数理解析研究所講究録
第 1875 巻 2014 年 1-18

1

On the -Boundedness of Solution Operators
for the weak Dirichlet-Neumann Problem$\mathcal{R}$Yoshihiro SHIBATA*

Department of Mathematics and Research Institute of Science and Engineering,
Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan.
e-mail address: yshibata@waseda.jp

1

Introduction

We consider an autonomous evolution equation:$Bu=g$ for $t>0,$$u_{t}-Au=f,$$u|_{t=0}=u_{0}$

.

(1.1)

Here, and $B$ are some linear operators and $Bu=g$ represents a non-homogeneous boundary
condition. Through the Laplace transform with respect to time variable, we have the corresponding generalized resolvent problem:$A$$\lambda v-Av=f, Bv=g$ .

(1.2)

Here, the reason why we call (1.2) a generalized resolvent problem is that we consider nonhomogeneous boundary condition. Let be represented by $v=R(\lambda)(f, g)$ with some solution
operatorto (1.2). When $g=0$ , ifsatisfies the condition of Hille-Yosida type, thengenerates a continuous semigroup, which gives us a unique solution to the Cauchy problem:$v$$R(\lambda)$$R(\lambda)$$A$$u_{t}-Au=0,$ $Bu=0$

for $t>0,$$u|_{t=0}=u_{0}.$Moreover, if
satisfies suitable multiplier conditions, the Laplace inverse transform of$R(\lambda)(f, g)$ gives us asolution to the non-homogeneous initial-boundary value problem:$R(\lambda)$$u_{t}-Au=f,$ $Bu=g$

for$t\in \mathbb{R}.$In addition, the condition and vanish for $t<0$ implies that also vanishes for $t<0$ , which
especially means that $u|_{t=0}=0$ . Combining these two results, we can solve (1.1). In fact,
Sakamoto [6] proved the unique existence of solutions to the initial-boundary mixed problem
for the general hyperbolic equations with the boundary condition satisfying uniform Lopatinski
conditions in rather general domains . Since her problem is hyperbolic, she considered the
problem the in theframework. Therefore, the boundedness of the operator norm of
implies the unique existence and suitable estimates of solutions to the evolution equations bymeans of the Plancherel formula.
$f$

$u$

$g$

$*$

$L_{2}$“Partially supported by JST CREST and JSPS Grant-in-aid for Scientific Research (S) $\neq 24224004$*Kreiss [5] treated the hyperbolic system case, but he proved a priori estimates only.

$R(\lambda)$

2
We wanted to extend Sakamoto’s approach to theframework for a long time and the Weis

$L_{p}$-boundedness $(1<p<\infty)$ of the operator valued Fourier multiplier theo-

theorem [10] of therem enables us to extend Sakamoto’s approach at least to the parabolic type equations including
Stokes equations for both of the compressible and incompressible fluid flows (cf. Enomoto andimplies not
Shibata [4] and Shibata [8] . In fact, the -boundedness of solution operator
maximal regularity by means of the
only the generation of analytic semigroup but also
Weis theorem.
In this paper, we explain how to prove the -boundedness of solution operators by treating
the following generalized resolvent problem for the weak Dirichlet-Neumann problem:$L_{p}$$R(\lambda)$$\mathcal{R}$$)$$L_{p}-L_{q}$$\mathcal{R}$$(\lambda u, \varphi)_{\Omega}+(\nabla u, \nabla\varphi)_{\Omega}=-(f, \nabla\varphi)_{\Omega}+(g, \varphi)_{\Omega}+<h_{n},$$\varphi>\Gamma_{2}$for any$\varphi\in W_{q,\Gamma_{1}}^{1}(\Omega)$

,

(1.3)

with boundary
domain in
subject to $u=h_{d}$ on . Here, is a uniform. For any domain in , we set $(a, b)_{G}= \int_{G}a(x)b(x)dx$ . WhenWe assume that
, we set $<a,$ $b> \Gamma=\int_{\Gamma}a(x)b(x)d\sigma.$
hypersurface with surface element
is adenotes the functional space:
Before stating our main results, first we introduce the Weis operator valued Fourier multiplier
theorem. For this purpose, we introduce the notion of boundedness of operator families.
$C^{1}$$\Omega$$\Gamma_{1}$$G$$\Gamma_{1}\cap\Gamma_{2}=\emptyset$$\Gamma$$C^{1}$$\mathbb{R}^{N}(N\geq 2)$$\Gamma_{1}\cup\Gamma_{2}.$$\mathbb{R}^{N^{1}}$$d\sigma$$W_{q,\Gamma_{1}}^{1}(\Omega)$$\{\varphi\in W_{q}^{1}(\Omega)|\varphi|r_{1}=0\}.$$\mathcal{R}$denotes the set of all bounded
Definition 1.1. Let $X$ and be two Banach spaces andbounded, if
is calledfamily of operators
linear operators from $X$ into $Y.$
$p\in[1,\infty)$numberany
natural
that
for
such
$C>0$
and
there exist constantsof independent, symmetric, $\{-1,1\}$ -valued random
and sequences
variables on $[0,1]$ there holds the inequality:$Y$$\mathcal{L}(X, Y)$$A$$\mathcal{R}$$\mathcal{T}\subset \mathcal{L}(X, Y)$$n,$$\{T_{j}\}_{j=1}^{n}\subset \mathcal{T},$$\{r_{j}(u)\}_{j=1}^{n}$$\{x_{j}\}_{j=1}^{n}\subset X$$\{\int_{0}^{1}\Vert\sum_{j=1}^{n}r_{j}(u)T_{j}x_{j}\Vert_{Y}^{p}du\}^{\frac{1}{p}}\leq C\{\int_{0}^{1}\Vert\sum_{j=1}^{n}r_{j}(u)x_{j}\Vert_{X}^{p}du\}^{\frac{1}{p}}$The smallest such$C$is called$\mathcal{R}$-bound of$\mathcal{T}$, which is denoted by$\mathcal{R}_{\mathcal{L}(X,Y)}(\mathcal{T})$.

functions having compact supports
be the set of all $X$ valued
andLet$X$ valued functions, respectively while $S’(\mathbb{R}, X)=$
decreasing
space
rapidlyofand the Schwartz, we define
being the set of all complex numbers. Given $M\in L_{1}$ ,locby:the operator$\mathcal{L}(\mathcal{S}(\mathbb{R}, \mathbb{C}), X),$$C^{\infty}$$S(\mathbb{R}, X)$$\mathcal{D}(\mathbb{R}, X)$$(\mathbb{R}\backslash \{0\}, X)$$\mathbb{C}$$T_{M}$$\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}, X)arrow S’(\mathbb{R}, Y)$$T_{M}\phi=\mathcal{F}^{-1}[M\mathcal{F}[\phi]], (\mathcal{F}[\phi]\in \mathcal{D}(\mathbb{R}, X)$

(1.4)

,

denote the Fourier transform and the Fourier inverse transform, respectively.
where andfollowing
theorem is obtained by Weis [10].
The$\mathcal{F}$$\mathcal{F}^{-1}$

Theorem 1.2. Let $X$ and $Y$ be two $UMD$ Banach spaces and $1<p<\infty$ . Let $M$ be a function
such thatin$C^{1}(\mathbb{R}\backslash \{0\}, \mathcal{L}(X, Y))$$\mathcal{R}_{\mathcal{L}(X,Y)}(\{(\tau\frac{d}{d\tau})^{\ell}M(\tau)|\tau\in \mathbb{R}\backslash \{0\}\}\leq\kappa<\infty (\ell=0,1)$

with some constantoperator from$\kappa$. Then, the operator
defined in (1.4) is extended to a bounded linear. Moreover, denoting this extension by , we haveinto$T_{M}$$L_{p}(\mathbb{R}, Y)$$L_{p}(\mathbb{R}, X)$$\Vert T_{M}\Vert_{\mathcal{L}(L_{p}(\mathbb{R},X),L_{p}(\mathbb{R},Y))}\leq C\kappa$

for some positive constant$C$depending on $p,$ $X$ and $Y.$

$T_{M}$

3
Remark 1.3. For the definition of UMD space, we refer to a book due to Amann [1]. And, for$1<q<\infty$ , Lebesgue spaceand Sobolev space
are both UMD spaces.
$L_{q}(\Omega)$$W_{q}^{7n}(\Omega)$

Secondly, we introduce the definition of uniform

Definition 1.4. Let$C^{1}$

domains.

be a domain inwith boundary. We say that is a uniform
domain if there exist positive constants
and $K$ such that for any $x_{0}=(x_{01}, \ldots, x_{0N})\in\partial\Omega$
there exist a coordinate number and afunction $h(x’)(x’=(x_{1}, \ldots, x_{j-1}, x_{j+1}, \ldots, x_{N}))$defined on(x\’o) with $x_{0}’=(x_{01}, \ldots, x_{0j-1}, x_{0j+1}, \ldots, x_{0N})$ andsuch that$\mathbb{R}^{N}$$\Omega$$\alpha,$$\partial\Omega$$C^{1}$$\Omega$$\beta$$C^{1}$$j$$B_{\alpha}’$$\Vert h\Vert_{W_{\infty}^{1}(B_{\alpha}’(x_{\acute{0}}))}\leq K$,$\Omega\cap B_{\beta}(x_{0})=\{x\in \mathbb{R}^{N}|x_{j}>h(x’)(x’\in B_{\alpha}’(x_{0}’))\}\cap B_{\beta}(x_{0})$.

$\partial\Omega\cap B_{\beta}(x_{0})=\{x\in \mathbb{R}^{N}|x_{j}=h(x’)(x’\in B_{\alpha}’(x_{0}’))\}\cap B_{\beta}(x_{0})$Here, $B_{\alpha}’(x_{0}’)=\{x’\in \mathbb{R}^{N-1}||x’-x_{0}’|<\alpha\},$$B_{\beta}(x_{0})=\{x\in \mathbb{R}^{N}||x-x_{0}|<\beta\}.$Thirdly, we recall some further symbols used throughout the paper. For any multi-index, we set. We write $\nabla u=(D_{1}u, \ldots, D_{N}u)$ with $D_{j}=$$L_{q}(G)$ and. For any domain in
denote the usual Lebesgue space andSobolev space, respectively, whileanddenote their norms, respectively. For$X$a Banach space with norm
denotes the -product space of $X$ , while
denotesalso the norm offor the sake of simplicity. For a domain
inbeing the set of allcomplex number, Anal $(U, X)$ denotes the set of all $X$ -valued holomorphic functions defined onU.
andare sets for the resolvent parameter defined by$\alpha=(\alpha_{1}, \ldots, \alpha_{N})$$D^{\alpha}h=\partial_{1}^{\alpha}1\ldots\partial_{N}^{\alpha_{N}}h$$G$$\partial/\partial x_{j}$$\mathbb{R}^{N},$$W_{q}^{m}(G)$$\Vert\cdot\Vert_{L_{q}(G)}$$\Vert\cdot\Vert_{X},$$\Vert\cdot\Vert_{W_{q}^{m}(G)}$$X^{d}$$d$$\Vert\cdot\Vert_{X}$$X^{d}$$\Sigma_{\epsilon}$$U$$\mathbb{C}$$\mathbb{C},$$\lambda$$\Sigma_{\epsilon,\lambda_{0}}$$\Sigma_{\epsilon}=\{\lambda\in \mathbb{C}\backslash \{0\}||\arg\lambda|\leq\pi-\epsilon\}, \Sigma_{\epsilon,\lambda_{0}}=\{\lambda\in\Sigma_{\epsilon}||\lambda|\geq\lambda_{0}\}.$The letter $C$ denotes generic constants andmeans that the constanton. The values of constants andmay change from line to line.
The following theorem is our main result in this paper.
$C_{a,b,c},\cdot\cdot$

$a,$

$b,$

$c,$

$C$

$\cdots$$C_{a,b,c},\cdots$

Theorem 1.5. Let $1<q<\infty$ and $0<\epsilon<\pi/2$ . Assume that is a uniform
and the boundary of consists of two
hypersurfaces
andwith$\Omega$$\mathbb{R}^{N}$$C^{1}$$\Omega$$X_{q}(\Omega)$and$\mathcal{X}_{q}(\Omega)$

depends$C_{a,b,c},\cdots$$\Gamma_{1}$$C^{1}$

domain in. Let$\Gamma_{1}\cap\Gamma_{2}=\emptyset$$\Gamma_{2}$be functional spaces defined by$X_{q}(\Omega)=\{(f, g, h_{d}, h_{n})|f\in L_{q}(\Omega)^{N}, g\in L_{q}(\Omega), h_{d}, h_{n}\in W_{q}^{1}(\Omega)\},$$\mathcal{X}_{q}(\Omega)=\{F=(F_{1}, \ldots, F_{6})|F_{1}, F_{4}, F_{6}\in L_{q}(\Omega)^{N}, F_{2}, F_{3}, F_{5}\in L_{q}(\Omega)\}.$

Then, there exists a
and an operator family
Anal
such
$(f,
g,
h_{d},
h_{n})\in
X_{q}(\Omega)u=\mathcal{A}(\lambda)F_{\lambda}(f,g, h_{d}, h_{n})$that for any
andis a unique solution
to (1.3), where we have set.
Moreover, there exists a constant such that$\lambda_{0}>0$$\mathcal{A}(\lambda)\in$$(\Sigma_{\epsilon,\lambda_{0}}, \mathcal{L}(\mathcal{X}_{q}(\Omega), W_{q}^{1}(\Omega)))$$\lambda\in\Sigma_{\epsilon,\lambda_{0}}$$F_{\lambda}(f, g, h_{d}, h_{n})=(f, \lambda^{-1/2}, \lambda^{1/2}h_{d}, \nabla h_{d}, h_{n}, \lambda^{-1/2}\nabla h_{n})$$\kappa$$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{q}(\Omega),L_{q}(\Omega)^{N+1})}(\{(\lambda\frac{d}{d\lambda})^{\ell}(\lambda^{1/2}, \nabla)\mathcal{A}(\lambda)|\lambda\in\Sigma_{\epsilon,\lambda_{0}}\})\leq\kappa (\ell=0,1)$.

