (5.4).

From (5.4) we immediately obtain

$||¥nabla Q_{h}(¥zeta U)||_{y}¥leqq C(||¥tilde{¥rho}F||+||¥tilde{¥rho}^{1-1/a}¥nabla U||)$ .

We can take $¥epsilon>0$ so that $¥epsilon-1/4<1-1/¥alpha$ $(<0)$ . For such $¥epsilon$ we choose the realnumbers $¥beta$ and $¥delta$ in Section 2 in such a way that (?1/2)βδ<ε?1/4. Then

$||¥tilde{¥rho}^{1-1/a}¥nabla U||¥leqq C||¥tilde{¥rho}^{1-1/¥alpha}¥nabla U||¥leqq C||¥tilde{¥rho}^{(-1/2)¥beta¥delta}¥nabla U||$

$¥leqq C|||U|||$ .

Applying Theorem 1, we have

$||¥nabla Q_{h}(¥zeta U)||_{y}¥leqq C||¥rho^{3/4}F||$ ,

where we have used the relation $¥tilde{¥rho}¥leqq C¥rho^{a}¥leqq C¥rho^{4/5}$ . Hence there are a subsequence$¥{h_{¥nu}¥}$ and a vector function $V¥in[H_{(0)}^{1}(¥Sigma)]^{N}$ such that $h_{¥nu}¥rightarrow+0$ $(¥nu¥rightarrow¥infty)$ and $¥{Q_{h_{¥mu}}(¥zeta U)¥}$

converges weakly to $V$ in $[H_{(0)}^{1}(¥Sigma)]^{N}$ as $ y¥rightarrow¥infty$ . At the same time, from Lemma 6,$¥{Q_{h_{p}}(¥zeta U)¥}$ converges strongly to $-¥tilde{¥rho}¥partial_{y_{n-1}}U$ in $[L^{2}(¥Sigma)]^{N}$ as $¥nu¥rightarrow¥infty$ . Thus $V=-¥tilde{¥rho}$ .

$¥partial_{y_{n-1}}(¥zeta U)$ and

$||¥nabla¥tilde{¥rho}¥partial_{y_{n-1}}(¥zeta U)||_{y}¥leqq C||¥rho^{3/4}F||$ .

On the other hand, we use the usual parallel translation for $¥mathrm{y}^{¥prime¥prime}$-direction. Since$¥tilde{¥rho}$ is independent of $y^{¥prime¥prime}$ , we have

$||¥tilde{¥rho}¥nabla(¥partial_{y^{¥prime¥prime}}(¥zeta U)||_{y}¥leqq C||¥tilde{¥rho}F||¥leqq C||¥rho^{3/4}F||$.

Accordingly we obtain

(5. 12) $||¥tilde{¥rho}¥partial¥partial_{x_{i}}(¥zeta U)||_{x}¥leqq C||¥rho^{3/4}F||$

for $i=1$,?, $n-l$ . From (2.4) it follows that $-¥Delta u_{i}+¥sum_{j}c_{ij}u_{j}=f_{i}$ in $¥Sigma$ . Hence

(5. 12) is also valid for $i=n$ .

On Systems of Variational Inequalities 137

Let $¥epsilon_{2}>0$ be given. In advance we take $¥zeta$ in such a way that $¥zeta=1$ in $¥Sigma_{¥epsilon_{2}}$ .

Let $t$ be any real number with $1<t<10/9$ . Then there is a real number $¥alpha$ with$¥mathit{4}/5<¥alpha<1$ such that $t<¥mathit{2}/(1+¥alpha)$, If we put $p=¥mathit{2}/t$ and $q=¥mathit{2}/(¥mathit{2}-t)$ , then $p$ , $q>1$

and $p^{-1}+q^{-1}=1$ . It is evident that $¥rho^{a}¥leqq C_{¥epsilon_{2}}¥tilde{¥rho}$ in $¥Sigma_{¥epsilon_{2}}$ . Hence by Holder’ $¥mathrm{s}$ inequalitywe have

$¥int_{¥Sigma_{¥mathit{6}2}}(¥partial^{2}u_{i})^{t}dx¥leqq(¥int_{¥Sigma ¥mathrm{e}¥mathrm{z}}¥tilde{¥rho}^{-tq}dx)^{1/q}(¥int_{¥Sigma¥epsilon}2(¥tilde{¥rho}¥partial^{2}u_{i})^{tp}dx)^{1/p}$

(5. 13)$¥leqq C_{¥epsilon_{2}}(¥int_{0}^{R}¥rho^{1-¥alpha tq}d¥rho)^{1/q}(¥int_{¥Sigma_{62}}(¥tilde{¥rho}¥partial^{2}u_{t})^{2}dx)^{1/p}$

Here $¥alpha tq<¥mathit{2}$ , because $t<¥mathit{2}/(1+¥alpha)$ . Thus, the first integral on the right-hand side of(5. 13) is finite, and, in view of (5. 12), the proof of Theorem 2 is completed. Q.E.D.

References

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138 K. HAYASIDA and H. NAGASE

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nuna adreso:Kazuya HAYASIDADepartment of MathematicsFaculty of ScienceKanazawa UniversityKanazawa, 920JapanHaruo NAGASESuzuka College ofTechnologySuzuka, 510-02Japan

(Ricevita la 1-an de februaro, 1984)(Reviziita la 27-an de aprilo, 1984)(Reviziita la 21-an de julio, 1984)