[This work is supported by the National
Science Foundation of China, grants no. 12071463 and Key Research Program of Frontier Sciences,
CAS, no. QYZDJ-SSW-SYS011.]

Abstract    We prove the regularity of solutions to the strain tensor equation on a region $S$ with the Gauss curvature changing sign.
Furthermore, we obtain the density property that smooth infinitesimal isometries are dense in the $W^{2,2}({S},\R^3)$ infinitesimal isometries. Finally, the matching property is established. Those results are  important tools in
obtaining recovery sequences ($\Ga$-lim sup inequality) for dimensionally-reduced shell theories
in elasticity.

Keywords    shell, nonlinear elasticity, Riemannian geometry, tensor analysis

Mathematics  Subject Classifications
(2010)   74K20(primary), 74B20(secondary).

§ INTRODUCTION AND MAIN RESULTS

Let $M\subset\R^3$ be a surface with a normal $\n$ and let the middle surface of a shell be  an open set $S\subset M.$ Let $T^kS$ denote  all the $k$-order tensor fields on $S$ for an integer $k\geq0.$ Let $T^2_{\sym}S$ be all the $2$-order symmetrical tensor fields on $S.$ For $y\in \WW^{1,2}(S,\R^3),$ we decompose it into $y=W+w\n,$ where $w=\<y,\n\>$ and $W\in TS.$
For $U\in T^2_\sym S$ given, linear strain tensor of a displacement $y\in\WW^{1,2}({S},\R^3)$ of the middle surface ${S}$ takes the form
DW+wΠ=Ux∈S,where $D$ is the connection of the induced metric in $M,$  $2\sym DW=DW+D^TW,$ and $\Pi$ is the second fundamental form of $M.$
Equation (<ref>) plays a fundamental role in the theory of thin shells, see [4, 6, 5, 7, 13, 15] and many others. When $U=0,$ a solution $y$ to (<ref>) is referred to as an infinitesimal isometry.

The type of equation (<ref>) depends on the sign of the curvature on the region S: It is elliptic if $S$ has positive curvature; it is parabolic if the curvature is zero but $\Pi\not=0$ on $S;$ it is hyperbolic if $S$ has negative curvature. When the curvature of the region $S$ changes its sign, problem (<ref>) is of changing type.

Here  we establish the regularity of solutions to (<ref>) when $S$ is a region with its curvature changing sign, which will be specified below.
Then it
is proved that smooth infinitesimal isometries are dense in the $W^{2,2}({S},\R^3)$ infinitesimal isometries on the region $S.$ Finally, the matching property is derived that
smooth enough infinitesimal isometries can be matched with higher
order infinitesimal isometries. Those results are  important tools in
obtaining recovery sequences ($\Ga$-lim sup inequality) for dimensionally-reduced shell theories
in elasticity, when the elastic energy density scales like $h^\b,$ $\b\in(2, 4),$ that is, intermediate
regime between "pure bending" ($\b=2$) and the von-Kármán regime ($\b=4$). Such results have been obtained for elliptic surfaces [7], developable surfaces [4], and  hyperbolic surfaces [13, 15]. Moreover, the case of degenerated hyperbolic surfaces has been studied in [1]. A survey on this topic is presented in [5].

Mixed type equations also arise naturally in many other areas.  A detailed account of the historical background and known results on mixed type equations and transonic flows is given in [8].
A detailed review on mixed type equations and Riemannian-Lorentzian metrics is presented in [11]. The most intensively studied equation of mixed type is the Tricomi equation [10].

In this paper we study the mixed type equation (<ref>) which is very different from all the above cases. Equation (<ref>) is equivalent to a mixed type scalar equation of the form
D^2w,Q^*Π+1κDw, X_0+κ(_gΠ)w=κf+1κX_0,F+DF,Q^*Πx∈S,$\kappa\not=0,$ where $w\in\LL^2(S)$ is the unknown, $f\in\LL^2(S)$ and $F\in\LL^2(S,TS)$ are given,
and $\kappa$ is the Gauss curvature. The type of (<ref>) is subject to the sign of $\kappa.$ When $\kappa=0,$ there are two terms degenerating in (<ref>): The coefficient of $\<Dw-F,X_0\>$ becomes
from $+\infty$ to $-\infty$ as a point crosses a zero curvature curve from the positive curvature to the negative. The coefficient of a second derivative of $w$ along the direction of the zero principal curvature changes from positive to negative. Those situations challenge the analysis of (<ref>).

We observe that problem (<ref>) is equivalent to a vector field system as

where $\nabla\n:$ $S_x\rw S_x$ is the second fundamental form operator, $(V,v)$ is the unknown where $V$ is a vector field and $v$ is function, and $(F,f)$ is as in (<ref>).
Problem (<ref>) changes  type when the curvature changes  sign. In order to have solutions to problem (<ref>), one normally starts to solve problem (<ref>) in the previous works as in [1, 4, 7, 13]. However, it makes the solvability of (<ref>) extremely difficult that  the coefficient of $\<Dw-F,X_0\>$ varies between $+\infty$ and $-\infty$ when the curvature changes  sign. Instead of analysing (<ref>), here we begin from (<ref>) and then obtain solutions to (<ref>). The hard  task is to achieve regularity of solutions to (<ref>) by some priori estimates near the zero curvature curve. Thanks to the Bochner technique and the tensor analysis,  the regularity analysis has been complete in Section 3.

We state our main results as follows.
Let $S\subset M$ be given by
S={ (t,s) | (t,s)∈[0,a)×(-b_0,b_1) },  a>0,  b_0>0,  b>0,where $\a:$ $[0,a)\times[0,b]\rw M$ is an imbedding map which is a family of regular curves with two parameters $t,$ $s$ such that
Π(_t(t,s),_t(t,s))>0,    (t,s)∈[0,a)×[-b_0,0],where $\a(\cdot,s)$ is a closed curve with  period $a$ for each $s\in[-b_0,b_1].$  Set

$$S=S^+\cup\Ga_0\cup S^-,$$

$$S^+=\{\,\a(t,s)\,|\,(t,s)\in[0,a)\times(0,b)\,\},\quad \Ga_0=\{\,\a(t,0)\,|\,t\in[0,a]\,\},$$

$$ S^-=\{\,\a(t,s)\,|\,(t,s)\in[0,a)\times(-b_0,0)\,\}.$$

Curvature assumptions    Let $\kappa$ be the Gaussian curvature function on $M.$ We assume that $S$ satisfies the following curvature conditions:
κ(x)>0x∈S^+∪_b_1;κ=0,  Dκ(x)≠0x∈_0;κ(x)<0x∈S^-∪_-b_0,where

$$\Ga_{b_1}=\{\,\a(t,b_1)\,|\,t\in[0,a)\,\},\quad \Ga_{-b_0}=\{\,\a(t,-b_0)\,|\,t\in[0,a)\,\}.$$

Our main results are the following.

Let ${S}$ be  of class $\CC^{m+5}$  for some  integer $m\geq1.$ For $U\in\WW^{m+1,2}({S},T^2_{\sym}S),$ there exists a solution $y=W+w\n\in\WW^{m,2}({S},\R^3)$ to equation $(\ref{01})$ satisfying

$$ \|W\|_{\WW^{m+1,2}({S},TS)}^2+\|w\|_{\WW^{m,2}({S})}^2\leq C\|U\|_{\WW^{m+1,2}({S},T^2_{\sym}S)}^2.$$

The curvature of $S$ affects the regularity of $(\ref{01})$ as

$$\begin{cases} \|y\|^2_{\WW^{m,2}(S,\R^3)}\leq C\|U\|_{\WW^{m,2}({S},T^2_{\sym}S)}^2\quad\mbox{if $\kappa>0$ on $\overline{S}$};\\
\|y\|^2_{\WW^{m,2}(S,\R^3)}\leq C\|U\|_{\WW^{m+1,2}({S},T^2_{\sym}S)}^2\quad\mbox{if $\kappa<0$ on $\overline{S}$.}
\end{cases}$$

By the imbedding theorem <cit.>, the following corollary is immediate.