Finally, we discuss the generation of analytic semigroup and maximal
regularity results
related to (1.3) as an application of Theorem 1.5. Let
be the dual space of.
It follow from the Hahn-Banach theorem that for any$f\in
L_{q}(\Omega)^{N}$
there exist
andsuch that$L_{p}-L_{q}$$W_{q,\Gamma_{1}}^{-1}(\Omega)$$W_{q,\Gamma_{1}}^{1}(\Omega)$$F\in W_{q,\Gamma_{1}}^{-1}(\Omega)$$g\in L_{q}(\Omega)$$F(\varphi)=-(f, \nabla\varphi)_{\Omega}+(g, \varphi)_{\Omega}$

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

.

(1.5)

4
Let

$A$

be an operator defined byAu

for any$(\varphi)=(\nabla u, \nabla\varphi)_{\Omega}$and$u\in W_{q,\Gamma_{1}}^{1}(\Omega)$$\varphi\in W_{q,\Gamma_{1}}^{1}(\Omega)$.

It follows from (1.3) and (1.5) that the resolvent problem: $\lambda u-Au=F$ is represented by$(\lambda u, \varphi)_{\Omega}+(\nabla u, \nabla\varphi)_{\Omega}=-(f, \nabla\varphi)_{\Omega}+(g, \varphi)_{\Omega}$

for any(1.6)$\varphi\in W_{q,\Gamma_{1}}^{1}(\Omega)$

subject to $u=0$ on . Since -boundedness implies boundedness, by Theorem 1.5 we see thatsatisfying the estimate:
the equation (1.6) admits a unique solution$\mathcal{R}$$\Gamma_{1}$$\in W_{q,\Gamma_{1}}^{1}(\Omega)$(1.7)$\Vert(\lambda^{1/2}u, \nabla u)\Vert_{L_{q}(\Omega)}\leq\kappa\Vert F\Vert_{W_{q,\Gamma_{1}}^{-1}(\Omega)}$

for any
we have$\lambda\in\Sigma_{\epsilon,\lambda_{0}}$and$F\in W_{q,\Gamma_{1}}^{-1}(\Omega)$. Here, we may aesume that$\lambda_{0}\geq 1$. In addition, by (1.6)$|(\lambda u, \varphi)_{\Omega}|\leq\Vert\nabla u\Vert_{L_{q}(\Omega)}\Vert\nabla\varphi\Vert_{L_{q’}(\Omega)}+\Vert f\Vert_{L_{q}(\Omega)}\Vert\nabla\varphi\Vert_{L_{q},(\Omega)}+|\downarrow g\Vert_{L_{q}(\Omega)}\Vert\varphi\Vert_{L_{q},(\Omega)}$$\leq C\kappa\Vert F\Vert_{W_{q,\Gamma_{1}}^{-1}(\Omega)}\Vert\varphi\Vert_{W_{q,\Gamma_{1}}^{1}(\Omega)},$which furnises that$\Vert\lambda u\Vert_{W_{q,\Gamma_{1}}^{-1}(\Omega)}\leq C\kappa\Vert F\Vert_{W_{q,\Gamma_{1}}^{-1}(\Omega)}$

Therefore,

generates an analytic semigroup$A$$\{T(t)\}_{t\geq 0}$

on

.

(1.8)

$W_{q,\Gamma_{1}}^{-1}(\Omega)$satisfying the estimate:(1.9)$\Vert T(t)F\Vert_{W_{q,\Gamma_{1}}^{-1}(\Omega)}+\Vert(t^{1/2}T(t)F, tT(t)F)\Vert_{L_{q}(\Omega)}\leq Ce^{\lambda_{0}t}\Vert F\Vert_{W_{q,\Gamma_{1}}^{-1}(\Omega)}$

for any $t>0$ with some constant $C.$
Next, we consider the evolution equation:$u_{t}-Au=F$

for any$t\in \mathbb{R}$in$\Omega,$. Applying the Laplace transform to (1.10), we have$(\lambda\hat{u}, \varphi)_{\Omega}+(\nabla\hat{u}, \nabla\varphi)_{\Omega}=-(\hat{f}, \nabla\varphi)_{\Omega}+(\hat{g}, \varphi)_{\Omega}$

subject to(1.10)$u|r_{1}=h_{d}|_{\Gamma_{1}}$$\hat{u}=\hat{h}_{d}$on$\Gamma_{1}$. Using the operator$\mathcal{A}(\lambda)$

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

given in Theorem 1.5,$\hat{u}$(1.11)

is represented by$\hat{u}=\mathcal{A}(\lambda)(\hat{f}, \lambda^{-1/2}\hat{g}, \lambda^{1/2}\hat{h}_{d}, \nabla\hat{h}_{d}, 0,0)$

. Let
with
to (1.10) is represented by$\lambda=\gamma+i\tau\in \mathbb{C}$$\mathcal{L}^{-1}$

be the inverse Laplace transform, and then a unique solution.

$u(t)=\mathcal{L}^{-1}[\mathcal{A}(\lambda)(f, \lambda^{-1/2}\hat{g}, \lambda^{1/2}\hat{h}_{d}, \nabla\hat{h}_{d}, 0,0)](t)$Therefore, by Theorem 1.2 we have$\Vert e^{-\gamma t}u_{t}\Vert_{L_{p}(\mathbb{R},W_{q,\Gamma_{1}}^{-1}(\Omega))}+\Vert e^{-\gamma t}(\Lambda_{\gamma}^{1/2}u, \nabla u)\Vert_{L_{p}(\mathbb{R},L_{q}(\Omega))}$$\leq C\kappa\Vert e^{-\gamma t}(f, \Lambda_{\gamma}^{-1/2}g, \Lambda_{\gamma}^{1/2}h_{d}, \nabla h_{d})\Vert_{L_{p}(\mathbb{R},L_{q}(\Omega))}$for any$\gamma\geq\lambda_{0}$. Namely, the operator$A$

has maximal$L_{p}-L_{q}$regularity. Here, we have set$\Vert e^{-\gamma t}v\Vert_{L_{p}(\mathbb{R},X)}=(\int_{-\infty}^{\infty}(e^{-\gamma t}\Vert v(t)\Vert_{X})^{p}dt)^{1/p},$$\Vert e^{-\gamma t}\Lambda_{\gamma}^{s}v\Vert_{L_{p}(\mathbb{R},X)}=(\int_{-\infty}^{\infty}(e^{-\gamma t}\Vert \mathcal{L}^{-1}[\lambda^{s}\hat{v}(\lambda)](t)\Vert_{X})^{p}dt)^{1/p}$

$u$

5
2

Model Problems

2.1

A Model Problem in the whole space$\mathbb{R}^{N}$

Let us consider the problem:$\lambda(u, \varphi)_{\mathbb{R}^{N}}+(\nabla u, \nabla\varphi)_{\mathbb{R}^{N}}=-(f, \nabla\varphi)_{\mathbb{R}^{N}}+(g, \varphi)_{\mathbb{R}^{N}}$for any$\varphi\in W_{q}^{1}(\mathbb{R}^{N})$

.

(2.1)

Instead of (2.1), we consider the equation: $(\lambda-\triangle)u=divf+g$ and then using the Fouriertransform and its inversion formula, we have$u(x)= \mathcal{F}_{\xi}^{-1}[\frac{\mathcal{F}[divf+g](\xi)}{\lambda+|\xi|^{2}}](x)=\sum_{j=1}^{N}\mathcal{F}_{\xi}^{-1}[\frac{i\xi_{j}\hat{f}_{j}(\xi)}{\lambda+|\xi|^{2}}](x)+\mathcal{F}_{\xi}^{-1}[\frac{\hat{g}(\xi)}{\lambda+|\xi|^{2}}](x)$Here and hereafter,
the Fourier inverse transform of$\mathcal{F}[f](\xi)=f(\xi)$

(2.2)

and
denote the Fourier transform of $f(x)$ and, respectively, which are defined exactly by$\mathcal{F}_{\xi}^{-1}[h(\xi)](x)$$h(\xi)$$\mathcal{F}[f](\xi)=f(\xi)=\int_{\mathbb{R}^{N}}e^{-ix\cdot\xi}f(x)dx, \mathcal{F}_{\xi}^{-1}[h(\xi)](x)=\frac{1}{(2\pi)^{N}}\int_{\mathbb{R}^{N}}e^{ix\cdot\xi}h(\xi)d\xi.$

To prove the boundedness of the operators defined by the Fourier transform in
the following lemma due to Enomoto-Shibata [4, Theorem 3.3].
$R$

Theorem 2.1. Let $1<q<\infty$ and let be a set insuch that for any multi-index
depending on and such that$\Lambda$$\Lambda\cross(\mathbb{R}^{N}\backslash \{0\})$$\mathbb{C}$. Let

be a function defined on
there exists a constant$C_{\alpha}$$\Lambda$(2.3)$|\partial_{\xi}^{\alpha}m(\lambda, \xi)|\leq C_{\alpha}|\xi|^{-|\alpha|}$$(\lambda, \xi)\in\Lambda\cross(\mathbb{R}^{N}\backslash \{0\})$Then, the set, we use$m(\lambda, \xi)$$\alpha\in \mathbb{N}_{0}^{N}(\mathbb{N}_{0}=\mathbb{N}\cup\{0\})$$\alpha$for any$\mathbb{R}^{N}$$\{K_{\lambda}|\lambda\in\Lambda\}$is$\mathcal{R}$. Let

be an operator defined by-bounded on
and$K_{\lambda}$$K_{\lambda}f=\overline{J^{-}}_{\xi}-1[m(\lambda, \xi)f(\xi)].$$\mathcal{L}(L_{q}(\mathbb{R}^{n}))$(2.4)$\mathcal{R}_{\mathcal{L}(L_{q}(\mathbb{R}^{N}))}(\{K_{\lambda}|\lambda\in\Lambda\})\leq C_{q,N} \max C_{\alpha}$$|\alpha|\leq N+2$

with some constant$C_{q,N}$

that depends solely on

Since
for anysatisfies the following multiplier conditions:$| \lambda+|\xi|^{2}|\geq 2\sin^{2}\frac{\epsilon}{2}(|\lambda|+|\xi|^{2})$$q$

and $N.$$\lambda\in\Sigma_{\epsilon}$and$\xi\in \mathbb{R}^{N}$, we see easily that$(\lambda+|\xi|^{2})^{-1}$$|\partial_{\xi}^{\alpha}[\lambda(\lambda+|\xi|^{2})^{-1}]|\leq C_{\alpha,\epsilon}(|\lambda|^{1/2}+|\xi|)^{-|\alpha|},$$|\partial_{\xi}^{\alpha}[(\lambda^{1/2}i\xi_{j})(\lambda+|\xi|^{2})^{-1}]|\leq C_{\alpha,\epsilon}(|\lambda|^{1/2}+|\xi|)^{-|\alpha|}$

,

(2.5)

$\partial_{\xi}^{\alpha}[(i\xi_{j}\xi_{k})(\lambda+|\xi|^{2})^{-1}]|\leq C_{\alpha,\epsilon}(|\lambda|^{1/2}+|\xi|)^{-|\alpha|},$

for

$j,$

$k=1,$$\ldots,$$N$

and any$\lambda\in\Sigma_{\epsilon}$and$\xi\in \mathbb{R}^{N}$. Since$\lambda^{\frac{1}{2}}u(x)=\sum_{j=1}^{N}\mathcal{F}_{\xi}^{-1}[\frac{(\lambda^{1/2})i\xi_{j}\hat{f}_{j}(\xi)}{\lambda+|\xi|^{2}}](x)+\overline{J^{-}}_{\xi}-1[\frac{\lambda \mathcal{F}[\lambda^{-\frac{1}{2}}g](\xi)}{\lambda+|\xi|^{2}}](x)$,$\frac{\partial u}{\partial x_{k}}(x)=\sum_{j=1}^{N}\mathcal{F}_{\xi}^{-1}[\frac{(i\xi_{j}\xi_{k})\hat{f}_{j}(\xi)}{\lambda+|\xi|^{2}}](x)+\mathcal{F}_{\xi}^{-1}[\frac{(\lambda^{\frac{1}{2}}i\xi_{k})\mathcal{F}[\lambda^{-\frac{1}{2}}g](\xi)}{\lambda+|\xi|^{2}}](x)$6Therefore, if we define an oprator$U_{0}(\lambda)$

by
(2.6)

$U_{0}( \lambda)(F_{1}, F_{2})=\sum_{j=1}^{N}\mathcal{F}_{\xi}^{-1}[i\xi_{j}\hat{F}_{1j}(\xi)(\lambda+|\xi|^{2})^{-1}](x)+\mathcal{F}_{\xi}^{-1}[\lambda^{1/2}\hat{F}_{2}(\xi)(\lambda+|\xi|^{2})^{-1}](x)$with $F=(F_{11}, \ldots, F_{1N})$ , then we have the following theorem.
Theorem 2.2. Let $1<q<\infty$ and $0<\epsilon<\pi/2$ . For any domain

$G$

in

$\mathbb{R}^{N}$, we set$X_{q0}(G)=\{(f,g)|f\in L_{q}(G)^{N}, g\in L_{q}(G)\},$$\mathcal{X}_{q0}(G)=\{(F_{1}, F_{2})|F_{1}=(F_{11}, \ldots, F_{1N})\in L_{q}(\mathbb{R}^{N}), F_{2}\in L_{q}(\mathbb{R}^{N})\}.$

Let
any

$U_{0}(\lambda)$

, for
Anal
be the operator defined by (2.6). Then,
and $(f, g)\in X_{q}(\mathbb{R}^{N})^{N}u(x)=U_{0}(\lambda)(f, \lambda^{-1/2}g)$ is a unique solution to (2.1), and$(\Sigma_{\epsilon}, \mathcal{L}(\mathcal{X}_{q0}(\mathbb{R}^{N}), W_{q}^{1}(\mathbb{R}^{N})))$$U_{0}(\lambda)\in$$\lambda\in\Sigma_{\epsilon}$(2.7)$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{0}(\mathbb{R}^{N})}, L_{q}(\mathbb{R}^{N})^{N+1})(\{(\lambda\frac{d}{d\lambda})^{\ell}U_{0}(\lambda)|\lambda\in\Sigma_{\epsilon}\})\leq\gamma_{0}$with some constant

2.2

$A$

$\gamma_{0}$depending solely on$\epsilon,$$q$

and $N.$model problem in the half space$\mathbb{R}_{+}^{N}$, Dirichlet condition case.

In this subsection we consider the weak Dirichlet problem in the half-space$(\lambda u, \varphi)_{\mathbb{R}_{+}^{N}}+(\nabla u, \nabla\varphi)_{\mathbb{R}_{+}^{N}}=-(f, \nabla\varphi)_{\mathbb{R}_{+}^{N}}+(g, \varphi)_{\mathbb{R}_{+}^{N}}$for any$\mathbb{R}_{+}^{N}$

:
(2.8)

$\varphi\in W_{q,0}^{1}(\mathbb{R}_{+}^{N})$being the boundary, where $W_{q,0}^{1}(G)=\{u\in W_{q}^{1}(G)|u|_{\partial G}=0\},$subject to $u=h_{d}$ on$\mathbb{R}_{+}^{N}=\{x=(x_{1},\ldots,x_{N})\in\mathbb{R}^{N}|x_{N}>0\}$$\mathbb{R}_{0}^{N}=\{x=(x_{1},\ldots,x_{N})\in\mathbb{R}^{N}|x_{N}=0\}.$, andof $G,$, andand, we may assume thatis dense in
Since
insteadsubject to $u=h_{d}$ on
we consider the strong equation: $(\lambda-\triangle)u=divf+g$ in
odd
and
the
of
extensioneven
thedenoteandondefinedof (2.8). Given function$u(x)$
byis given
extension of to $x_{N}<0$ , respectively. unique solution$\partial G$$\mathbb{R}_{0}^{N}$$f\in C_{0}^{\infty}(\mathbb{R}_{+}^{N})^{N}$$L_{q}(\mathbb{R}_{+}^{N})$$C_{0}^{\infty}(\mathbb{R}_{+}^{N})$$g\in C_{0}^{\infty}(\mathbb{R}_{+}^{N})$$\mathbb{R}_{0}^{N}$$\mathbb{R}_{+}^{N}$$h$$\mathbb{R}_{+}^{N},$$h^{e}$$h^{o}$

$h$

$A$

$h$

$u(x)= \mathcal{F}_{\xi}^{-1}[\frac{\mathcal{F}[(divf)^{0}](\xi)}{\lambda+|\xi|^{2}}](x)+\mathcal{F}_{\xi}^{-1}[\frac{\mathcal{F}[g^{o}](\xi)}{\lambda+|\xi|^{2}}](x)+\mathcal{F}_{\xi’}^{-1}[e^{-\omega_{\lambda}(\xi’)x_{N}}\mathcal{F}_{\xi’}[h_{d}](\xi’, 0)](x’)$

and. Here,withinverse fourier transform defined by$\omega_{\lambda}(\xi’)=\sqrt{\lambda+|\xi’|^{2}}$$\overline{J_{\xi’}-}$$\mathcal{F}_{\xi}^{-1}$$\mathcal{F}_{\xi’}[h_{d}](\xi’, y_{N})=\int_{\mathbb{R}^{N-1}}e^{-ix’\cdot\xi’}h_{d}(x’, y_{N})dx’,$(2.9)

denote the partial Fourier transform and partial$\mathcal{F}_{\xi’}^{-1}[g(\xi’)](x’)=\frac{1}{(2\pi)^{N-1}}\int_{N^{N-1}}e^{ix’\cdot\xi’}g(\xi’)d\xi’$with $x’=(X_{1}, \ldots, x_{N-1})$ and $\xi’=(\xi_{1}, \ldots, \xi_{N-1})$ . To obtain$u(x)= \sum_{j=1}^{N-1}\sqrt-1\xi[\frac{i\xi_{j}\mathcal{F}[f_{j}^{o}](\xi)}{\lambda+|\xi|^{2}}](x)+\mathcal{F}_{\xi}^{-1}[\frac{i\xi_{N}\mathcal{F}[f_{N}^{e}](\xi)}{\lambda+|\xi|^{2}}](x)$$+ \mathcal{F}_{\xi}^{-1}[\frac{\lambda^{1/2}\mathcal{F}[\lambda^{-1/2}g^{o}](\xi)}{\lambda+|\xi|^{2}}](x)+\int_{0}^{\infty}\mathcal{F}_{\xi’}^{-1}[e^{-\omega_{\lambda}(\xi’)(x_{N}+y_{N})}\mathcal{F}_{\xi’}[D_{N}h_{d}](\xi’, y_{N})](x’)dy_{N}$(2.10)$+ \sum_{j=1}^{N-1}\int_{0}^{\infty}\overline{J_{\xi’}^{-1}-}[e^{-\omega_{\lambda}(\xi’)(x_{N}+y_{N})}\frac{i\xi_{j}}{\omega_{\lambda}(\xi’)}\mathcal{F}_{\xi’}[D_{j}h_{d}](\xi’, y_{N})](x’)dy_{N}$$- \int_{0}^{\infty}e^{-\omega_{\lambda(\xi’)(x_{N}+y_{N})}}\frac{\lambda^{1/2}}{\omega_{\lambda}(\xi)}\mathcal{F}_{\xi’}[\lambda^{1/2}h_{d}](\xi’,y_{N})](x’)dy_{N},$7
we use the formula: $( divf)^{o}=\sum_{j=1}^{N-1}D_{j}(f_{j}^{O})+D_{N}(f_{N}^{e})$ with$\sum_{j=1}^{N-1}(i\xi_{j})(i\xi_{j})\omega_{\lambda}(\xi’)^{-1}$$D_{j}=\partial/\partial x_{j},$$\omega_{\lambda}(\xi’)=\lambda\omega_{\lambda}(\xi’)^{-1}-$

and the Volevich trick:$e^{-\omega_{\lambda}(\xi’)x_{N}} \mathcal{F}_{\xi’}^{-1}[h_{d}](\xi’, 0)=-\int_{0}^{\infty}\frac{\partial}{\partial y_{N}}[e^{-\omega_{\lambda}(\xi’)(x_{N}+y_{N})}\mathcal{F}_{\xi’}^{-1}[h_{d}](\xi’, y_{N})]dy_{N}.$In view of (2.10), we define an operator$\mathcal{S}_{d}(\lambda)$by$\mathcal{S}_{d}(\lambda)(F_{1}, F_{2}, F_{3}, F_{4})=\sum_{j=1}^{N-1}\mathcal{F}_{\xi}^{-1}[\frac{i\xi_{j}\mathcal{F}[F_{1j}^{o}](\xi)}{\lambda+|\xi|^{2}}](x)+\overline{J^{-}}_{\xi}-1[\frac{i\xi_{N}\mathcal{F}[F_{1N}^{e}](\xi)}{\lambda+|\xi|^{2}}](x)$$+ \mathcal{F}_{\xi}^{-1}[\frac{\lambda^{1/2}\mathcal{F}[F_{2}^{o}](\xi)}{\lambda+|\xi|^{2}}](x)+\int_{0}^{\infty}\mathcal{F}_{\xi’}^{-1}[e^{-\omega_{\lambda}(\xi’)(x_{N}+y_{N})_{\overline{J^{-}}_{\xi’}}}[F_{4N}](\xi’, y_{N})](x’)dy_{N}$(2.11)$+ \sum_{j=1}^{N-1}\int_{0}^{\infty}\overline{J^{-}}_{\xi’}-1[e^{-\omega_{\lambda}(\xi’)(x_{N}+y_{N})}\frac{i\xi_{j}}{\omega_{\lambda}(\xi’)}\mathcal{F}_{\xi’}[F_{4j}](\xi’, y_{N})](x’)dy_{N}$$- \int_{0}^{\infty}e^{-\omega_{\lambda}(\xi’)(x_{N}+y_{N})}\frac{\lambda^{1/2}}{\omega_{\lambda}(\xi)}\mathcal{F}_{\xi’}[F_{3}](\xi’, y_{N})](x’)dy_{N}$withwe have$F_{2},$$F_{3}\in L_{q}(\mathbb{R}_{+}^{N})$and$F_{1},$$F_{4}=(F_{41}, \ldots, F_{4,N})\in L_{q}(\mathbb{R}_{+}^{N})^{N}$$u(x)=S_{d}(\lambda)F_{\lambda}^{d}(f, g, h_{d})$. Combining (2.10) and (2.11),.

(2.12)

with. To prove the -boundedness offollowing lemma due to Shibata and Shimizu [9, Lemma 5.4]
$F_{\lambda}^{d}(f, g, h_{d})=(f, \lambda^{-1/2}g, \lambda^{1/2}h_{d}, \nabla h_{d})$$\mathcal{R}$Lemma 2.3. Let $0<\epsilon<\pi/2$ and $1<q<\infty$ . Letthat satisfy the multiplier conditions:$m_{1}$

and$m_{2}$$S_{d}(\lambda)$, we use thebe functions defined on$\Sigma_{\epsilon}\cross \mathbb{R}^{N-1}\backslash \{0\}$$| \partial_{\xi}^{\alpha’},[(\lambda\frac{d}{d\lambda})^{\ell}m_{1}(\lambda, \xi’)]|\leq C_{\alpha’}(|\lambda|^{1/2}+|\xi’|)^{-|\alpha’|} (\ell=0,1)$,(2.13)$| \partial_{\xi}^{\alpha’},[(\lambda\frac{d}{d\lambda})^{\ell}m_{2}(\lambda, \xi’)]|\leq C_{\alpha’}|\xi’|^{-|\alpha’|} (\ell=0,1)$

for any$\alpha’=(\alpha_{1}, \ldots, \alpha_{N-1})\in \mathbb{N}_{0}^{N-1\uparrow}and$$(\lambda, \xi’)\in\Sigma_{\epsilon}\cross \mathbb{R}^{N-1}\backslash \{0\}$, where$\partial_{\xi}^{\alpha’},=\partial_{\xi_{1}}^{\alpha_{1}}\cdots\partial_{\xi_{N-1}}^{\alpha_{N-1}}$

Let $K_{j}(\lambda)(j=1,2)$ be operators defined by$[K_{1}( \lambda)g](x)=\int_{0}^{\infty}\mathcal{F}_{\xi’}^{-1}[m_{1}(\lambda, \xi’)\lambda^{1/2}e^{-\omega_{\lambda}(\xi’)(x_{N}+y_{N})}\mathcal{F}_{\xi’}[g](\xi’, y_{N})](x’)dy_{N},$$[K_{2}( \lambda)g](x)=\int_{0}^{\infty}\mathcal{F}_{\xi’}^{-1}[m_{2}(\lambda, \xi’)|\xi’|e^{-\omega_{\lambda}(\xi’)(x_{N}+y_{N})}\mathcal{F}_{\xi’}[g](\xi’, y_{N})](x’)dy_{N}.$Then, there exists a constant$\beta_{0}$depending on$\epsilon,$$q$

and $N$ such that$\mathcal{R}_{\mathcal{L}(L_{q}(\mathbb{R}_{+}^{N}))}(\{(\lambda\frac{d}{d\lambda})^{\ell}K_{j}(\lambda)|\lambda\in\Sigma_{\epsilon}\})\leq\beta_{0} (\ell=0,1, j=1,2)$

.