Let $S$ be of ${S}\in\CC^{m+5}$ for some integer $m\geq0.$ Then problem $(\ref{01})$ admits a solution $y=W+w\n\in\CC^m_B(S,\R^3)$ satisfying

$$ \|W\|_{\CC^{m+1}_B({S},TS)}+\|w\|_{\CC^{m}_B(S)}\leq C\|U\|_{\CC^{m+3}_B({S},T^2_{\sym}S)},$$

$$\CC^m_B(S,\R^3)=\{\,y\in\CC^m(S,\R^3)\,|\,D^\a y\in\LL^\infty(S,\R^3)\,\,\mbox{for}\,\,|\a|\leq m\,\}.$$

For $y\in\WW^{1,2}({S},\R^3),$ we denote the left hand side of equation (<ref>) by $\sym\nabla y.$

$$\V({S},\R^3)=\{\,y\in\WW^{2,2}({S},\R^3)\,|\,\sym\nabla y=0\,\}.$$

Let ${S}$ be  of class $\CC^{m+5,1}.$  Then, for every $y\in\V({S},\R^3)$
there exists a sequence $\{\,y_k\,\}\subset\V({S},\R^3)\cap \CC^m_B({S},\R^3)$ such that

A one parameter family $\{\,y_\varepsilon\,\}_{\varepsilon>0} \subset\CC^1_B(\overline{{S}},\R^3)$ is said to be a (generalized)
$m$th order infinitesimal isometry if the change of metric induced by $y_\varepsilon$ is of order $\varepsilon^{m+1},$ that is,

$$\|\nabla^Ty_\varepsilon\nabla y_\varepsilon-g\|_{L^\infty({S},T^2)}=\O(\varepsilon^{m+1})\quad\mbox{as}\quad \varepsilon\rw0,$$
where $g$ is the induced metric of $M$ from $\R^3,$ see [4].
A given $m$th order infinitesimal isometry can be modified by higher order
corrections to yield an infinitesimal isometry of order $m_1>m,$ a property to which we
refer to by matching property of infinitesimal isometries, [4, 7].

Let ${S}$ be  of class $\CC^{4m,1}.$ Given $y\in\V({S},\R^3)\cap\CC^{4m-2}_B({S},\R^3),$
there exists a family $\{\,z_\varepsilon\,\}_{\varepsilon>0}\subset\CC^2_B({S},\R^3),$ equi-bounded in $\CC^2_B({S},\R^3),$ such
that for all small $\varepsilon>0$  the family:

$$ y_\varepsilon=\id+\varepsilon y+\varepsilon^2z_\varepsilon$$

is a  $m$th order infinitesimal isometry of class $\CC^2_B({S},\R^3).$

Application to elasticity of thin shells
Let $\n$ be the normal field of surface $M.$ Consider a family $\{\,S_h\,\}_{h>0}$ of thin shells of thickness $h$ around $S,$

$$S_h=\{\,x+t\n(x)\,|\,x\in S,\,\,|t|<h/2\,\},\quad 0<h<h_0,$$
where $h_0$ is small enough so that the projection map $\pi:$  $S_h\rw S,$ $\pi(x+t\n)=x$ is well defined.
For a $\WW^{1,2}$ deformation $u_h:$ $S_h\rw\R^3,$ we assume that its elastic energy (scaled per
unit thickness) is given by the nonlinear functional:

$$E_h(u_h)=\frac{1}{h}\int_{S_h}W(\nabla u_h)dz.$$

The stored-energy density function $W:$ $\R^3\times\R^3\rw\R$ is $\CC^2$ in an open neighborhood of
SO(3), and it is assumed to satisfy the conditions of normalization, frame indifference and
quadratic growth: For all $F\in\R^3\times\R^3,$ $R\in\SO(3),$

$$W(R)=0,\quad W(RF)=W(F),\quad W(F)\geq C\dist^2(F,\SO(3)),$$

with a uniform constant $C>0.$ The potential $W$ induces the quadratic forms ([2])

$$\Q_3(F)=D^2W(Id)(F,F),\quad\Q_2(x,F_{\tan})=\min\{\,Q_3(\hat F)\,|\,\hat F=F_{\tan}\,\}.$$

Let $A:$ $S\rw\R^{3\times3}$ be a matrix field. We define $A\in T^2S$ by

$$A(\a,\b)=\<A(x)\a,\b\>\qfq\a,\,\,\b\in T_xS,\quad x\in S.$$

For given $V\in\V(\Om,\R^3),$ there exists a unique $A\in\WW^{1,2}(S,T^2)$ such that
∇_V=A(x)∈T_xS,     A(x)=-A^T(x),  x∈.

We shall consider a sequence $e_h>0$ such that:
0<lim_h0e_h/h^<̱∞  Let

Recall the following
Let $S$ be a surface embedded in $\R^3,$ which is compact, connected, oriented,
of class $\CC^{1,1},$ and whose boundary $\pl  S$ is the union of finitely many Lipschitz curves.
Let $u_h\in\WW^{1,2}(S_h,\R^3)$ be a sequence of deformations whose scaled energies $E_h(u_h)/e_h$ are
uniformly bounded. Then there exist a sequence $Q_h\in\SO(3)$ and $c_h\in\R^3$ such that for the
normalized rescaled deformations

$$y_h(z)=Q_hu_h(x+\frac{h}{h_0}t\n(x))-c_h,\quad z=x+t\n(x)\in S_{h_0},$$
the following holds.

$(i)$ $y_h$ converge to $\pi$ in $\WW^{1,2}(S_{h_0},\R^3).$

$(ii)$ The scaled average displacements

$$ V_h(x)=\frac{h}{h_0\sqrt{e_h}}\int_{-h_0/2}^{h_0/2}[y_h(x+t\n)-x]dt$$

converge to some $V\in\V(S,\R^3).$

$(iii)$ $\lim\inf_{h\rw0} E_h(u_h)/e_h\geq I(V),$ where
I(V)=1/24∫_S_2(x,(∇(A)-A∇)_tan)dg,where $A$ is given in $(\ref{1.5}).$

The above result proves the lower bound for the $\Ga$-convergence. With Theorems <ref> and <ref> a recover sequence can be constructed in the $\Ga$-limit for thin shells as in [4, 7, 13] such that the following theorem holds true. The details of the proof are omitted.

Let $S\subset M$ be  of class $\CC^{5,1}$ given in $(\ref{Q0})$ with $(\ref{Pi2.801})-(\ref{kappa-}).$ Suppose that

Then for every $V\in\V(S,\R^3)$ there exists a sequence of deformations $\{\,u_h\,\}\subset\WW^{1,2}(S,\R^3)$ such that $(i)$ and $(ii)$ of Theorem $\ref{t1.4}$ hold. Moreover,
lim_h01/e_hE_h(u_h)=I(V),where $I(V)$ is given in $(\ref{1.4}).$

§ $\LL^2$ SOLUTIONS OF THE TENSOR EQUATION OF MIXED TYPE

Let $\nabla$ and $D$ denote the connection of $\R^3$ in the Euclidean metric and the one of $M$ in the reduced metric,
respectively. We have to treat the relationship between
$\nabla$ and $D$ carefully.

Let $m\geq1$ be an integer. Let $T\in T^mM$ be a $m$th order tensor field on $M.$ We define a $m-1$th order tensor field by

$$\ii_YT(Y_1,\cdots,Y_{m-1})=T(Y,Y_1,\cdots,Y_{m-1})\qfq Y_1,\,\,\cdots,\,Y_{m-1}\in TM,$$
which is called an inner product of $T$ with $Y.$ For any $T\in T^2S$ and $\a\in T_xM,$

$$\tr_g\ii_\a  DT$$
is a linear functional on $T_xM,$ where $\tr_g\ii_\a DT$ is the trace of the 2-order tensor field $\ii_\a DT$ in the induced metric $g.$ Thus there is a vector, denoted by $\div_gT,$ such that

$$\<\div_gT,\a\>=\tr_g\ii_\a DT\qfq \a\in T_xM,\,\,x\in M.$$
Clearly, the above formula defines a vector field $\div_gT\in TM.$

We need a linear operator $Q$ ([13], [15]) as follows. For each point
$p\in M$, the Riesz representation theorem implies that there
exists an isomorphism $Q:$ $T_pM\rw T_pM$ such that

,Q=(,,̱n⃗(p)), ∈̱T_pM.Let $e_1,$ $e_2$ be an orthonormal basis of $T_pM$ with
positive orientation, that is,

Then $Q$ can be expressed explicitly by
Q=,e_2e_1-,e_1e_2    ∈T_pM.Clearly, $Q$ satisfies

$$ Q^T=-Q,\quad Q^2=-\Id.$$

Operator $Q$ plays an important role in our analysis.

Here we sum up some formulas about operator $Q$ and their proofs have been given in [1].
The following identities hold.
QD_XY=D_X(QY)X, Y∈TM.X,YZ=Z,YX+Z,QXQYX, Y, Z∈TM.

Let $P\in T^2M.$
Let $X$ and $Y$ be vector fields and $f$ be a function. Then

Let $P\in T^2M$ and
let $p\in M$ be given. Then
Qv,wQP+Pv,w𝕀=Qv⊗QP w+Pv⊗w v,  w∈T_pM.