Since
and
satisfy the multiplier conditions (2.13), respectively, byLemma 2.3 and Theorem 3.4, we have the following theorem.
$\lambda^{1/2}/\omega_{\lambda}(\xi’)$$\dagger_{\mathbb{N}}$$i\xi_{j}/\omega_{\lambda}(\xi’)$

denotes the set of all natural numbers and$\mathbb{N}_{0}=\mathbb{N}\cup\{0\}.$8Theorem 2.4. Let $1<q<\infty$ and $0<\epsilon<\pi/2$ . For any domain

$G$

in

$\mathbb{R}^{N}$, we set$X_{qd}(G)=\{(f,g, h_{d})|f\in L_{q}(G)^{N}, g\in L_{q}(G), h_{d}\in W_{q}^{1}(G)\},$$\mathcal{X}_{qd}(G)=\{(F_{1}, F_{2}, F_{3}, F_{4})|F_{1}, F_{4}\in L_{q}(G)^{N}, F_{2}, F_{3}\in L_{q}(G)\}.$

Let
any

$S_{d}(\lambda)$

be the operator defined in (2.11). Then,and$\lambda\in\Sigma_{\epsilon}$$\mathcal{S}_{d}(\lambda)\in$$(f, g, h_{d})\in X_{qd}(\mathbb{R}_{+}^{N})u=S_{d}(\lambda)F_{\lambda}^{d}(f, g, h_{d})$

, for
Anal
is a unique solution to (2.8), and$(\Sigma_{\epsilon}, \mathcal{L}(X_{qd}(\mathbb{R}_{+}^{N}), W_{q}^{1}(\mathbb{R}_{+}^{N})))$$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{qd}(\mathbb{R}_{+}^{N}),L_{q}(\mathbb{R}_{+}^{N})^{N+1})}(\{(\lambda\frac{d}{d\lambda})^{\ell}(\lambda^{1/2}, \nabla)S_{d}(\lambda)|\lambda\in\Sigma_{\epsilon}\})\leq\beta_{0} (\ell=0,1)$

with some constant

2.3

$A$

$\beta_{0}$depending on$\epsilon,$$q$

and $N.$model problem in the half space$\mathbb{R}_{+}^{N}$, Neumann condition case.

In this subsection we consider the weak Neumann problem in the half-space$\mathbb{R}_{+}^{N}:$(2.14)$(\lambda u, \varphi)_{\mathbb{R}_{+}^{N}}+(\nabla u, \nabla\varphi)_{\mathbb{R}_{+}^{N}}=-(f, \nabla\varphi)_{\mathbb{R}_{+}^{N}}+(g, \varphi)_{\mathbb{R}_{+}^{N}}+<h_{n}, \varphi>_{\mathbb{R}_{0}^{N}}$for any, where $<a,$$(\lambda-\Delta)u=divf+g$ insubject to $D_{N}u=h_{n}$ on
solution is given by$b>= \mathbb{R}_{0}^{N}\int_{\mathbb{R}^{N-1}}a(x’)b(x’)dx’$$\varphi\in W_{q}^{1},(\mathbb{R}_{+}^{N})$$\mathbb{R}_{+}^{N}$$\mathbb{R}_{0}^{N}$. We consider the strong equation:

instead of (2.14). Then, its unique. $(2.15)$$u(x)= \mathcal{F}_{\xi}^{-1}[\frac{\mathcal{F}[(divf)^{e}](\xi)}{\lambda+|\xi|^{2}}](x)+\mathcal{F}_{\xi}^{-1}[\frac{\mathcal{F}[g^{e}](\xi)}{\lambda+|\xi|^{2}}](x)+\mathcal{F}_{\xi}^{-1}[\frac{e^{-\omega_{\lambda}(\xi’)x_{N}}}{\omega_{\lambda}(\xi’)}\overline{J_{\xi’}-}[h_{n}](\xi’, 0)](x’)$

Since we may assume that$f\in C_{0}^{\infty}(\mathbb{R}_{+}^{N})^{N}$, we have $( divf)^{e}=\sum_{j=1}^{N-1}D_{j}(f_{j}^{e})+D_{N}(f_{N}^{o})$ , so that$u(x)= \sum_{j=1}^{N-1}\mathcal{F}_{\xi}^{-1}[\frac{i\xi_{j}\mathcal{F}[f_{j}^{e}](\xi)}{\lambda+|\xi|^{2}}](x)+\mathcal{F}_{\xi}^{-1}[\frac{i\xi_{N}\mathcal{F}[f_{N}^{o}](\xi)}{\lambda+|\xi|^{2}}](x)$$+ \mathcal{F}_{\xi}^{-1}[\frac{\lambda^{1/2}\mathcal{F}[\lambda^{-1/2}g^{e}](\xi)}{\lambda+|\xi|^{2}}](x)$(2.16)$+ \int_{0}^{\infty}\mathcal{F}_{\xi’}^{-1}[e^{-\omega_{\lambda}(\xi’)(x_{N}+y_{N})}\frac{\lambda^{1/2}}{\omega_{\lambda}(\xi)}\mathcal{F}_{\xi’}[\lambda^{-1/2}D_{N}h_{n}](\xi’, y_{N})](x’)dy_{N}$$- \int_{0^{e^{-\omega_{\lambda}(\xi’)(x_{N}+y_{N})_{\sqrt{}}}}}^{\infty}\xi’[h_{n}](\xi’, y_{N})](x’)dy_{N}.$In view of (2.16), we define an operator$S_{n}(\lambda)$by$S_{n}( \lambda)(F_{1}, F_{2}, F_{5}, F_{6})=\sum_{j=1}^{N-1}\mathcal{F}_{\xi}^{-1}[\frac{i\xi_{j}\mathcal{F}[F_{1j}^{e}](\xi)}{\lambda+|\xi|^{2}}](x)+\overline{J_{\xi}^{-1}-}[\frac{i\xi_{N}\mathcal{F}[F_{1N}^{o}](\xi)}{\lambda+|\xi|^{2}}](x)$$+ \mathcal{F}_{\xi}^{-1}[\frac{\lambda^{1/2}\mathcal{F}[F_{2}^{e}](\xi)}{\lambda+|\xi|^{2}}](x)+\int_{0}^{\infty}\mathcal{F}_{\xi}^{-1}[e^{-\omega_{\lambda}(\xi’)(x_{N}+y_{N})}\frac{\lambda^{i/2}}{\omega_{\lambda}(\xi)}\mathcal{F}_{\xi},[F_{6N}](\xi’, y_{N})](x’)dy_{N}$(2.17)$- \int_{0}^{\infty}e^{-\omega_{\lambda}(\xi’)(x_{N}+y_{N})}\mathcal{F}_{\xi’}[F_{5}](\xi’, y_{N})](x’)dy_{N}$withwe have$F_{2},$$F_{5}\in L_{q}(\mathbb{R}_{+}^{N})$and$F_{1},$$F_{6}=(F_{61}, \ldots, F_{6N})\in L_{q}(\mathbb{R}_{+}^{N})^{N}$$u(x)=S_{n}(\lambda)F_{\lambda}^{n}(f,g, h_{n})$withthe following theorem.

$F_{\lambda}^{n}(f, g, h_{n})=(f, \lambda^{-1/2}g, h_{n}, \lambda^{-1/2}\nabla h_{n})$. Combining (2.16) and (2.17),(2.18). Applying Lemma 2.3 and Theorem 3.4, we have

9
Theorem 2.5. Let $1<q<\infty$ and.$O<\epsilon<\pi/2$ . For any domain

$G$

in

$\mathbb{R}^{N}$, we set$X_{qn}(G)=\{(f, g, h_{n})|f\in L_{q}(G)^{N}, g\in L_{q}(G), h_{n}\in W_{q}^{1}(G)\},$$\mathcal{X}_{qn}(G)=\{(F_{1}, F_{2}, F_{5}, F_{6})|F_{1}, F_{6}\in L_{q}(G)^{N}, F_{2}, F_{5}\in L_{q}(G)\}.$

Let$S_{n}(\lambda)$

for anybe the operator defined in (2.17). Then,and$S_{n}(\lambda)\in$

Anal$(f, g, h_{n})\in X_{qn}(\mathbb{R}_{+}^{N})u=S_{n}(\lambda)F_{\lambda}^{n}(f_{9}, h_{n})$$\lambda\in\Sigma_{\epsilon}$and$(\Sigma_{\epsilon}, \mathcal{L}(X_{qn}(\mathbb{R}_{+}^{N}), W_{q}^{1}(\mathbb{R}_{+}^{N})))$,

is a unique solution to (2.14),$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{qn}(\mathbb{R}_{+}^{N}),L_{q}(\mathbb{R}_{+}^{N})^{N+1})}(\{(\lambda\frac{d}{d\lambda})^{\ell}(\lambda^{1/2}, \nabla)S_{n}(\lambda)|\lambda\in\Sigma_{\epsilon}\})\leq\beta_{0} (\ell=0,1)$

with some constant3$\mathcal{R}$Let:$\beta_{0}$depending on$\epsilon,$$q$

and $N.$-boundedness of solution operators in a bent-half space

be a bijection ofclass and let
be its inverse map. We assume thatand $\nabla\Phi^{-1}=\mathcal{A}_{-1}+B_{-1}(x)$ , where andare orthonormal matrices withconstant coefficients and $B(x)$ and $B_{-1}(x)$ are matrices of functions insuch that$\Phi$$\mathbb{R}^{N}arrow \mathbb{R}^{N}$$C^{1}$$\Phi^{-1}$$\nabla\Phi=\mathcal{A}+B(x)$$\mathcal{A}$$\mathcal{A}_{-1}$$L_{\infty}(\mathbb{R}^{N})$$\Vert(B, B_{-1})\Vert_{L_{\infty}(\mathbb{R}^{N})}\leq M_{1}$

.

(3.1)

We will choose
small enough eventually, so that we may assume that $0<M_{1}\leq 1$ in thefollowing. Set
and. Let be a function defined by
We choose $0<M_{1}\leq 1$ so small that$M_{1}$$\Omega+=\Phi(\mathbb{R}_{+}^{N})$$r_{+}=\Phi(\mathbb{R}_{0}^{N})$$\det(\nabla\Phi)=1+\mathfrak{g}.$$\mathfrak{g}$(3.2)$\Vert \mathfrak{g}\Vert_{L_{\infty}(\mathbb{R}^{N})}\leq C_{N}M_{1}$with some constant depending solely on $N$ . In this section, we consider the weak Dirichletproblem and the weak Neumann problem on$\Omega_{+}.$

3.1

Dirichlet boundary condition caseIn this subsection, we consider the variational problem:$(\lambda u, \varphi)_{\Omega+}+(\nabla u, \nabla\varphi)_{\Omega+}=-(f, \nabla\varphi)_{\Omega+}+(g, \varphi)_{\Omega+}$

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

,

(3.3)

subject to $u=h_{d}$ on
By the change of variable: $y=\Phi(x)$ , we transform (3.3) into thehalf-space problem. Setting $uo\Phi(x)=v(x)$ and $\varphi 0\Phi(x)=\psi(x)$ and using the formula:withand $B_{-1}(x)=(B_{kj}(x))$ , we have$\Gamma+\cdot$$\frac{\partial x_{k}}{\partial y_{j}}=\mathcal{A}_{jk}+B_{kj}(x)$$\mathcal{A}_{-1}=(\mathcal{A}_{kj})$$( \nabla u, \nabla\varphi)_{\Omega+}=\sum_{j,k,\ell=1}^{N}\int_{\mathbb{R}_{+}^{N}}(\mathcal{A}_{kj}+B_{kj}(x))(\mathcal{A}_{\ell j}+B_{\ell j}(x))\frac{\partial v(x)}{\partial x_{j}}\frac{\partial\psi(x)}{\partial x\ell}(1+\mathfrak{g}(x))dx$$=(\nabla v, \nabla\psi)_{\mathbb{R}_{+}^{N}}+(\mathcal{P}\nabla v, \nabla\psi)_{\mathbb{R}_{+}^{N}},$with$( \mathcal{P}\nabla v, \nabla\psi)_{\mathbb{R}_{+}^{N}}=(\mathfrak{g}\nabla v, \nabla\psi)_{\mathbb{R}_{+}^{N}}+((1+\mathfrak{g})\{\sum_{j=1}^{N}(\mathcal{A}_{kj}B_{\ell j}+B_{kj}\mathcal{A}_{\ell j}+B_{kj}B_{\ell j})\}\frac{\partial v}{\partial x_{k}}, \frac{\partial\psi}{\partial x_{\ell}})_{\mathbb{R}_{+}^{N}}.$10
In the similar way, we have$(f, \nabla\varphi)_{\Omega+}=\sum_{j,k=1}^{N}((1+\mathfrak{g})f_{j}\circ\Phi, (\mathcal{A}_{kj}+B_{kj})\frac{\partial\psi}{\partial x_{k}})_{\mathbb{R}_{+}^{N}}=(F, \nabla\psi)_{\mathbb{R}_{+}^{N}},$

where we have set $F=(F_{1}, \ldots, F_{N})$ andand $H_{d}=h_{d}\circ\Phi$ , finally we arrive at the variational equation:$F_{k}= \sum_{j=1}^{N}(1+\mathfrak{g})(\mathcal{A}_{kj}+B_{kj})f_{j}o\Phi$. Setting $G=$$(1+\mathfrak{g})go\Phi$(3.4)$(\lambda v, \psi)_{\mathbb{R}_{+}^{N}}+(\lambda \mathfrak{g}v, \psi)_{\mathbb{R}_{+}^{N}}+(\nabla v, \nabla\psi)_{\mathbb{R}_{+}^{N}}+(\mathcal{P}\nabla v, \nabla\psi)_{\mathbb{R}_{+}^{N}}=(F, \nabla\psi)_{\mathbb{R}_{+}^{N}}+(G, \psi)_{\mathbb{R}_{+}^{N}}$, subject to $v=H_{d}$ on
for any
2.4. Inserting the formula:$\psi\in W_{q,0}^{1}(\mathbb{R}_{+}^{N})$$\mathbb{R}_{0}^{N}$. Let$v=S_{d}(\lambda)F_{\lambda}^{d}(F, G, H_{d})$be the operator given in Theorem
into (3.4), we have
$S_{d}(\lambda)$$(\lambda v, \psi)_{\mathbb{R}_{+}^{N}}+(\lambda \mathfrak{g}v, \psi)_{\mathbb{R}_{+}^{N}}+(\nabla v, \nabla\psi)_{\mathbb{R}_{+}^{N}}+(P\nabla v, \nabla\psi)_{\mathbb{R}_{+}^{N}}$(3.5)$=-(F-\mathcal{P}\nabla S_{d}(\lambda)F_{\lambda}^{d}(F, G, H_{d}), \nabla\psi)_{\mathbb{R}_{+}^{N}}+(G+\lambda gS_{d}(\lambda)F_{\lambda}^{d}(F, G, H_{d}), \psi)_{\mathbb{R}_{+}^{N}}$for any$\psi\in W_{q,0}^{1}(\mathbb{R}_{+}^{N})$$\mathcal{F}_{2}(\lambda)F^{d}=\lambda \mathfrak{g}S_{d}(\lambda)F^{d}$, subject to $v=H_{d}$ on$\mathbb{R}_{0}^{N}$. Settingand$\mathcal{F}_{1}(\lambda)F^{d}=-\mathcal{P}\nabla S_{d}(\lambda)F^{d}$with $F^{d}=(F_{1}, F_{2}, F_{3}, F_{4})$ , we write (3.5) ae follows:$(\lambda(1+\mathfrak{g})v, \psi)_{\mathbb{R}_{+}^{N}}+((I+\mathcal{P})\nabla v,\nabla\psi)_{\mathbb{R}_{+}^{N}}$$=-(F+\mathcal{F}_{1}(\lambda)F_{\lambda}^{d}(F, G, H_{d}), \nabla\psi)_{\mathbb{R}_{+}^{N}}+(G+\overline{f}_{2}(\lambda)F_{\lambda}^{d}(F, G, H_{d}), \psi)_{\mathbb{R}_{+}^{N}}$subject to $v=H_{d}$ on$\mathbb{R}_{0}^{N}$. Setting$\mathcal{F}(\lambda)F^{d}=(\mathcal{F}_{1}(\lambda)F^{d}, \mathcal{F}_{2}(\lambda)F^{d}, 0)$for any$\psi\in W_{q}^{1},(\mathbb{R}_{+}^{N})$, we have(3.6)$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{q}(\mathbb{R}_{+}^{N}))}(\{(\lambda\frac{d}{d\lambda})^{\ell}F_{\lambda}^{d}\mathcal{F}(\lambda)|\lambda\in\Sigma_{\epsilon}\})\leq C_{N}M_{1}\beta_{0} (\ell=0,1)$

where$\beta_{0}$is the same constant as in Theorem 2.4. To prove (3.6), we use the following lemmas.

be Banach spaces, and let$\mathcal{T}+S=\{T+S|T\in\mathcal{T}, S\in S\}$ is also an. Then,Lemma 3.1. (1) Let $X$ and$\mathcal{L}(X, Y)$,$Y$$\mathcal{T}$and be -bounded families inand-bounded family in$S$$\mathcal{R}$$\mathcal{L}(X, Y)$$\mathcal{R}$$\mathcal{R}_{\mathcal{L}(X,Y)}(\mathcal{T}+S)\leq \mathcal{R}_{\mathcal{L}(X,Y)}(\mathcal{T})+\mathcal{R}_{\mathcal{L}(X,Y)}(S)$.

and $S$ be -bounded families in(2) Let $X,$ $Y$ andbe Banach spaces, and let$\mathcal{S}\mathcal{T}=\{ST|T\in\mathcal{T},S\in S\}$ is also an-bounded family in
Then,,respectively.
and$Z$$\mathcal{L}(X, Y)$$\mathcal{R}$$\mathcal{T}$$\mathcal{R}$$\mathcal{L}(Y, Z)$$\mathcal{L}(X, Z)$and.

$\mathcal{R}_{\mathcal{L}(X,Z)}(S\mathcal{T})\leq \mathcal{R}_{\mathcal{L}(X,Y)}(\mathcal{T})\mathcal{R}_{\mathcal{L}(Y,Z)}(\mathcal{S})$Lemma 3.2. Let $1<p,$ $q<\infty$ and let $D$ be a domain in(1) Letbe a bounded function defined on a subset in a complex plane and let
be a multiplication operator withdefined by $M_{m}(\lambda)f=m(\lambda)f$ for any $f\in L_{q}(D)$ . Then,$\mathbb{R}^{N}.$$m(\lambda)$$M_{m}(\lambda)$$\mathbb{C}$$\Lambda$$m(\lambda)$$\mathcal{R}_{\mathcal{L}(L_{q}(D))}(\{M_{m}(\lambda)|\lambda\in\Lambda\})\leq C_{N,q,D}\Vert m\Vert_{L_{\infty}(\Lambda)}.$andthat satisfies the conditions:be afuntion defined on$\gamma>0$
operatorvalued
.
be
an
Let
any
with some constantfor. Then,Fourier multiplier defined byfor any with
into itself. Moreover, denoting thisis extended to a bounded linear operator fromextension also by , we have(2) Let$n(\tau)$$C^{1}$$|n(\tau)|\leq\gamma$$\mathbb{R}\backslash \{0\}$$|\tau n’(\tau)|\leq\gamma$$\tau\in \mathbb{R}\backslash \{0\}$$T_{n}f=\mathcal{F}^{-1}[n\mathcal{F}[f]]$$f$$L_{p}(\mathbb{R}, L_{q}(D))$$T_{n}$$\Vert T_{n}\Vert_{\mathcal{L}(L_{p}(\mathbb{R},L_{q}(D)))}\leq C_{p,q,D}\gamma.$$T_{n}$$\mathcal{F}[\phi]\in \mathcal{D}(\mathbb{R}, L_{q}(D))$$T_{n}$

11
Remark 3.3. For proofs of Lemma 3.1 and Lemma 3.2, we refer to [3, p.28, 3.4.Propositionand p.27, 3.2.Remarks (4) (cf. also Bourgain [2]), respectively.
$]$

For any natural number
and sequenceofindependent, symmetric, $\{-1,1\}$ -valued random variable on $[0,1]$ , using Theorem 2.4 we have$n,$$\{\lambda_{l}\}_{\ell=1}^{n}\subset\Sigma_{\epsilon},$$\{F_{\ell}\}_{\ell=1}^{n}\subset \mathcal{X}_{q}(\mathbb{R}_{+}^{N})$$\{r_{l}(u)\}_{\ell-1}^{n}$$\int_{0}^{1}\Vert\sum_{\ell=1}^{n}r_{\ell}(u)\mathcal{F}_{1}(\lambda_{\ell})F_{\ell}\Vert_{L_{q}(\mathbb{R}_{+}^{N})}^{q}du\leq(C_{N}M_{1})^{q}\int_{0}^{1}\Vert\sum_{\ell=1}^{n}r_{\ell}(u)\nabla S_{d}(\lambda_{\ell})F_{\ell}\Vert_{L_{q}(\mathbb{R}_{+}^{N})}^{q}du$$\leq(C_{N}M_{1}\beta_{0})^{q}\int_{0}^{1}\Vert\sum_{\ell=1}^{n}r_{\ell}(u)F_{\ell}\Vert_{L_{q}(\mathbb{R}_{+}^{N})}^{q}du,$$\int_{0}^{1}\Vert\sum_{\ell=1}^{n}r_{\ell}(u)\lambda_{p}^{-1/2}\mathcal{F}_{2}(\lambda_{\ell})F_{\ell}\Vert_{L_{q}(\mathbb{R}_{+}^{N})}^{q}du\leq(C_{N}M_{1})^{q}\int_{0}^{1}\Vert\sum_{\ell=1}^{n}r_{\ell}(u)\lambda_{\ell}^{1/2}S_{d}(\lambda_{\ell})F_{\ell}\Vert_{L_{q}(\mathbb{R}_{+}^{N})}^{q}du$$\leq(C_{N}M_{1}\beta_{0})^{q}\int_{0}^{1}\Vert\sum_{\ell=1}^{n}r_{\ell}(u)F_{\ell}\Vert_{L_{q}(\mathbb{R}_{+}^{N})}^{q}du,$Note thatonfor$\Vert F_{\lambda}^{d}(F, G, H_{d})\Vert_{L_{q}(\mathbb{R}_{+}^{N})}=\Vert(F, \lambda^{-1/2}G, \lambda^{1/2}H_{d}, \nabla H_{d})\Vert_{L_{q}(\mathbb{R}_{+}^{N})}$$X_{q}(\mathbb{R}_{+}^{N})$$\lambda\neq 0$

give us equivalent norms. Since$\Vert F_{\lambda}^{d}\mathcal{F}(\lambda)F_{\lambda}^{d}(F, G, H_{d})\Vert_{L_{q}(\mathbb{R}_{+}^{N})}\leq C_{N}M_{1}\beta_{0}\Vert F_{\lambda}^{d}(F, G, H_{d})\Vert_{L_{q}(\mathbb{R}_{+}^{N})}$as follows from (3.6) (cf. the definition of -boundeness with $\ell=1$ in Definition 1.1), choosing$0<M_{1}\leq 1$ so small that
, we see thatexists infor$\mathcal{R}$$C_{N}M_{1}\beta_{0}\leq 1/2$

, and therefore
any
Moreover, we have$\lambda\in\Sigma_{\epsilon}$$(I+F_{\lambda}^{d})^{-1}\mathcal{F}(\lambda)$$v=S_{d}(\lambda)F_{\lambda}^{d}(I+\mathcal{F}(\lambda)F_{\lambda}^{d})^{-1}(F, G, H_{d})$$\mathcal{L}(X_{q}(\mathbb{R}_{+}^{N}))$is a unique solution to (3.5).

$F_{\lambda}^{d}(I+ \mathcal{F}(\lambda)F_{\lambda}^{d})^{-1}=F_{\lambda}^{d}+\sum_{\ell=1}^{\infty}(-1)^{\ell}F_{\lambda}^{d}(\mathcal{F}(\lambda)F_{\lambda}^{d})^{\ell}=(I+F_{\lambda}^{d}\mathcal{F}(\lambda))^{-1}F_{\lambda}^{d},$which furnishes that
$S_{d}(\lambda)(I+F_{\lambda}^{d}\mathcal{F}(\lambda))^{-1}$$\mathcal{S}_{d}(\lambda)F_{\lambda}^{d}(I+\mathcal{F}(\lambda)F_{\lambda}^{d})^{-1}=S_{d}(\lambda)(I+F_{\lambda}^{d}\mathcal{F}(\lambda))^{-1}F_{\lambda}^{d}$, by (3.6) and Theorem 2.4 we have

. Setting$S_{bd}(\lambda)=$$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{q}(\mathbb{R}_{+}^{N}),L_{q}(\mathbb{R}_{+}^{N})^{N+1})}(\{(\lambda\frac{d}{d\lambda})^{\ell}S_{bd}(\lambda)|\lambda\in\Sigma_{\epsilon}\})\leq 2\beta_{0} (\ell=0,1)$

and the solution to (3.4) is represented by
we have the following theorem.
$v$$v=\mathcal{S}_{bd}(\lambda)F_{\lambda}^{d}(F, G, H_{d})$. By the change of variable:$x=\Phi^{-1}(y)$Theorem 3.4. Let $1<q<\infty$ and $0<\epsilon<\pi/2$ . Then, there exists a constantwithdepending on $N$ and such that if the condition (3.1) holds, then the following
assertion holds: There exists an operator family
Anal
such.
that
is a unique solution to (3.3) for any $(f, g, h_{d})\in X_{qd}(\Omega_{+})$ andand$M_{1}$$0<M_{1}\leq 1$$q,$$\epsilon$$\mathcal{T}_{d}(\lambda)\in$$(\Sigma_{\epsilon}, \mathcal{L}(\mathcal{X}_{qd}(\Omega_{+}), W_{q}^{1}(\Omega_{+})))$$u=\mathcal{T}_{d}(\lambda)F_{\lambda}^{d}(f, g, h_{d})$$\lambda\in\Sigma_{\epsilon},$$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{qd}(\Omega_{+}),L_{q}(\Omega_{+})^{N+1})}(\{(\lambda\frac{d}{d\lambda})^{\ell}(\lambda^{1/2}, \nabla)\mathcal{T}_{d}(\lambda)|\lambda\in\Sigma_{\epsilon}\})\leq\beta_{1}$with some constant$\beta_{1}$depending on$\beta_{0},$$q,$$\epsilon$

and $N.$12
3.2

Neumann boundary condition caseIn this subsection, we consider the variational problem:(3.7)$(\lambda u, \varphi)_{\Omega_{+}}+(\nabla u, \nabla\varphi)_{\Omega_{+}}=-(f, \nabla\varphi)_{\Omega_{+}}+(g, \varphi)_{\Omega_{+}}+<h_{n}, \varphi>r_{+}$

for any$dS$ being the surface element of, where $<h_{n},$Employing the same argument as in Subsec. 3.1, we transfer (3.7) to the half-space problem:$\varphi>r_{+}=\int_{r_{+}}h_{n}\varphi dS,$$\varphi\in W_{q}^{1},(\Omega_{+})$$\Gamma_{+}.$(3.8)$(\lambda(1+\mathfrak{g})v, \psi)_{\mathbb{R}_{+}^{N}}+((I+\mathcal{P})\nabla v,\nabla\varphi)_{\mathbb{R}_{+}^{N}}=-(F, \nabla\psi)_{\mathbb{R}_{+}^{N}}+(G, \psi)_{\mathbb{R}_{+}^{N}}+<H_{n},$ $\psi>_{\mathbb{R}_{0}^{N}}$for any$\psi\in W_{q}^{1},(\mathbb{R}_{+}^{N})$. Let$v=S_{n}(\lambda)F_{\lambda}^{n}(F, G, H_{n})$be the operator given in Theorem 2.5. Inserting the formula:into (3.8), we have
$S_{n}(\lambda)$$(\lambda(1+\mathfrak{g})v, \psi)_{\mathbb{R}_{+}^{N}}+((I+\mathcal{P})\nabla v,\nabla\psi)_{\mathbb{R}_{+}^{N}}=-(F-\mathcal{P}\nabla S_{n}(\lambda)F_{\lambda}^{n}(F, G, H_{n}), \nabla\psi)_{\mathbb{R}_{+}^{N}}$$+(G+\lambda \mathfrak{g}S_{n}(\lambda)F_{\lambda}^{n}(F, G, H_{n}), \psi)_{\mathbb{R}_{+}^{N}}+<H_{n},$Setting
have$\mathcal{F}(\lambda)F^{n}=(\mathcal{P}\nabla S_{n}(\lambda)F^{n}, \lambda \mathfrak{g}S_{n}(\lambda)F^{n}, 0)$$\psi>_{\mathbb{R}_{O}^{N}}$for any$\psi\in W_{q}^{1},(\mathbb{R}_{+}^{N})$

(3.9)

.

with $F^{n}=(F_{1}, F_{2}, F_{5}, F_{6})$ , by Theorem 2.5 we.