For any $X,$ $Y\in TS,$

For any $Z,$ $X,$ $V,$ $W\in TM,$
X,ZW,QXQ∇V+W,∇XV,  Z

Using (<ref>) where $Y=Z$ and $Z=V,$ we have

$$\<X,Z\>V=\<V,Z\>X+\<V,QX\>QZ\qfq p\in M.$$

$$\<X,Z\>W=\<W,Z\>X+\<W,QX\>QZ\qfq p\in M.$$
Thus we obtain

=W,QXQ∇(X,Z V)+X,ZW,∇XV

=W,QXQ∇(V,ZX+V,QXQZ)+W,ZX+W,QXQZ,  ∇XV

+W,QXQZ,∇XVp∈M.It follows by $Q^T=-Q$ and $\<QZ,Z\>=0$ that
X,Z(W,QXQ∇V+W,∇XV),  Z

For given $\varepsilon>0$ small, let

$$\Ga_{-\varepsilon}=\{\,\a(t,-\varepsilon)\,|\,t\in[0,a)\,\},\quad \Ga_{b_1}=\{\,\a(t,b_1)\,|\,t\in[0,a)\,\}.$$

Let $X\in TS$ be a given vector field and $\nu$ be the outside normal of $\Om_{-\varepsilon}.$ We consider solvability of system

W,QX|__-ε=q,  W,ν|__b_1=p,.where $V\in T\Om_{-\varepsilon}$ is the unknown, and $f_i\in\LL^2(\Om_{-\varepsilon}),$
$q\in\LL^2(\Ga_{-\varepsilon}),$ and $p\in\LL^2(\Ga_{b_1})$ are given.

Let  $Y\in TS$ be a given vector field. In order to solve problem (<ref>), we introduce an operator $\L_Y:$ $\LL^2(S,T)\rw\LL^2(S,T)$ by
=e^-κ[(÷_g Q∇V+V,Y-ηQX)QX+(÷_g V-ηV,∇X)∇X],D(Ł_Y)={ V∈^2(S,T)| ÷_g V∈^2(S), ÷_gQ∇V∈^2(S) },where $\g\in[0,\infty)$ and $\eta\in\CC^m(\Om_{-\varepsilon}).$

For $V,$ $W\in\WW^{1,2}(S,T),$ we compute
W,Ł_YV=e^-κ[(÷_g Q∇V+V,Y-ηQX)W,QX

+(÷_g V-ηV,∇X)W,∇X]

+e^-κ(W,QXQ∇V,Dκ+W,∇XV,Dκ)-e^-κ(Q∇V, DW,QX

+e^-κV, ∇QDW,QX-DW,∇X

Ł_Y^*W      =e^-κ[∇QDW,QX-DW,∇X-(W,QX∇QDκ-W,∇XDκ)

Let $X,$ $Y\in TS$ be given. Suppose that $\Om\subset S$ be a subregion of $S$ such that

$$\<X,\nu\>\not=0\qfq x\in\pl\Om,$$
where $\nu$ is the outside normal of $\Om.$  Then
(W,Ł_YV)_^2(,T)      =(V,Ł_Y^*W)_^2(,T)

+∫_(_1V,νW,ν-_2V,QXW,QX)d,for $V,$ $W\in T\Om,$ where
_1=e^-κ/X,νΠ(X,X),  _2=e^-κ/X,νΠ(Qν,Qν) x∈.

We integrate (<ref>) over $\Om$ and use (<ref>), where $Z=\nu,$ to have (<ref>).

Let $X,$ $Y\in TS$ be given. Let $\L_Y$ and $\L_Y^*$ be given in $(\ref{2.10})$ and $(\ref{2.13}),$ respectively. Then

=(W,QXW,∇QDκ-W,∇XW, Dκ)+W, D_W(∇X)

+W,  D_Q∇W(QX)x∈_-ε.

From (<ref>) and (<ref>), we have
e^-κW,Ł_YW      =(÷_g Q∇W+W,Y-ηQX)W,QX

+(÷_g W-ηW,∇X)W,∇X

=DW,  W,QXQ∇+W,∇X𝕀-η(W,QX^2+W,∇X^2)

+W,Y+÷_gQ∇W,QXp∈_-ε,where formula $\div_gW=\<DW,\id\>$ has been used.
In addition, it follows from (<ref>) that
e^-κW,Ł_Y^*W      =W,  ∇QDW,QX-DW,∇X

-(W,QXW,∇QDκ-W,∇XW, Dκ)

+W,QXW, Y-ηQX-ηW,∇X^2

=-DW,  QX⊗Q∇W+∇X⊗W-W,  D_Q∇W(QX)+D_W(∇X)

-(W,QXW,∇QDκ+W,∇XW, Dκ)

+W,QXW, Y-η(W, QX^2+W,∇X^2)p∈_-ε.Moreover, formula (<ref>) yields

$$\<W,QX\>Q\nabla\n+\<W,\nabla\n X\>\id=QX\otimes Q\nabla\n W+\nabla\n X\otimes W.$$

Thus (<ref>) follows from (<ref>) and (<ref>).

For given $\varepsilon>$ small, set

$$S_\varepsilon=\{\,\a(t,s)\in \,|\,(t,s)\in[0,a)\times(-\varepsilon,\varepsilon)\,\}.$$

Let $x\in\Ga_0.$ Since $\kappa(x)=0$ and $D\kappa(x)\not=0,$ from <cit.>, there exist vector fields $X_1,$ $X_2$ in a neighborhood of $x$
satisfying $\nabla\n X_i=\lam_iX_i,$ where $\lam_i$ are the principal curvatures. Clearly we may extend the vector fields $X_i$ to the region $\overline{S}_{\varepsilon}$ when $\varepsilon>0$ is given small.
We assume that $X_i$ are vector fields such that
∇X_i=_iX_i,  |X_i|=1,  X_1,X_2=0x∈S_ε,where

$$\lam_1>0\qfq x\in S_\varepsilon,\quad\lam_2=0\qfq x\in\Ga_0,$$

$$\lam_2>0\qfq x\in S^+\cap \overline{S}_\varepsilon,\quad \lam_2<0\qfq x\in S^-\cap\overline{S}_{\varepsilon}.$$

For given $\varepsilon>0$ small

$$X_2(\lam_2)\not=0\qfq x\in S_{\varepsilon}.$$

It will suffice to prove

$$X_2(\lam_2)\not=0\qfq x\in\Ga_0.$$

First, we claim that $\<Q\a_t,X_2\>(x)\not=0$ for $x\in\Ga_0.$ If not, then $\<Q\a_t,X_2\>(x)=0$ implies that $X_2=\eta \a_t$ with $\eta\not=0,$ and thus

which contradicts  assumption $\Pi(\a_t,\a_t)\not=0.$

In addition, assumption (<ref>) implies
Dκ,_t=0x∈_0,  Dκ=ιQ_tx∈_0,for some $\iota\not=0.$
Thus we obtain

From (<ref>) and (<ref>), we have
Π(_t,_t)>0x∈S.We assume that for each $s\in[-b_0,b_1]$ the closed curve $\a(\cdot,s)$ goes in anticlockwise with $t\in[0,a)$ increasing.
We further assume that
X_2(_2)>0S_ε,for given $\varepsilon>0$ small. For otherwise, we replace $X_2$ with $-X_2.$
Furthermore, we assume that $X_1,$ $X_2$ has positive orientation. For otherwise, we replace $X_1$ with $-X_1.$ Thus
QX_2=X_1,  QX_1=-X_2x∈S_ε.

We have
{X_2,Q_t<0    (_t,_s,)>0,

X_2,Q_t>0    (_t,_s,)<0.x∈_0.

Let $\det(\a_t,\a_s,\n)<0.$ The case of $\det(\a_t,\a_s,\n)>0$ can be treated similarly.

$$\frac{Q\a_t}{|\a_t|},\quad \frac{\a_t}{|\a_t|}$$
forms an orthonormal basis with positive orientation,  the curvature assumptions (<ref>)-(<ref>) yield

$$\<Q\a_t,D\kappa\>>0\qfq x\in\Ga_0.$$

It follows from (<ref>) that $\iota>0$ for $x\in\Ga_0.$ Thus in the case of $\det(\a_t,\a_s,\n)<0$ (<ref>) follows from (<ref>) and (<ref>).