(3.10)$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{q}(\mathbb{R}_{+}^{N}))}(\{(\lambda\frac{d}{d\lambda})^{\ell}F_{\lambda}^{n}\mathcal{F}(\lambda)|\lambda\in\Sigma_{\epsilon}\})\leq C_{N}M_{1}\beta_{0} (\ell=0,1)$

We choose $M_{1}\in(0,1] in such a way that C_{N}M_{1}\beta_{0}\leq 1/2.$ Since
give us equivalent norms on, and therefore
we see thatexists for any$\Vert F_{\lambda}(F, G, H_{n})\Vert_{L_{q}(\mathbb{R}_{+}^{N})}=$$X_{q}(\mathbb{R}_{+}^{N})$$\Vert(F, \lambda^{-1/2}G, H_{n}, \lambda^{-1/2}\nabla H_{n})\Vert_{L_{q}(\mathbb{R}_{+}^{N})}$$(I+F_{\lambda}^{n}\mathcal{F}(\lambda))^{-1}$for$\lambda\neq 0$, by (3.10)$\lambda\in\Sigma_{\epsilon}$$v=S_{n}(\lambda)F_{\lambda}^{n}(I+\mathcal{F}(\lambda)F_{\lambda}^{n})^{-1}(F, G, H_{n})$is a unique solution to (3.8). Moreover, we have, by (3.10) and Theorem 2.5, we have
Therefore, setting$F_{\lambda}^{n}(I+\mathcal{F}(\lambda)F_{\lambda}^{n})^{-1}=(I+F_{\lambda}^{n}\mathcal{F}(\lambda))^{-1}F_{\lambda}^{n}.$$\mathcal{S}_{bn}(\lambda)=S_{n}(\lambda)(I+F_{\lambda}^{n}\mathcal{F}(\lambda))^{-1}$$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{q}(\mathbb{R}_{+}^{N}),L_{q}(\mathbb{R}_{+}^{N})^{N+1})}(\{(\lambda\frac{d}{d\lambda})^{\ell}(\lambda, \nabla)S_{bn}(\lambda)|\lambda\in\Sigma_{\epsilon}\})\leq 2\beta_{0} (\ell=0,1)$

and the solution to (3.8) is represented by
we have the following theQrem.
$v$$v=S_{bn}(\lambda)F_{\lambda}^{n}(F, G, H_{n})$. By the change of variable:$x=\Phi^{-1}(y)$Theorem 3.5.. Let $1<q<\infty$ and $0<\epsilon<\pi/2$ . Then, there exists a constantwith$N$ and$0<M_{1}\leq 1$ depending onsuch that if the condition (3.1) holds, then the following
assertion holds: There exists an operator family
Anal
such
that
is a unique solution to (3.3) for any $(f, g, h_{n})\in X_{qn}(\Omega_{+})$ and$M_{1}$$q,$$\epsilon$$\mathcal{T}_{n}(\lambda)\in$$(\Sigma_{\epsilon}, \mathcal{L}(\mathcal{X}_{qn}(\Omega_{+}), W_{q}^{1}(\Omega_{+})))$$u=\mathcal{T}_{n}(\lambda)F_{\lambda}^{n}(f, g, h_{n})$$\lambda\in\Sigma_{\epsilon},$and$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{qn}(\Omega_{+}),L_{q}(\Omega_{+})^{N+1})}(\{(\lambda\frac{d}{d\lambda})^{\ell}(\lambda^{1/2}, \nabla)\mathcal{T}_{n}(\lambda)|\lambda\in\Sigma_{\epsilon}\})\leq\beta_{1}$with some constant4$A$$\beta_{1}$depending on$\beta_{0},$$q,$$\epsilon$

and $N.$

proof of Theorem 1.5

First, we state some properties of uniform$C^{1}$

domain.

13
Proposition 4.1. Let be a uniform. Let
domain in
be a positive number given in
Theorem 3.4 and Theorem 3.5. Then, there exists positive constants $d_{i}(i=0,1,2)$ and , atmost countably many $N$ -vector of functions
and points
andsuch that the following assertions hold:$C^{1}$$\Omega$$\mathbb{R}^{N}$$M_{1}$$c_{0}$$\Phi_{j}^{i}\in C^{1}(\mathbb{R}^{N})(i=1,2)$$x_{j}^{0}\in\Omega,$$x_{j}^{1}\in\Gamma_{1}$$x_{j}^{2}\in\Gamma_{2}$The map:(i)(ii)$\mathbb{R}^{N}\ni x\mapsto\Phi_{j}^{i}(x)\in \mathbb{R}^{N}(i=1,2)$$\Omega=(\bigcup_{j-1}^{\infty}B_{d^{0}}(x_{j}^{0}))\cup(\bigcup_{i=1}^{2}\bigcup_{j=1}^{\infty}(\Phi_{j}^{i}(\mathbb{R}_{+}^{N})\cap B_{d^{i}}(d_{j}^{i}))),$$\Phi_{j}^{i}(\mathbb{R}_{+}^{N})\cap B_{d^{i}}(x_{j}^{i})=\Omega\cap B_{d^{i}}(x_{j}^{i})$(iii)

There exist$\Vert\zeta_{j}^{i}\Vert_{W_{\infty}^{1}(\mathbb{R}^{N})},$

$1$

(iv)

are bijective.

$C^{\infty}$functions$\zeta_{j}^{i}$,and$(i=1,2)$ .

$\Phi_{j}^{i}(\mathbb{R}_{0}^{N})\cap B_{d^{i}}(x_{j}^{i})=\Gamma_{i}\cap B_{d^{i}}(x_{j}^{i})$$\tilde{\zeta}_{j}^{i}$such that$\Vert\tilde{\zeta}_{j}^{i}\Vert_{W_{\infty}^{1}(\mathbb{R}^{N})}\leq c_{0},\tilde{\zeta}_{j}^{i}=1$

on $\Gamma_{i}(i=1,2)$ .

on$0\leq\zeta_{j}^{i},\tilde{\zeta}_{j}^{i}\leq 1,$$supp\zeta_{j}^{i},$$\nabla\Phi_{j}^{i}=\mathcal{A}_{j}^{i}+B_{j}^{i}(x),$$supp\zeta_{j}^{i},$ $supp\tilde{\zeta}_{j}^{i}\subset B_{d^{i}}(x_{j}^{i})$$\sum_{i=0}^{2}\sum_{j=1}^{\infty}\zeta_{j}^{i}=1$

For $i=1,2$ and,-are $N\cross N$ constant orthonormal matrices, andcontinous functions defined onsuch that$j\in \mathbb{N},$$B_{d^{0}}(x_{j}^{0})\subset\Omega,$

on , and$\nabla(\Phi_{j}^{i})^{-1}=\mathcal{A}_{j,-}^{i}+B_{j,-}^{i}$$\mathcal{A}_{j}^{i}$$B_{j}^{i}$and$B_{j}^{i}$,$\overline{\Omega}$,-are$\sum_{j=1}^{\infty}\zeta_{j}^{i}=$, where$N\cross N$

andmatrices of$\mathcal{A}_{j}^{i}$$\mathbb{R}^{N}$$\Vert(B_{j}^{i}, B_{j,-}^{i})\Vert_{L_{\infty}(\mathbb{R}^{N})}\leq M_{1}.$(v)

There exists a natural number $L\geq 2$ such that any $L+1$ distinct sets ofhave an empty intersection.

$0,1,2,$$\{B_{d^{i}}(x_{j}^{i})|i=$$j\in \mathbb{N}\}$In the following, we write
for the sake of simplicity. By the finite intersection
property stated in Proposition 4.1 (v) for any $r\in[1, \infty)$ there exists a constantsuch that$B_{j}^{i}=B_{d^{i}}(x_{j}^{i})$$C_{r,L}$$[ \sum_{i=0}^{2}\sum_{j=1}^{\infty}\Vert f\Vert_{L_{r}(\Omega\cap B_{j}^{i})}]^{1/r}\leq C_{r,L}\Vert f\Vert_{L_{r}(\Omega)}.$The following propositions were proved in Shibata [7, 8].
Proposition 4.2. Let $1<q<\infty,$ $q’=q/(q-1)$ and $i=0,1,2$ . Then, the following assertions
hold.

(i) Letbe a sequence inAssume that$\{f_{j}\}_{j=1}^{\infty}$$\sum_{j=1}^{\infty}g_{j}^{q}<\infty$with some constantstrong topology of$M_{3}$

and$L_{q}(\Omega)$

and let$\{g_{j}\}_{j=1}^{\infty}$$|(f_{j}, \varphi)_{\Omega}|\leq M_{3}g_{j}\Vert\varphi\Vert_{L_{q}(\Omega\cap B_{j}^{i})}$independent of $j=1,2,3,$$L_{q}(\Omega),$

be a sequence of positive real numbers.

$(f, \varphi)_{\Omega}=\sum_{j=1}^{\infty}(f_{j}, \varphi)_{\Omega}$$\ldots$

for any. Then,for any$\varphi\in L_{q’}(\Omega)$$f= \sum_{j=1}^{\infty}f_{j}$$\varphi\in L_{q’}(\Omega)$(4.1)

exists in the, and$1 f1_{L_{q}(\Omega)}\leq C_{q}M_{3}(\sum_{j=1}^{\infty}g_{j}^{q})^{\frac{1}{q}}$(ii) Let$\{f_{j}\}_{j=1}^{\infty}$

be a sequence in$W_{q}^{1}(\Omega)$such that$\sum_{j=1}^{\infty}\Vert f_{j}\Vert_{W_{q}^{1}(\Omega)}^{q}<\infty$and$|(f_{j}, \varphi)_{\Omega}|\leq M_{3}\Vert f_{j}\Vert_{L_{q}(\Omega)}\Vert\varphi\Vert_{L_{q’}(\Omega\cap B_{j}^{i})}, |(D_{\ell}f_{j}, \varphi)_{\Omega}|\leq M_{3}\Vert D_{\ell}f_{j}\Vert_{L_{q}(\Omega)}\Vert\varphi\Vert_{L_{q’}(\Omega\cap B_{j}^{i})}$for any$\varphi\in L_{q’}(\Omega)$$W_{q}^{1}(\Omega)$withand $\ell=1,$$\ldots$, N. Then,$f= \sum_{j=1}^{\infty}f_{j}$exists in the strong topology of$\Vert f\Vert_{L_{q}(\Omega)}\leq C_{q}M_{3}(\sum_{j=1}^{\infty}\Vert f_{j}\Vert_{L_{q}(\Omega)}^{q})^{\frac{1}{q}}, \Vert\nabla f\Vert_{L_{q}(\Omega)}\leqC_{q}M_{3}(\sum_{j=1}^{\infty}\Vert\nabla f_{j}\Vert_{L_{q}(\Omega)}^{q})^{\frac{1}{q}}$

14
(iii)

Let
positive numbers. Let
is satisfied with$\{f_{j}^{(i)}\}_{j=1}^{\infty}(i=1,2)$

be sequences ofand letbe sequences inand be any complex numbers. Assume that the condition (4.1). In addition, we assume that
and$\{g_{j}^{(i)}\}_{j=1}^{\infty}(i=1,2)$$L_{q}(\Omega)$

$b$

$a$

$f_{j}=f_{j}^{(i)}$$g_{j}=g_{j}^{(i)}$$|(af_{j}^{(1)}+bf_{j}^{(2)}, \varphi)_{\Omega}|\leq M_{3}g_{j}^{(3)}\Vert\varphi\Vert_{L_{q’}(\Omega\cap B_{j}^{i})}$with some sequenceThen,$\{g_{j}^{(3)}\}_{j=1}^{\infty}$of positive numbers satisfying condition:$af^{(1)}+bf^{(2)}= \sum_{j=1}^{\infty}(af_{j}^{(1)}+bf_{j}^{(2)})\in L_{q}(\Omega)$$\sum_{j=1}^{\infty}(g_{j}^{(3)})^{q}<\infty.$,$\Vert af^{(1)}+bf^{(2)}\Vert_{L_{q}(\Omega)}\leq C_{q}M_{3}(\sum_{j=1}^{\infty}(g_{j}^{(3)})^{q})^{\frac{1}{q}}$In the following, we write
simplicity. The following proposition is used to define the infinite sum of
defined on$\mathcal{H}_{j}^{0}=\mathbb{R}^{N},$$\mathcal{H}_{j}^{i}=\Phi_{j}^{i}(\mathbb{R}_{+}^{N}),$for the sake of-bounded operators$\partial \mathcal{H}_{j}^{i}=\Phi_{j}^{i}(\mathbb{R}_{0}^{N})(i=1,2)$$\mathcal{R}$$\mathcal{H}_{j}^{i}.$Proposition 4.3. Let $1<q<\infty,$ $q’=q/(q-1)$ and $i=0,1,2$ . Let
Then, the following assertions hold.

$\Lambda$

be a domain in

and letbe an operator family in(i) Let. Assume that there exist constants
operator families in Anal
independent of $j=1,2,3,$
such that$\mathcal{F}(\lambda)(\lambda\in\Lambda)$$\mathcal{G}_{k}(\lambda)(k=1, \ldots, K)$$\mathcal{L}(L_{q}(\mathcal{H}_{j}^{i}))$$(\Lambda, \mathcal{L}(L_{q}(\mathcal{H}_{j}^{i})))$

and$M_{4}$$\mathbb{C}.$be$M_{5,k}$$\ldots$$|( \sum_{\ell=1}^{n}a_{\ell}\mathcal{F}(\lambda_{\ell})f_{\ell}, \varphi)_{\mathcal{H}_{j}^{i}}|\leq M_{4}(\sum_{k=1}^{K}\Vert\sum_{\ell=1}^{n}a\ell \mathcal{G}_{k}(\lambda_{\ell})f_{\ell}\Vert_{L_{q}(\mathcal{H}_{j}^{i})})\Vert\varphi\Vert_{L_{q}(\mathcal{H}_{j}^{i})},$$\mathcal{R}_{\mathcal{L}(L_{q}(\mathcal{H}_{j}^{i}))}(\{(\lambda\frac{d}{d\lambda})^{\ell}\mathcal{G}_{k}(\lambda)|\lambda\in\Lambda\})\leq M_{5,k} (\ell=0,1, k=1, \ldots, K)$

for any$L_{q}(\mathcal{H}_{j}^{i})$$\varphi\in L_{q’}(\mathcal{H}_{j}^{i})$. Then,and for any integer $n,$
and
Anal

$\mathcal{F}(\lambda)\in$$\{a_{\ell}\}_{\ell=1}^{n}\subset \mathbb{C},$$\{\lambda_{\ell}\}_{\ell=1}^{n}\subset\Lambda$

and$\mathcal{R}_{\mathcal{L}(L_{q}(\mathcal{H}^{i}\prime))J}(\{(\lambda\frac{d}{d\lambda})^{\ell}\mathcal{F}(\lambda)|\lambda\in\Lambda\})\leq C_{q}M_{4}(\sum_{k=1}^{K}M_{5,k}^{q})^{1/q} (\ell=0,1)$

be a sequence in Anal$K)$ be sequences in Anal
$M_{7,K}$ independentof $j=1,2,3\ldots$ such that(ii) Let$1,$$\{f_{\ell}\}_{\ell=1}^{n}\subset$$(\Lambda, \mathcal{L}(L_{q}(\mathcal{H}_{j}^{i})))$and let. Assume that there exist constants$(\Lambda, \mathcal{L}(L_{q}(\mathcal{H}_{j}^{i}), L_{q}(\Omega)))$$\{\mathcal{F}_{j}(\lambda)\}_{j=1}^{\infty}$$(\Lambda, \mathcal{L}(L_{q}(\mathcal{H}_{j}^{i})))$$\ldots,$.

$\{\mathcal{G}_{jk}(\lambda)\}_{j=1}^{\infty}(k=$$\mathcal{R}_{\mathcal{L}(L_{q}(\mathcal{H}_{j}^{i}))}(\{(\lambda\frac{d}{d\lambda})^{\ell}\mathcal{G}_{jk}(\lambda)|\lambda\in\Lambda\})\leq M_{7,k} (\ell=0,1, k=1, \ldots, K)$$M_{6}$

and,$|( \sum_{\ell=1}^{n}a_{\ell}\mathcal{F}_{j}(\lambda_{\ell})f_{\ell}, \varphi)_{\Omega}|\leq M_{6}(\sum_{k=1}^{K}\Vert\sum_{\ell=1}^{n}a\ell \mathcal{G}_{jk}(\lambda_{\ell})f_{\ell}\Vert_{L_{q}(\mathcal{H}_{j}^{i})})\Vert\varphi\Vert_{L_{q’}(\Omega\cap B_{j}^{1)}}$for any integer$L_{q’}(\Omega)$. Let$\theta_{j}^{i}$$n,$$\{a_{\ell}\}_{\ell=1}^{n}\subset \mathbb{C},$be operators in$\{\lambda_{\ell}\}_{\ell=1}^{n}\subset\Lambda$

andand for any
such that$\{f_{\ell}\}_{\ell=1}^{n}\subset L_{q}(\mathcal{H}_{j}^{i})$$\mathcal{L}(L_{q}(\Omega), L_{q}(\mathcal{H}_{j}^{i}))(j=1,2,3, \ldots)$$\varphi\in$$\Vert\theta_{j}^{i}f\Vert_{L_{q}(\mathcal{H}_{\dot{j}})}\leq$15with some constantfor any $f\in L_{q}(\Omega)$ . Then, there exists an operAnal
such thatin the strong
andfor any$M_{8}$$M_{8}\Vert f\Vert_{L_{q}(\Omega\cap B_{j}^{i})}$ator
topology of$\mathcal{F}(\lambda)\in$$(\Lambda, \mathcal{L}(L_{q}(\Omega), L_{q}(\Omega)))$$\mathcal{F}(\lambda)f=\sum_{j=1}^{\infty}\mathcal{F}_{j}(\lambda)\theta_{j}^{i}f$$f\in L_{q}(\Omega)$$L_{q}(\Omega)$$\mathcal{R}_{\mathcal{L}(L_{q}(\Omega),L_{q}(\Omega))}(\{(\lambda\frac{d}{d\lambda})^{p}\mathcal{F}(\lambda)|\lambda\in\Lambda\})\leq C_{q}M_{6}(\sum_{k=1}^{K}M_{7,k}^{q})^{1/q}M_{8} (\ell=0,1)$.

Using Theorem 2.2, Theorem 3.4 and Theorem 3.5, we construct a parametrix. For $f\in$$g\in L_{q}(\Omega),$
, let
and
be solutions to thefollowing variational equations:$E_{q}(\Omega)^{N},$$h_{d}\in W_{q}^{1}(\Omega)$$h_{n}\in W_{q}^{1}(\Omega)$$u_{j}^{i}\in W_{q}^{1}(\mathcal{H}_{j}^{i})$for any$(\lambda u_{j}^{0}, \varphi)_{\mathcal{H}_{j}^{0}}+(\nabla u_{j}^{0}, \nabla\varphi)_{\mathcal{H}_{j}^{0}}=-(\tilde{\zeta}_{j}^{0}f, \nabla\varphi)_{\mathcal{H}_{j}^{0}}+(\tilde{\zeta}_{j}^{0}g, \varphi)_{\mathcal{H}_{j}^{0}}$$(\lambda u_{j}^{1}, \varphi)_{\mathcal{H}_{j}^{1}}+(\nabla u_{j}^{1}, \nabla\varphi)_{\mathcal{H}_{j}^{1}}=-(\tilde{\zeta}_{j}^{1}f, \nabla\varphi)_{\mathcal{H}_{j}^{1}}+(\tilde{\zeta}_{j}^{1}g, \varphi)_{\mathcal{H}_{J}^{1}}$,subject to$u_{j}^{1}=\tilde{\zeta}_{j}^{1}h_{d}$on$\partial \mathcal{H}_{j}^{1}$for any$\varphi\in W_{q}^{1},(\mathcal{H}_{j}^{0})$,$\varphi\in W_{q_{\}}0}^{1}(\mathcal{H}_{j}^{1})$(4.4)$\varphi>\partial \mathcal{H}_{j}^{2}$$T_{j}^{2}(\lambda)\in$

(4.3)

, and

$(\lambda u_{j}^{2}, \varphi)_{\mathcal{H}_{j}^{2}}+(\nabla u_{j}^{2}, \nabla\varphi)_{\mathcal{H}_{j}^{2}}=-(\tilde{\zeta}_{j}^{2}f, \nabla\varphi)_{\mathcal{H}_{j}^{2}}+(\tilde{\zeta}_{j}^{2}g, \varphi)_{\mathcal{H}_{j}^{2}}+<\tilde{\zeta}_{j}^{2}h_{n},$for anyfamilies(4.2). By Theorem 2.2, Theorem 3.4 and Theorem 3.5 there exist operator
Anal
Anal
and
such that$\varphi\in W_{q}^{1},(\mathcal{H}_{j}^{2})$$T_{j}^{0}(\lambda)\in$

Anal$(\Sigma_{\epsilon}, \mathcal{L}(\mathcal{X}_{q0}(\mathcal{H}_{j}^{0}), W_{q}^{1}(\mathcal{H}_{j}^{0}))),$$T_{j}^{1}(\lambda)\in$$(\Sigma_{\epsilon}, \mathcal{L}(\mathcal{X}_{qd}(\mathcal{H}_{j}^{1}), W_{q}^{1}(\mathcal{H}_{j}^{1})))$$(\Sigma_{\epsilon}, \mathcal{L}(\mathcal{X}_{qn}(\mathcal{H}_{j}^{2}), W_{q}^{1}(\mathcal{H}_{j}^{2})))$$u_{j}^{0}=T_{j}^{0}(\lambda)(\tilde{\zeta}_{j}^{0}f, \lambda^{-1/2}\tilde{\zeta}_{j}^{0}g)$,$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{q0}(\mathcal{H}_{j}^{O}),L_{q}(\mathcal{H}_{j}^{0})^{N+1})}(\{(\lambda\frac{d}{d\lambda})^{\ell}(\lambda^{1/2}, \nabla)T_{j}^{0}(\lambda)|\lambda\in\Sigma_{\epsilon}\})\leq\beta_{2} (\ell=0,1)$$u_{j}^{1}=T_{j}^{1}(\lambda)F_{\lambda}^{d}(\tilde{\zeta}_{j}^{1}f,\tilde{\zeta}_{j}^{1}g,\tilde{\zeta}_{j}^{1}h_{d})$

(4.5)

,

(4.6)

,

$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{qd}(\mathcal{H}_{j}^{1}),L_{q}(\mathcal{H}_{j}^{1})^{N+1})}(\{(\lambda\frac{d}{d\lambda})^{\ell}(\lambda^{1/2}, \nabla)T_{j}^{1}(\lambda)|\lambda\in\Sigma_{\epsilon}\})\leq\beta_{2} (\ell=0,1)$$u_{j}^{2}=T_{j}^{2}(\lambda)F_{\lambda}^{n}(\tilde{\zeta}_{j}^{2}f,\tilde{\zeta}_{j}^{2}g,\tilde{\zeta}_{j}^{2}h_{n})$,,$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{qn}(\mathcal{H}_{j}^{2}),L_{q}(\mathcal{H}_{j}^{2})^{N+1})}(\{(\lambda\frac{d}{d\lambda})^{\ell}(\lambda^{1/2}, \nabla)T_{j}^{2}(\lambda)|\lambda\in\Sigma_{\epsilon}\})\leq\beta_{2} (\ell=0,1)$

with some constant
independent of and . Set-boundedness implies the boundedness, by (4.5), (4.6) and (4.7) we have$i$$\beta_{2}$$j$$u= \sum_{i=0}^{2}\sum_{j=1}^{\infty}\zeta_{j}^{i}u_{j}^{i}$(4.7). Noting that the$\mathcal{R}$$\Vert(\lambda^{1/2}u_{j}^{i}, \nabla u_{j}^{i})\Vert_{L_{q}(\mathcal{H}_{j}^{i})}\leq\beta_{2}\{\begin{array}{l}\Vert(\tilde{\zeta}_{j}^{0}f, \lambda^{-1/2}\tilde{\zeta}_{j}^{0}g)\Vert_{L_{q}(\mathcal{H}_{j}^{0})},\Vert(\tilde{\zeta}_{j}^{1}f, \lambda^{-1/2}\tilde{\zeta}_{j}^{1}g, \lambda^{1/2}\tilde{\zeta}_{j}^{1}h_{d}, \nabla(\tilde{\zeta}_{j}^{1}h_{d}))\Vert_{L_{q}(\mathcal{H}_{j}^{1})},\Vert(\tilde{\zeta}_{j}^{2}f, \lambda^{-1/2}\tilde{\zeta}_{j}^{2}g,\tilde{\zeta}_{j}^{2}h_{n}, \lambda^{-1/2}\nabla(\tilde{\zeta}_{j}^{2}h_{n}))\Vert_{L_{q}(\mathcal{H}_{J}^{2})}\prime.\end{array}$By Proposition 4.2, we have$u\in W_{q}^{1}(\Omega)$

and$\Vert(\lambda^{1/2}u, \nabla u)\Vert_{L_{q}(\Omega)}\leq C_{q,\Omega}\beta_{2}\Vert(f, \lambda^{-1/2}g, \lambda^{1/2}h_{d}, \nabla h_{d}, h_{n}, \lambda^{-1/2}\nabla h_{n})\Vert_{L_{q}(\Omega)}$