We set
X__0={ _0_t-Q_t     (_t,_s,)>0,

_0_t+Q_t     (_t,_s,)<0, .where $\si_0$ is a constant satisfying

We have

$$\Pi(X_{\si_0},X_{\si_0})>0\qfq x\in\overline{S^+},$$

X__0,ν<0x∈_0,  X__0,ν>0x∈_b_1,where $\nu$ is the outside normal of region $S^+.$

It is easy to check that

$$\left\{\begin{array}{l} \nu=\frac{Q\a_t}{|\a_t|}\qfq x\in\Ga_0,\\
\nu=-\frac{Q\a_t}{|\a_t|}\qfq x\in\Ga_{b_1}, \end{array}\right.\quad\mbox{if}\quad \det(\a_t,\a_s,\n)>0$$

$$\left\{\begin{array}{l} \nu=-\frac{Q\a_t}{|\a_t|}\qfq x\in\Ga_0,\\
\nu=\frac{Q\a_t}{|\a_t|}\qfq x\in\Ga_{b_1}, \end{array}\right.\quad\mbox{if}\quad \det(\a_t,\a_s,\n)<0.$$

Thus (<ref>) follows. Moreover, we have

$$\Pi(X_{\si_0},X_{\si_0})\geq\Pi(\a_t,\a_t)(\si_0\pm\frac{\Pi(\a_t,Q\a_t)}{\Pi(\a_t,\a_t)})^2>0\qfq x\in\Ga_0.$$

By  Lemma <ref> we fix $\varepsilon>0$ small such that
{X_2,Q_t<0    (_t,_s,)>0,

X_2,Q_t>0    (_t,_s,)<0.x∈_-ε.Moreover, since

$$\<\a_t,X_1\>\not=0\qfq x\in\Ga_0,$$
we assume that the above $\varepsilon$ is such that

$$\<\a_t,X_1\>\not=0\qfq x\in\overline{S}_{2\varepsilon}.$$

{ ν=Q_t/|_t|x∈_-ε,    (_t,_s,)>0,

ν=-Q_t/|_t|x∈_-ε,     (_t,_s,)>0..Thus, by (<ref>),

We take  $\zeta\in\CC_0^\infty(-\infty,\infty)$ such that

$$0\leq\zeta\leq1;\quad\zeta=1\qfq s\in(-\varepsilon,\varepsilon);\quad \zeta=0\qfq s\geq2\varepsilon.$$
Then we take
η=sη_0(s)in (<ref>)  where
$\eta_0\in\CC^\infty_0(-\infty,\infty)$  such that

$$0\leq\eta_0\leq1;\quad \eta_0=0\qfq s\leq \varepsilon/2;\quad \eta_0=1\qfq s\geq\varepsilon.$$
In addition, we take $X$ in (<ref>) by
X=ζ[ε^1/3(_t,X_1) X_1+X_2]+(1-ζ)X__0x∈_-ε,where $\sign\<\a_t,X_1\>$ is the sign function for $x\in\overline{S}_{2\varepsilon},$ $X_{\si_0}$ is given in (<ref>), and $\si_0$ is given such that
Π(X,X)>0,  X,ν<0x∈_-ε,when $\varepsilon>0$ is small enough.

Let $\eta$ and $X$ be  given in $(\ref{2.32*})$ and $(\ref{2.32}),$ respectively. Let $Y\in\CC^0(\overline{\Om}_{-\varepsilon},T).$  Then there are constants $\si>0,$ $\g>0,$ and $s>0$  such that
-W,  Ł_YW+Ł_Y^*W≥|W|^2W∈T_-ε,  x∈_-ε, for given $\varepsilon>0$ small.

Step 1.    First, we prove
|X|>0x∈S^+∖S_ε.If not, there were a point $x\in\overline{S^+\setminus S_\varepsilon}$ such that

If $\zeta=0,$ then

which is not possible. Thus $\zeta\not=0.$ By (<ref>),

implies that

which contradicts $0\leq\zeta\leq1$ when given $\varepsilon>0$ is small.

Since $\kappa>0$ for $x\in\overline{S^+\setminus S_\varepsilon},$ $X,$ $\nabla\n X$ forms an vector field basis on $\overline{S^+\setminus S_\varepsilon}.$ Then there exists $\si>0$ such that

Step 2.   Consider the region $S_\varepsilon.$ Noting that $\zeta=1$ for $x\in S_\varepsilon,$ we have

$$X=\varepsilon^{1/3}(\sign\<\a_t,X_1\>) X_1+X_2\qfq x\in\overline{S}_\varepsilon.$$
Denote $\<W,X_i\>=W_i$ for $i=1,$ $2.$ We compute
(W,QX_2W,∇QDκ-W,∇X_2W, Dκ)+W, D_W(∇X_2)

-W,2Y+÷_gQ∇W,QX_2+W,  D_Q∇W(QX_2)

=(W, X_1W,∇QDκ-_2W,X_2W, Dκ)+W, D_W(_2 X_2)

-W,2Y+÷_gQ∇W,X_1+W,  D_Q∇W(X_1)

-C(W_1^2+|W_1||W_2|)x∈S_ε.Similarly, we have
(W,QX_1W,∇QDκ-W,∇X_1W, Dκ)+W, D_W(∇X_1)

-W,2Y+÷_gQ∇W,QX_1+W,  D_Q∇W(QX_1)

-[(|_2||X_1(κ)|+_1X_2(κ))+|X_2,D_X_2(_1X_1)|+|÷_gQ∇,X_2|]W_2^2.    It follows from (<ref>), (<ref>), and (<ref>) that
-W,  Ł_YW+Ł_Y^*≥[_1X_2(κ)-|_2||X_1(κ)|-ε_1(|X_1(κ)|+X_2(κ))]W_1^2

-ε[(|_2||X_1(κ)|+_1X_2(κ))+|X_2,D_X_2(_1X_1)|+|÷_gQ∇,X_2|]}W_2^2,      for $x\in\overline{S}_\varepsilon,$
where $C>0$ is independent of $W\in TS_\varepsilon.$ Noting that $X_2(\kappa)=\lam_1X_2(\lam_2)>0$ for $x\in\Ga_0$ and

$$|\lam_2|=\O(\varepsilon)\qfq x\in \overline{S}_\varepsilon,$$
we assume that the $\varepsilon>0$ has been given so small such that

$$\lam_1X_2(\kappa)-|\lam_2||X_1(\kappa)|-\varepsilon\lam_1(|X_1(\kappa)|+X_2(\kappa))>0\qfq x\in\overline{S}_\varepsilon.$$
Then we fix $\g>0$ such that

$$\g[\lam_1X_2(\kappa)-|\lam_2||X_1(\kappa)|]-\varepsilon\lam_1(|X_1(\kappa)|+X_2(\kappa))-C>0\qfq x\in\overline{S}_\varepsilon.$$

Finally, we move the $\varepsilon>0$ to zero again such that

-ε[(|_2||X_1(κ)|+_1X_2(κ))+|X_2,D_X_2(_1X_1)|+|÷_gQ∇,X_2|]}>0for $x\in\overline{S}_\varepsilon.$
Thus for the above given $\g>0$ and  $\varepsilon>0,$ from (<ref>),  we have obtained $\si>0$ such that
-e^κW,Ł_YW+Ł_Y^*W≥|W|^2W∈TS_ε,  x∈S_ε.

Step 3.   Consider region $S^+\setminus S_\varepsilon.$ From (<ref>) and (<ref>), we have
-e^κW,Ł_YW+Ł_Y^*W      ≥sη_0(W,QX^2+W,∇X^2)-C|W|^2

≥(s-C)|W|^2s>0,   W∈TS^+∖S_ε,for $x\in\overline{S^+\setminus S_\varepsilon}.$
Thus (<ref>) follows from (<ref>) and (<ref>) after we take $s=C+\si$ in (<ref>).

Next, we consider solvability of problem (<ref>) on region $\Om_{-\varepsilon}.$ Let $F\in\LL^2(\Om_{-\varepsilon},T),$  $p\in\LL^2(\Ga_{b_1}),$ and $q\in\LL^2(\Ga_{-\varepsilon})$ be given. Clearly, system (<ref>) is equivalent to problem
{Ł_Y V=Fx∈_-ε,

V,QX=qx∈_-ε,  V,ν=px∈_b_1, .where $\L_Y$ is defined in (<ref>) and vector field $X$ is given in (<ref>).