for any
have for any$\lambda\in\Sigma_{\epsilon,1}$. Using Proposition 4.2 and noting that$\tilde{\zeta}_{j}^{i}=1$on$supp\zeta_{j}^{i}$, by $(4.2)-(4.7)$ we$\varphi\in W_{q,\Gamma_{1}}^{1}(\Omega)$$( \lambda u, \varphi)_{\Omega}+(\nabla u, \nabla\varphi)_{\Omega}=\sum_{i=0}^{2}\sum_{j=1}^{\infty}[(\lambda\zeta_{j}^{i}u_{j}^{i}, \varphi)_{\mathcal{H}_{j}^{i}}+(\zeta_{j}^{i}\nabla u_{j}^{i}, \nabla\varphi)_{\mathcal{H}_{j}^{i}}+((\nabla\zeta_{j}^{i})u_{j}^{i}, \nabla\varphi)_{\mathcal{H}_{j}^{i}}]$16$= \sum_{i=0}^{2}\sum_{j=1}^{\infty}[(\lambda u_{j}^{i}, \zeta_{j}^{i}\varphi)_{\mathcal{H}_{j}^{t}}+(\nabla u_{j}^{i}, \nabla(\zeta_{j}^{i}\varphi))_{\mathcal{H}_{j}^{i}}-((\nabla\zeta_{j}^{i})\cdot\nabla u_{j}^{i}, \varphi)_{\mathcal{H}_{j}^{i}}-(div((\nabla\zeta_{j}^{i})u_{j}^{i}), \varphi)_{\mathcal{H}_{j}^{i}}\cdot]$$+ \sum_{j=1}^{\infty}<(\nabla\zeta_{j}^{2})u_{j}^{2},$$\varphi>a\mathcal{H}_{j}^{2}$$= \sum_{i=0}^{2}\sum_{j=1}^{\infty}[-(f, \nabla(\zeta_{j}^{i}\varphi))_{\mathcal{H}_{j}^{i}}+(g, \zeta_{j}^{i}\varphi)_{\mathcal{H}_{j}^{i}}-(2(\nabla\zeta_{j}^{i})\cdot(\nabla u_{j}^{i})+(\Delta\zeta_{j}^{i})u_{j}^{i}, \varphi)_{\mathcal{H}_{j}^{i}}$$+ \sum_{j=1}^{\infty}>$$=-(f, \nabla\varphi)_{\Omega}+(g+R_{1}(f, g, h_{d}, h_{n}), \varphi)_{\Omega}+<h_{n}+R_{2}(f,g, h_{d}, h_{n}),$ $\varphi>r_{2}$where we have set$R_{1}(f,g, h_{d}, h_{n})=- \sum_{i=0}^{2}\sum_{j=1}^{\infty}\{2(\nabla\zeta_{j}^{i})\cdot(\nabla u_{j}^{i})+(\Delta\zeta_{j}^{i})u_{j}^{i}\},$. (4.8)$R_{2}(f, g, h_{d}, h_{n})= \sum_{j=1}^{\infty}(\nabla\zeta_{j}^{2})u_{j}^{2}$

Noting that$\nabla(\zeta_{j}^{1}h_{d})=\zeta_{j}^{1}\nabla h_{d}+(\nabla\zeta_{j}^{1})h_{d}, \lambda^{-1/2}\nabla(\zeta_{j}^{1}h_{d})=\zeta_{j}^{1}(\lambda^{-1/2}\nabla h_{d})+\lambda^{-1/2}(\nabla\zeta_{j}^{1})h_{d},$for$F=(F_{1}, F_{2}, F_{3}, F_{4}, F_{5}, F_{6})\in \mathcal{X}_{q}(\Omega)$

we define an operator$U(\lambda)$by$U( \lambda)F=\sum_{j=1}^{\infty}\zeta_{j}^{0}T_{j}^{0}(\tilde{\zeta}_{j}^{0}F_{1},\tilde{\zeta}_{j}^{0}F_{2})$$+ \sum_{j=1}^{\infty}\zeta_{j}^{1}\{T_{j}^{1}(\lambda)(\tilde{\zeta}_{j}^{1}F_{1},\tilde{\zeta}_{j}^{1}F_{2},\tilde{\zeta}_{j}^{i}F_{3},\tilde{\zeta}_{j}^{1}F_{4})+\lambda^{-1/2}T_{j}^{1}(\lambda)(0,0,0, (\nabla\tilde{\zeta}_{j}^{1})F_{3})\}$$+ \sum_{j=1}^{\infty}\zeta_{j}^{2}\{T_{j}^{1}(\lambda)(\tilde{\zeta}_{j}^{2}F_{1},\tilde{\zeta}_{j}^{2}F_{2},\tilde{\zeta}_{j}^{2}F_{5},\tilde{\zeta}_{j}^{2}F_{6})+\lambda^{-1/2}T_{j}^{2}(\lambda)(0,0,0, (\nabla\tilde{\zeta}_{j}^{2})F_{5})\}$

By Proposition 4.3 and $(4.5)-(4.7)$ , we have$u=U(\lambda)F_{\lambda}(f, g, h_{d}, h_{n})$

,
(4.9)

$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{q}(\Omega),W_{q}^{1}(\Omega)^{N+1})}(\{(\lambda\frac{d}{d\lambda})^{\ell}U(\lambda)|\lambda\in\Sigma_{\epsilon,1}\})\leqC_{q,\Omega}\beta_{2}.$In view of (4.8), we define operators$V^{1}(\lambda)$

and$V^{2}(\lambda)$

as follows:$V_{1}( \lambda)F=-\sum_{j=1}^{\infty}\{2(\nabla\zeta_{j}^{0})\cdot(\nabla T_{j}^{0}(\lambda)(\tilde{\zeta}_{j}^{0}F_{1},\tilde{\zeta}_{j}^{0}F_{2}))+(\triangle\zeta_{j}^{0})T_{j}^{0}(\lambda)(\tilde{\zeta}_{j}^{0}F_{1},\tilde{\zeta}_{j}^{0}F_{2}))\}$$-2 \sum_{j=1}^{\infty}(\nabla\zeta_{j}^{1})\cdot\{\nabla T_{j}^{1}(\lambda)(\tilde{\zeta}_{j}^{1}F_{1},\tilde{\zeta}_{j}^{1}F_{2},\tilde{\zeta}_{j}^{1}F_{3},\tilde{\zeta}_{j}^{1}F_{4})+\lambda^{-1/2}\nabla T_{j}^{1}(\lambda)(0,0,0,(\nabla\tilde{\zeta}_{j}^{1})F_{3})\}$$- \sum_{j=1}^{\infty}(\Delta\zeta_{j}^{1})\{T_{j}^{1}(\lambda)(\tilde{\zeta}_{j}^{1}F_{1},\tilde{\zeta}_{j}^{1}F_{2},\tilde{\zeta}_{j}^{1}F_{3},\tilde{\zeta}_{j}^{1}F_{4})+\lambda^{-1/2}T_{j}^{1}(\lambda)(0,0,0, (\nabla\tilde{\zeta}_{j}^{1})F_{3})\}$

17$-2 \sum_{j=1}^{\infty}(\nabla\zeta_{j}^{2})\cdot\{\nabla T_{j}^{2}(\lambda)(\tilde{\zeta}_{j}^{2}F_{1},\tilde{\zeta}_{j}^{22}F_{2}, \zeta_{j}^{A}F_{5},\tilde{\zeta}_{j}^{2}F_{6})+\lambda^{-1/2}\nabla T_{j}^{2}(\lambda)(0,0,0, (\nabla\tilde{\zeta}_{j}^{2})F_{5})\}$$- \sum_{j=1}^{\infty}(\triangle\zeta_{j}^{2})\{T_{j}^{1}(\lambda)(\tilde{\zeta}_{j}^{1}F_{1},\tilde{\zeta}_{j}^{2}F_{2},\tilde{\zeta}_{j}^{2}F_{5},\tilde{\zeta}_{j}^{2}F_{6})+\lambda^{-1/2}T_{j}^{2}(\lambda)(0,0,0, (\nabla\tilde{\zeta}_{j}^{2})F_{5})\},$$V_{2}( \lambda)F=\sum_{j=1}^{\infty}(\nabla\zeta_{j}^{2})\{T_{j}^{2}(\lambda)(\tilde{\zeta}_{j}^{2}F_{1},\tilde{\zeta}_{j}^{2}F_{2},\tilde{\zeta}_{j}^{2}F_{5},\tilde{\zeta}_{j}^{2}F_{6})+\lambda^{-1/2}T_{j}^{2}(\lambda)(0,0,0, (\nabla\tilde{\zeta}_{j}^{2})F_{5})\}.$

By Proposition 4.3, Proposition 3.1, Propsition 3.2 and $(4.5)-(4.8)$ , we have$V_{1}(\lambda)F_{\lambda}(f, g, h_{d}, h_{n})=R_{1}(f, g, h_{d}, h_{n})$,$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{q}(\Omega),L_{q}(\Omega))}(\{(\lambda\frac{d}{d\lambda})^{\ell}\lambda^{-1/2}V_{1}(\lambda)|\lambda\in\Sigma_{\epsilon,\lambda_{0}}\})\leq C_{q,\Omega}\lambda_{0}^{-1/2} (\ell=0,1)$$V_{2}(\lambda)F_{\lambda}(f, g, h_{d}, h_{n})=R_{2}(f, g, h_{d}, h_{n})$,(4.10)$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{q}(\Omega),L_{q}(\Omega))}(\{(\lambda\frac{d}{d\lambda})^{\ell}V_{2}(\lambda)|\lambda\in\Sigma_{\epsilon,\lambda_{0}}\})\leq C_{q,\Omega}\lambda_{0}^{-1/2} (\ell=0,1)$$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{q}(\Omega),L_{q}(\Omega))}(\{(\lambda\frac{d}{d\lambda})^{\ell}\lambda^{-1/2}\nabla V_{2}(\lambda)|\lambda\in\Sigma_{\epsilon,\lambda_{0}}\})\leq C_{q,\Omega}\lambda_{0}^{-1/2} (\ell=0,1)$

with some constantthen by (4.10)$C_{q,\Omega}$

depending solely on

$q$

and

$\Omega.$$If\cdot we$set.

$\mathcal{V}(\lambda)F=(0, V_{1}(\lambda)F, 0, V_{2}(\lambda)F)$.

(4.11)$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{q}(\Omega),L_{q}(\Omega))}(\{(\lambda\frac{d}{d\lambda})^{\ell}F_{\lambda}\mathcal{V}(\lambda)|\lambda\in\Sigma_{\epsilon,\lambda_{0}}\})\leq C_{q,\Omega}\lambda_{0}^{1/2} (\ell=0,1)$

Therefore, choosing$\lambda_{0}$so large that

$C_{q,\Omega}\lambda_{0}^{-1/2}\leq 1/2$

, then$(I+F_{\lambda}\mathcal{V}(\lambda))^{-1}$exists and$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{q}(\Omega),L_{q}(\Omega))}(\{(\lambda\frac{d}{d\lambda})^{p}F_{\lambda}(I+F_{\lambda}\mathcal{V}(\lambda))^{-1}|\lambda\in\Sigma_{\epsilon,\lambda_{0}}\})\leq 2 (\ell=0,1)$

If we set $R(f, g, h_{d}, h_{n})=(0, R_{1}(f, g, h_{d}, h_{n}),$$0,$ $R_{2}(f, g, h_{d}, h_{n})$.

(4.12), then by (4.11)$\Vert F_{\lambda}R(f, g, h_{d}, h_{n})\Vert_{L_{q}(\Omega)}\leq(1/2)\Vert F_{\lambda}(f, g, h_{d}, h_{n})\Vert_{L_{q}(\Omega)}.$Since$\Vert F_{\lambda}R(f, g, h_{d}, h_{n})\Vert_{L_{q}(\Omega)}=\Vert(f, \lambda^{-1/2}g, \lambda^{1/2}h_{d}, \nabla h_{d}, h_{n}, \nabla^{-1/2}\nabla h_{n})\Vert_{L_{q}(\Omega)}$give us equivalent norms on$(I+R)^{-1}$ exists inforfor any
which combined with (4.10) furnishes that $u=U(\lambda)F_{\lambda}(I+R)^{-1}(f, g, h_{d}, h_{n})$ is a unique solution to (1.3). Here, the uniqueness follows from the existence theorem for the dual problem. By (4.9) and (4.11), and therefore $F_{\lambda}(I+R)^{-1}=$, which furnishes that
. Setting, by (4.10) and (4.12) we have$X_{q}(\Omega)$$\lambda\neq 0,$$\mathcal{L}(X_{q}(\Omega)$$\lambda\in\Sigma_{\epsilon,\lambda_{0}},$$R(f, g, h_{d}, h_{n})=\mathcal{V}(\lambda)F_{\lambda}(f, g, h_{d}, h_{n})$$(I+F_{\lambda}\mathcal{V}(\lambda))^{-1}F_{\lambda}$$u=U(\lambda)(I+F_{\lambda}\mathcal{V}(\lambda))^{-1}F_{\lambda}(f, g, h_{d}, h_{n})$$\mathcal{A}(\lambda)=U(\lambda)(I+F_{\lambda}\mathcal{V}(\lambda))^{-1}$$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{q}(\Omega),L_{q}(\Omega))}(\{(\lambda\frac{d}{d\lambda})^{p}(\lambda^{1/2}, \nabla)\mathcal{A}(\lambda)|\lambda\in\Sigma_{\epsilon,\lambda_{0}}\})\leq 2C_{q,\Omega}\beta_{2} (\ell=0,1)$

which completes the proof of Theorem 1.5.

,

,

18
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