Let $m\geq0$ be an integer. Set

We denote the completions of $\CC^m(\Om_{-\varepsilon},T)$ by $\LL^2_{-\varepsilon,b_1}(\Om_{-\varepsilon},T)$ and $\LL^2_{b_1,-\varepsilon}(\Om_{-\varepsilon},T)$ in the norms of

Let vector field $X$ be given in $(\ref{2.32})$ and $Y\in\CC^0(\overline{\Om}_\varepsilon, T).$  Let $F\in\LL^2(\Om_{-\varepsilon},T),$  $p\in\LL^2(\Ga_{b_1}),$ and $q\in\LL^2(\Ga_{-\varepsilon})$ be given. Then problem $(\ref{2.40})$ admits a unique solution $V\in D(\L_Y)\cap\LL^2_{-\varepsilon,b_1}(\Om_{-\varepsilon},T),$ which satisfies
(V,Ł_Y^*W)_^2(_-ε,T)      =(W,F)_^2(_-ε,T)+∫__-ε_2qW,QXd-∫__b_1_1pW,νd      for $W\in \CC^1_{b_1,-\varepsilon}(\Om_{-\varepsilon},T),$ where $\pounds_1$ and $\pounds_2$ are given in $(\ref{n2.17x}).$ Furthermore, the following estimate holds true.
V^2_^2(_-ε,T)      +V^2_^2(_-ε,T)+V^2_^2(_b_1,T)

Step 1.   Denote the completion of $\CC^1_{b_1,-\varepsilon}(\Om_{-\varepsilon},T)$ in the norm of $\WW^{1,2}(\Om_{-\varepsilon},T)$ by $\WW^{1,2}_{b_1,-\varepsilon}(\Om_{-\varepsilon},T).$

We define a bilinear functional,

$$\a:\quad\LL^2_{-\varepsilon,b_1}(\Om_{-\varepsilon},T)\times\WW^{1,2}_{b_1,-\varepsilon}(\Om_{-\varepsilon},T)\rw \R,$$

Clearly, we have

$$|\a(V,W)|\leq C\|V\|_{\LL^2(\Om_{-\varepsilon},T)}\|W\|_{\WW^{1,2}(\Om_{-\varepsilon},T)}$$
for $(V,W)\in\LL^2_{-\varepsilon,b_1}(\Om_{-\varepsilon},T)\times\WW^{1,2}_{b_1,-\varepsilon}(\Om_{-\varepsilon},T).$ On the other hand, it follows from (<ref>) and (<ref>) that
X,ν<0x∈_-ε,  X,ν>0x∈_b_1.Thus,
from (<ref>) and (<ref>), we have

≥W^2_^2(_-ε, T)-∫__-ε_2W,QX^2d+∫__b_1_1W,ν^2d

Next, we define a functional on $\LL^2_{-\varepsilon,b_1}(\Om_{-\varepsilon},T)$ by

clearly, $\F$ is bounded on $\LL^2_{-\varepsilon,b_1}(\Om_{-\varepsilon},T).$  By Theorem A given in Appendix there exists a $V\in \LL^2_{-\varepsilon,b_1}(\Om_{-\varepsilon},T)$
such that
-2(V,Ł_Y^*W)_^2(_-ε,T)      =-2(W,F)_^2(_-ε,T)-2∫__-ε_2qW,QXd

+2∫__b_1_1pW,νdfor all $W\in\CC^1_{b_1,-\varepsilon}(\Om_{-\varepsilon},T),$ which yield, by (<ref>) again,

+2∫__-ε_2(V,QX-q)W,QXd+2∫__b_1_1(p-V,ν)W,νdfor all $W\in\CC^1_{b_1,-\varepsilon}(\Om_{-\varepsilon},T).$ Thus $V\in\LL^2_{-\varepsilon,b_1}(\Om_{-\varepsilon},T)$ solves problem (<ref>) and identity (<ref>) holds.

Noting that $\zeta=1$ for $x\in\overline{S}_\varepsilon,$ from (<ref>), we have

$$|\nabla\n X|^2=\varepsilon^2\lam_1^2+\lam_2^2>0\qfq x\in\overline{S}_\varepsilon.$$
Then, by (<ref>), $QX,$ $\nabla\n X$ forms a vector field basis on $\overline{\Om}_\varepsilon.$ Thus
$V\in D(\L_Y).$

Step 2. From (<ref>) and (<ref>), we obtain (<ref>). Thus the uniqueness follows.

We consider a duality problem of system (<ref>)
{Ł_Y^* V=Fx∈_-ε,

V,ν=px∈_-ε,   V,QX=qx∈_b_1.  .

By a duality of operators $\L_Y$ and $\L_Y^*$ a similar argument as for Theorem <ref> yields the following.

Let vector field $X$ be given in $(\ref{2.32})$ and $Y\in\CC^0(\overline{\Om}_{-\varepsilon}, T).$  Let $F\in\LL^2(\Om_{-\varepsilon},T),$  $p\in\LL^2(\Ga_{-\varepsilon}),$ and $q\in\LL^2(\Ga_{b_1})$ be given. Then problem $(\ref{2.45})$ admits a unique solution $V\in D(\L_Y^*)\cap\LL^2_{b_1,-\varepsilon}(\Om_{-\varepsilon},T),$ which satisfies
(V,Ł_YW)_^2(_-ε,T)      =(W,F)_^2(_-ε,T)+∫__-ε_1pW,νd

-∫__b_1_2qW,QXdfor $W\in \CC^1_{-\varepsilon,b_1}(\Om_{-\varepsilon},T),$ where $\pounds_1$ and $\pounds_2$ are given in $(\ref{n2.17x}).$ Furthermore, the following estimate holds true.
V^2_^2(_-ε,T)      +V^2_^2(_-ε,T)+V^2_^2(_b_1,T)

§ $\WW^{M,2}$ SOLUTIONS OF THE TENSOR EQUATION OF MIXED TYPE

Consider problem
{Ł_0 V=Fx∈_-ε,

V,QX=px∈_-ε,  V,ν=qx∈_b_1, .where $X$ is given in (<ref>) and $\L_0$ is given in (<ref>) with $Y=0.$

In this section, we shall establish the following.

Let $S$ be of $\CC^{m+3}.$ Let $F\in\WW^{m,2}(\Om_{-\varepsilon},T),$ $p\in\WW^{m,2}(\Ga_{-\varepsilon}),$ and $q\in\WW^{m,2}(\Ga_{b_1})$ be given. Then problem $(\ref{3.1})$ admits a unique solution
$V\in\WW^{m,2}(\Om_{-\varepsilon},T)$ satisfying
V^2_^m,2(_-ε,T)≤C( F^2_^m,2(_-ε,T)+ q^2_^m,2(_b_1)+ p^2_^m,2(_-ε)).

By a similar argument and the duality, we have
Let $S$ be of $\CC^{m+3}.$ Let $F\in\WW^{m,2}(\Om_{-\varepsilon},T),$ $p\in\WW^{m,2}(\Ga_{b_1}),$ and $q\in\WW^{m,2}(\Ga_{-\varepsilon})$ be given. Then problem
{Ł_0^* V=Fx∈_-ε,

V,ν=px∈_-ε,  V,QX=qx∈_b_1, .admits a unique solution
$V\in\WW^{m,2}(\Om_{-\varepsilon},T)$ satisfying $(\ref{V3.2}).$

Let $S$ be of $\CC^3.$ Let $p\in\WW^{1,2}(\Ga_{-\varepsilon})$ and $q\in\WW^{1,2}(\Ga_{b_1})$ be given. Suppose that $F\in\LL^2(\Om_{-\varepsilon},T)$ be given such that

$$\Sigma_{s_1}^{s_2}=\{\,\a(t,s)\,|\,(t,s)\in[0,a)\times(s_1,s_2)\,\}\qfq -b_0\leq s_1<s_2\leq b_1.$$

Then problem $(\ref{3.1})$ admits a unique solution $V\in\WW^{1,2}(\Om_{-\varepsilon},T)$ satisfying
V^2_^1,2(_-ε,T)      ≤C(F^2_^2(_-ε,T)+F^2_^1,2(Σ_-ε^ε/2,T)

+p^2_^1,2(_-ε)+q^2_^1,2(_b_1)).If $S$ is of $\CC^{m+3},$ then the above $V$ satisfies
V^2_^m,2(_-ε,T)      ≤C(F^2_^m-1,2(_-ε,T)+F^2_^m,2(Σ_-ε^ε/2,T)

By the duality and a similar argument in the proof of Theorem <ref>, we have the following. The proof is omitted.

Let $S$ be of $\CC^3.$ Let $p\in\WW^{1,2}(\Ga_{-\varepsilon})$ and $q\in\WW^{1,2}(\Ga_{b_1})$ be given. Suppose that $F\in\LL^2(\Om_{-\varepsilon},T)$ be given such that

Then problem $(\ref{4*.9})$ admits a unique solution $V\in\WW^{1,2}(\Om_{-\varepsilon},T)$ satisfying $(\ref{4.101}).$
If $S$ is of $\CC^{m+3},$ then the above $V$ satisfies $(\ref{4.102}).$

The proofs of Theorems <ref> and <ref> will be given in the end of this section.

We  assume that $V\in\CC^1(\Om_{-\varepsilon},T)$ satisfies equation
Ł_0V=Fx∈_-ε.For convenience, we denote
{÷_g Q∇V-ηV,QX=f_1x∈_-ε,

÷_g V-ηV,∇X=f_2x∈_-ε..Then

$$F=e^{-\g\kappa}(f_1QX+f_2\nabla\n X).$$

Let $\Phi\in TS$ be given such that

$$\<\nabla\n\Phi,\Phi\>\not=0\qfq x\in \overline{S}.$$
We define an operator $\RR_\Phi:$ $\LL^2(S,T)\rw\LL^2(S,T)$ by
_ΦV=[Φ,Q∇V]-Q∇[Φ,V]V∈TS. Set
h_1=_ΦΦ,QΦ/∇Φ,Φ,  h_2=_ΦQΦ,QΦ

We have
(_Φ-h_1Q∇-h_2𝕀)Φ,QΦ      =_ΦΦ,QΦ-h_1Q∇Φ,QΦ-h_2Φ,QΦ

=_ΦQΦ,QΦ-h_1∇QΦ,Φ-(_ΦQΦ,QΦ-h_1∇QΦ,Φ)=0.Since $\frac{Q\Phi}{|\Phi|},$ $\frac{\Phi}{|\Phi|}$ forms an orthonormal basis, it follows from (<ref>) and (<ref>) that

$$\<(\RR_\Phi-h_1Q\nabla\n-h_2\id)W, Q\Phi\>=0\qfq W\in TS.$$

for all $W\in TS,$ that is, (<ref>).

For given $V\in\CC^1(S,T),$ we have
Ł_Z[Φ,V]=F_Φ,where $Z$ is given in $(\ref{2.96})$ and
=\Phi f_1-h_1f_1-h_2f_2+\<V,H_1\>,\quad f_{2\Phi}=\Phi f_2+\<V,H_2\>,$$

H_1      =∇QD÷_gΦ+[Φ(η)-h_1η] QX-h_2η∇X

$$\quad H_2
=\Phi(\eta)\nabla\n X-D\div_g\Phi+\eta D\Phi(\nabla\n X,\cdot).$$

From (<ref>), (<ref>), (<ref>),  (<ref>), and (<ref>), we have
÷_gQ∇[Φ, V]      =÷_g[Φ,Q∇V]-÷_g_ΦV

=Φ(÷_gQ∇V)-Q∇V(÷_gΦ)-DV, _Φ-V,÷_g_Φ

=Φ(÷_gQ∇V)-Q∇V,D÷_gΦ-DV, h_1Q∇+h_2𝕀+Z⊗Φ-V,÷_g_Φ

=Φ(÷_gQ∇V)+V, ∇QD÷_gΦ-h_1(÷_gQ∇V-V,÷_gQ∇)

-h_2÷_gV-D_ΦV, Z-V,÷_g_Φ

=Φf_1-h_1f_1-h_2f_2+[Φ,V], ηQX-Z-D_VΦ,Z

+V, ∇QD÷_gΦ+[Φ(η)-h_1η] QX-h_2η∇X

$$f_{1\Phi}=\div_gQ\nabla\n[\Phi, V]+\<[\Phi,V], Z-\eta QX\>=\Phi f_1-h_1f_1-h_2f_2+\<V,H_1\>.$$

Similarly, we compute

$$\div_g[\Phi,V]=\Phi\div_gV-V\div_g\Phi=\Phi f_2+\eta\<[\Phi,V],\nabla\n X\>+\<V,H_2\>.$$

Consider an operator on $\LL^2(\Om_{-\varepsilon},T):$
=e^-κ[(÷_g Q∇V-V,ηQX)QX+(÷_g V-ηV,∇X)∇X],D(Ł_0)      ={ V∈^2(_-ε,T)| ÷_g V∈^2(_-ε), ÷_gQ∇V∈^2(_-ε),

V,QX|__-ε=V,ν|__b_1=0 }.By Theorem <ref>, $\L_0^{-1}:$ $\LL^2(\Om_{-\varepsilon},T)\rw\LL^2(\Om_{-\varepsilon},T)$ is bounded. Similarly, Theorem <ref> implies
that  ${\L_0^*}^{-1}:$ $\LL^2(\Om_{-\varepsilon},T)\rw\LL^2(\Om_{-\varepsilon},T)$ is bounded, where the domain of operator $\L_0^*$ is

$$ D(\L_0^*)=\{\,V,\,\,\L_0^*V\in \LL^2(\Om_{-\varepsilon},T)\big|\,\<V,QX\>|_{\Ga_{b_1}}=\<V,\nu\>|_{\Ga_{-\varepsilon}}=0\,\},$$

where $\L_0^*$ is given in (<ref>) with $Y=0.$

Let $X$ be given in (<ref>). Then $QX,$ $\nabla\n X$ forms a vector field basis on $\overline{\Om}_{-\varepsilon}.$ Denote its conjugate basis by $F_1,$ $F_2,$ where
F_1=Q∇X/∇X,X,  F_2=X/∇X,Xx∈_-ε.We define operator $\K:$ $\LL^2(\Om_{-\varepsilon},T)\rw \LL^2(\Om_{-\varepsilon},T)$ by

$$\M=[D_\Phi F_1+(\g\<\Phi,D\kappa\>-h_1)F_1-h_2F_2]\otimes QX
+(D_\Phi F_2+\g\<\Phi,D\kappa\>F_2)\otimes \nabla\n X,$$

$$\N=e^{-\g\kappa}(H_1\otimes QX+H_2\otimes \nabla\n X).$$

Let $\K$ be given in $(\ref{K3.12}).$ Then
(F,G)_^2(_-ε,T)=(F,^* G)_^2(_-ε,T)+∫_∂_-εF,GΦ,νd,where
=-D_ΦG-(÷_g Φ) G+^TG+Ł_0^*^-1(^TG).

We have
F,G      =D_ΦF+F+Ł_0^-1(F), G

=÷_g(F,GΦ)+F,-(÷_gΦ)G-D_ΦG+F,^TG+Ł_0^-1F,^TG.We integrate the above identity over $\Om_{-\varepsilon}$ to have (<ref>).

We have
Ł_Z[Φ,V]=Ł_0VV∈^1,2(_-ε),where $Z$ is given in $(\ref{2.96}).$

For given $V\in\WW^{1,2}(\Om_{-\varepsilon}),$ set

$$\div_g Q\nabla\n V-\<V,\eta QX\>=f_1,\quad\div_g V-\eta\<V,\nabla\n X\>=f_2.$$

By Lemma <ref>

where $F_\Phi$ is given in Lemma <ref>.

Next, we shall prove

Noting that

$$\L_0V=e^{-\g\kappa}(f_1QX+f_2\nabla\n X),$$
we have

$$D_\Phi\L_0V=-\g\<\Phi,D\kappa\>\L_0V+e^{-\g\kappa}(\Phi f_1QX+\Phi f_2\nabla\n X)+e^{-\g\kappa}[f_1D_\Phi(QX)+f_2D_\Phi(\nabla\n X)].$$

Since $F_1,$ $F_2$ is the conjugate basis of $QX,$ $\nabla\n X,$ it follows that
Ł_0V      =D_ΦF_1+(Φ,Dκ-h_1)F_1-h_2F_2, Ł_0V QX
+D_ΦF_2+Φ,DκF_2, Ł_0V ∇

=e^-κ[-f_1D_Φ(QX)-f_2D_Φ(∇X)]+Φ,DκŁ_0V-e^-κ(h_1f_1+h_2f_2)QX.Thus, by (<ref>), we obtain
Ł_0V      =D_ΦŁ_0V+Ł_0V+Ł_0^-1(Ł_0V)

Now we construct a special $\Phi$ as follows.
Define a function $\psi$ on $S$ by

$$\psi(\a(t,s))=s\qfq x=\a(t,s)\in S.$$

$$\<D\psi,\a_t\>=0,\quad\<D\psi,\a_s\>=1\qfq x\in S.$$

$$\Phi_0=QD\psi\qfq x\in S.$$

$$\<\Phi_0,\nu\>=\<QD\psi,\pm\frac{Q\a_t}{|\a_t|}\>=0\qfq x\in\pl\Om_{-\varepsilon}.$$

$$\Phi=e^\var\Phi_0\qfq x\in\Om_{\varepsilon},$$
where $\var$ is a function given in Lemma <ref> below. Since
Dψ=Dψ,Q_t/|_t|^2Q_tx∈_-εwith $\<D\psi,Q\a_t\>\not=0,$ from (<ref>), we have

$$\<\nabla\n\Phi,\Phi\>=e^{2\var}\<\nabla\n\Phi_0,\Phi_0\>=e^{2\var}\frac{\<D\psi,Q\a_t\>^2}{|\a_t|^4}\Pi(\a_t,\a_t)>0\qfq x\in\overline{\Om}_{-\varepsilon}.$$

There exists a function $\var$ on $\overline{\Om}_{-\varepsilon}$ such that
D_ΦDψ+Dψ, QDψ=0x∈_-ε∖S^+,[Φ,X],QX=0x∈_b_1,where $X$ is given in $(\ref{2.32}).$ Furthermore, for $W\in\CC^1(\Om_{-\varepsilon},T),$ we have

Noting that $\Phi=e^\var\Phi_0=e^\var QD\psi,$ we have
D_ΦDψ+Dψ, QDψ      =e^D_Φ_0Dψ,Φ_0+D_ΦF_2+Φ,DκF_2,DψQDψ,∇X

+D_ΦF_1+(Φ,Dκ-h_1)F_1-h_2F_2, DψDψ, QX

=e^h-(h_1F_1,Dψ+h_2F_2, Dψ)Dψ, QX,where
h      =D_Φ_0Dψ,Φ_0+D_Φ_0 F_2+Φ_0,DκF_2,DψQDψ,∇X

+D_Φ_0 F_1+Φ_0,DκF_1, DψDψ,QX.

On the other hand, from (<ref>), we have
_ΦΦ,QΦ      =D_Φ(Q∇Φ)-D_Q∇ΦΦ, QΦ

=e^3[_Φ_0 Φ_0,QΦ_0+Φ_0()∇Φ_0,Φ_0].Similarly, we obtain

$$ \<\RR_\Phi Q\Phi,Q\Phi\>=e^{3\var}[\<\RR_{\Phi_0}Q\Phi_0,Q\Phi_0\>+(Q\Phi_0)(\var)\<\nabla\n \Phi_0,\Phi_0\>].$$

h_2      =e^/|Φ_0|^2{_Φ_0QΦ_0,QΦ_0-_Φ_0Φ_0,QΦ_0/∇Φ_0,Φ_0∇QΦ_0,Φ_0

-(QΦ_0)()∇Φ_0,Φ_0+Φ_0()∇QΦ_0,Φ_0}.We then obtain
h_1F_1,Dψ+h_2F_2, Dψ      =e^{ĥ+Φ_0()(F_1,Dψ+∇QΦ_0,Φ_0/|Φ_0|^2F_2,Dψ)

ĥ      =_Φ_0Φ_0,QΦ_0/∇Φ_0,Φ_0F_1,Dψ+F_2,Dψ/|Φ_0|^2[_Φ_0QΦ_0,QΦ_0-_Φ_0Φ_0,QΦ_0/∇Φ_0,Φ_0∇QΦ_0,Φ_0].It follows from (<ref>) that
D_ΦDψ+Dψ, QDψ=e^[h-ĥ+H()],where

$$H=(\<F_1,D\psi\>-\frac{\<\nabla\n D\psi,QD\psi\>}{|D\psi|^2}\<F_2,D\psi\>)QD\psi+\frac{\<\nabla\n QD\psi,QD\psi\>\<F_2,D\psi\>}{|D\psi|^2}D\psi.$$

By (<ref>) and (<ref>),

$$\<QD\psi,\nu\>=0\qfq x\in\Ga_{-\varepsilon}.$$

From (<ref>), (<ref>), and (<ref>), we have

$$\<H,\nu\>=\frac{\<D\psi,Q\a_t\>^2}{|\a_t|^4\Pi(X,X)}\Pi(\a_t,\a_t)\<X,\nu\><0\qfq x\in\Ga_{-\varepsilon}.$$

Thus, when given $\varepsilon>0$ is small, the integral curves of the vector field $H$ initiating from $\Ga_{-\varepsilon}$ direct to the interior of $\Om_{-\varepsilon}$ can across $\Ga_0.$ We solve equation (<ref>) along those integral curves to have $\var$ to satisfy (<ref>).

By a similar argument, we obtain a function $\var$ in a neighborhood $\Ga_{b_1}$ such that (<ref>) holds.

Next, we prove (<ref>) and (<ref>). By the symmetry of $D^2\psi, $ we have

$$\<[\Phi,W],D\psi\>=\Phi\<W,D\psi\>-\<W,D_\Phi D\psi\>-W\<\Phi,D\psi\>+\<\Phi,D_WD\psi\>=\Phi\<W,D\psi\>.$$
Moreover, since $\frac{QX}{|X|},$ $\frac{X}{|X|}$ forms an orthonormal frame, it follows from
(<ref>) that
[Φ,W],QX      =[Φ,W,X/|X|^2X],QX+[Φ,W,QX/|X|^2QX],QX

Let $k>0$ be given large and $W\in\CC^1(\Om_{-\varepsilon},T).$ Then

$$\<\K^*[\Phi,W]+kW,\nu\>|_{\Ga_{-\varepsilon}}=0\quad\Rightarrow\quad \<W,\nu\>|_{\Ga_{-\varepsilon}}=\<[\Phi,W],\nu\>|_{\Ga_{-\varepsilon}}=0,$$

$$\<\K^* [\Phi,W]+kW,QX\>|_{\Ga_{b_1}}=0\quad\Rightarrow\quad  \<W,QX\>\big|_{\Ga_{b_1}}=\<[\Phi,W],QX\>\big|_{\Ga_{b_1}}=0,$$
where operator $\K^*$ is given in $(\ref{G3.15}).$

Since $\frac{QD\psi}{|D\psi|},$  $\frac{D\psi}{|D\psi|}$ forms an orhonormal frame, from (<ref>), we have

$$D_\Phi D\psi+\M D\psi=\frac{\<D_\Phi D\psi+\M D\psi,D\psi\>}{|D\psi|^2}D\psi.$$

Noting that $D\psi|_{\Ga_{-\varepsilon}}=\pm\frac{\<D\psi,Q\a_t\>}{|\a_t|}\nu,$ we have

It follows from (<ref>) that
^*[Φ,W],Dψ      =-D_Φ[Φ,W]-(÷_g Φ)[Φ,W]+^T[Φ,W], Dψ

=-Φ^2W,Dψ-(÷_g Φ)W,Dψ+[Φ,W], D_ΦDψ+Dψ

=-Φ^2W,Dψ-[÷_g Φ-D_ΦDψ+Dψ,Dψ/|Dψ|^2]W,Dψ.Thus we obtain

We integrate the above inequality over $\Ga_{-\varepsilon}$ to have

for given $k>0$ large, that is what we want.

A similar computation yields the second conclusion.

We introduce a Hilbert space by

Let $W_0\in\CC^1(\Om_{-\varepsilon},T)$ be given with $\<W_0,\nu\>|_{\Ga_{-\varepsilon}}=0$ and $\<W_0,QX\>|_{\Ga_{b_1}}=0.$ Then there exists a unique $W\in\V_\Phi$ such that

$$\K^* [\Phi,W]+kW=W_0\qfq x\in\Om_{-\varepsilon}$$
for given $k>0$ large.

Define a bilinear form $\a:$ $\V_{\Phi}\times\WW^{1,2}_0(\Om,T)\rw\R$ by

$$\a(W,V)=(\K^* [\Phi,W]+kW,V)_{\LL^2(\Om_{-\varepsilon})}.$$
Since $\<\Phi,\nu\>|_{\pl\Om}=0,$ from (<ref>), we have
(W,V)      =([Φ,W],V)_^2(_-ε)+k(W,V)_^2(_-ε)which yield

$$|\a(W,V)|\leq C\|W\|_{\V_\Phi}\|V\|_{\WW^{1,2}(\Om,T)}\qfq W\in\V_\Phi,\quad W\in\WW^{1,2}_0(\Om,T).$$

By (<ref>), we have
(W,W)      =([Φ,W],W)_^2(_-ε)+kW^2_^2(_-ε)

Let $\F(V)=(W_0,V)$ for $V\in\V_\Phi.$ Then  $\F$ is bounded on $\V_\Phi.$ By Theorem A given in Appendix there exists $W\in\V_\Phi$ such that

$$\F(V)=\a(W,V)\qfq V\in\WW_0^{1,2}(\Om_{-\varepsilon},T),$$
which implies

$$\K^* [\Phi,W]+kW=W_0\qfq x\in\Om_{-\varepsilon}.$$

Moreover, a similar computation as in (<ref>) gives

$$(\K^* [\Phi,W]+kW,W)_{\LL^2(\Om_{-\varepsilon})}\geq\|[\Phi,W]\|^2_{\LL^2(\Om_{-\varepsilon})}+(k-C)\|W\|^2_{\LL^2(\Om_{-\varepsilon})}.$$
Thus the uniqueness follows.

Let $F\in\LL^2(\Om_{-\varepsilon},T)$ be given with $[\Phi,F]\in\LL^2(\Om_{-\varepsilon},T).$
Let  $p\in\WW^{1,2}(\Ga_{-\varepsilon})$ and $q\in\WW^{1,2}(\Ga_{b_1})$ be given. Then problem $(\ref{3.1})$ admits a unique
solution $V\in\V_\Phi$ satisfying

Let $k>0$ be given large.
_0={ W∈^1(_-ε,T) | W,ν|__-ε=W,QX|__b_1=0 },
$$\W=\{\,W\in\V_\Phi\,|\,\mbox{there is $W_0\in\D_0$ such that $\K^* [\Phi,W]+kW=W_0$}\,\}.$$

By Lemma <ref>, we introduce an inner product on $\W$ by

where $\K^* [\Phi,V]+kV=V_0$ and $\K^* [\Phi,W]+kW=W_0.$ Then $\W\subset\V_\Phi$ is an inner product space. It follows from Lemma <ref> that
W,QX|__b_1=[Φ,W],QX|__b_1=W,ν|__-ε=[Φ,W],ν|__-ε=0for $W\in\W.$

Step 1.   We define a bilinear form $\a:$ $\V_\Phi\times\W\rw\R$ by

$$\a(V,W)=-(V,\L_0^*(\K^* [\Phi,W]+kW))_{\LL^2(\Om_{-\varepsilon})}.$$
Clearly, we have

$$|\a(V,W)|\leq C\|V\|_{\Phi}\|W\|_\W\qfq V\in\V_\Phi,\quad W\in\W.$$

Next, we prove the  coerciveness. From (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) (again), we compute
-2(W,Ł_0^*(^* [Φ,W]+kW))_^2(_-ε)=-2(Ł_0W,^*[Φ,W]+kW)_^2(_-ε)

+2∫__b_1_1W,ν^* [Φ,W]+kW,νd

-2∫__-ε_2W,QX^* [Φ,W]+kW,QXd

+2∫__b_1_1W,ν^* [Φ,W]+kW,νd

-2∫__-ε_2W,QX^* [Φ,W]+kW,QXd

$$\Ga_{b_1}(W)=\int_{\Ga_{b_1}}|\pounds_1|[2\<W,\nu\>\<\K^* [\Phi,W]+kW,\nu\>-\<[\Phi,W],\nu\>^2-k\<W,\nu\>^2]d\Ga,$$

$$\Ga_{-\varepsilon}(W)=\int_{\Ga_{-\varepsilon}}|\pounds_2|[2\<W,QX\>\<\K^* [\Phi,W]+kW,QX\>-\<[\Phi,W],QX\>^2-k\<W,QX\>^2]d\Ga.$$

We estimate $\Ga_{-\varepsilon}(W)$ and $\Ga_{b_1}(W)$ below. By (<ref>), $QX,$ $\nu$ forms a vector field basis along $\Ga_{-\varepsilon}.$ It follows from (<ref>) that

$$W=\O(\<W,QX\>),\quad [\Phi,W]=\O(\<[\Phi,W],QX\>)\qfq x\in\Ga_{-\varepsilon}.$$

Set $\tau=\Phi/|\Phi|.$   Then

Noting that

$$\<{\L_0^*}^{-1}(\N^T[\Phi,W]),QX\>=0\qfq x\in\Ga_{-\varepsilon},$$
from (<ref>), we obatin
|_2|[2W,QX^* [Φ,W]+kW,QX-[Φ,W],QX^2-kW,QX^2]

=2|_2|W,QX-D_Φ[Φ,W]-(÷_g Φ)[Φ,W]+^T[Φ,W]+Ł_0^*^-1(^T[Φ,W]),QX

$$\Ga_{-\varepsilon}(W)\geq\si\|\<W,QX\>\|^2_{\WW^{1,2}(\Ga_{-\varepsilon})}\qfq W\in\W$$
for given $k>0$ large. By (<ref>) a similar argument yields

$$\Ga_{b_1}(W)\geq \si\|\<W,\nu\>\|^2_{\WW^{1,2}(\Ga_{b_1})}\qfq W\in\W$$
for given $k>0$ large.

Using (<ref>) in (<ref>), we obtain

$$\a(W,W)\geq\si\|W\|^2_{\V_\Phi}\qfq W\in\W.$$

Step 2.   Let $F\in\LL^2(\Om_{-\varepsilon},T)$ be given with $[\Phi,F]\in\LL^2(\Om_{-\varepsilon},T).$ Set
(W)      =-(F,^* [Φ,W]+kW)_^2(_-ε,T)
+∫__-ε|_2| p^* [Φ,W]+kW,QXd

+∫__b_1|_1|q^* [Φ,W]+kW,νdW∈_Φ.Since $(F,\K^* [\Phi,W])_{\LL^2(\Om_{-\varepsilon},T)}=(\K F,[\Phi,W])_{\LL^2(\Om_{-\varepsilon},T)},$ it is easy to check that $\F$ is bounded on $\V_\Phi.$  By Theorem A given in Appendix there exists a $V\in\V_\Phi$ such that

Step 3.   We shall show that $V\in\V_\Phi$ given in (<ref>) is the solution to problem (<ref>). Let $W_0\in\CC^1(\Om_{-\varepsilon},T)$ be given with $\<W_0,\nu\>|_{\Ga_{-\varepsilon}}=\<W_0,QX\>|_{\Ga_{b_1}}=0.$ By Lemma <ref>, there is $W\in\V_\Phi$ such that $\K^*[\Phi,W]+kW=W_0.$ Then (<ref>) becomes
(V,Ł_0^*W_0)_^2(_-ε)      =(F,W_0)_^2(_-ε,T)-∫__-ε|_2| pW_0,QXd

-∫__b_1|_1|qW_0,νdW_0∈_0,where $\D_0$ is given in (<ref>). Using (<ref>) in (<ref>), one has

+∫__b_1|_1|(q-V,ν)W_0,νd+∫__-ε|_2|(p-V,QX)W_0,QXdfor all $W_0\in\D_0.$ Thus $V\in\V_\Phi$ solves problem (<ref>).
Finally, (<ref>) follows from (<ref>).

Let $Y\in TM$ be with $|Y|=1.$ Then
_YDZ=DZ,Y-Z,QY[QY,Y]Z∈TM.In addition,

$|Y|=1$ implies that $QY,$ $Y$ is an orthonormal frame. Then

$$\ii_ZDY=\<D_ZY,QY\>QY=\<\ii_{QY}DY,Z\>QY\qfq Z\in TM.$$

we have
DZ,Y      =QYZ,YQY+YZ,YY

=_ZDY+Z,QY[QY,Y],that is (<ref>).

Let $E_1,$ $E_2$ be an orthonormal frame. Leting $v=E_i,$ $w=QE_j,$ and $P=\nabla\n$ in (<ref>) yields

$$\<E_i,E_j\>Q\nabla\n+\<\nabla\n E_i,QE_j\>\id=QE_i\otimes Q\nabla\n QE_j+\nabla\n E_i\otimes QE_j. $$

$$\<E_i,E_j\>Q\nabla\n Z+\<\nabla\n E_i,QE_j\>Z=\<Z,QE_i\> Q\nabla\n QE_j+\<Z,\nabla\n E_i\> QE_j. $$

$$\<E_i,E_j\>\<Q\nabla\n Z,E_k\>-\<Q\nabla\n E_i,E_j\>\<Z,E_k\>=-\<QZ,E_i\>\< Q\nabla\n QE_j,E_k\>+\<\nabla\n Z, E_i\> \<QE_j,E_k\> $$

for all $1\leq i,$ $j,$ $k\leq2,$ that is (<ref>).

Let $\Om\subset\Om_{-\varepsilon}$ be a subregion and let $\theta\in\CC^1(\Om).$ Let $E\in T\Om$ be given with $|E|=1.$ Let $\si>0$ be given small.

-∫_θ^2∇QE,QE[1/2|V|^2ν∧[QE,E],+V,QEν∧dV,E, ]dfor all $V\in\WW^{1,2}(\Om,T),$ where $\nu$ is the outside normal of $\Om$ and $\E$ is the volume element of $M.$

Let $E_1,$ $E_2$ be an orthonormal frame. For given $W\in TM,$ using (<ref>), we have

=DV⊗W, -E⊗Q∇Q-∇QE⊗Q

=∇Q_EDV,QW+_∇QEDV,QW.Letting $W=\ii_EDV$ in the above identity gives

=Q^*Π(_EDV,_EDV)+_∇QEDV,Q_EDV.Since $QE,$ $E$ forms an orthonormal frame, we obtain
_∇QEDV,Q_EDV=∇QE,QE_ QEDV,Q_EDV

=∇QE,QEDV(QE,E)E+DV(QE,QE)QE, DV(E,E)QE-DV(E,QE)E

=∇QE,QE_QEDV∧_EDV, .On the other hand, it follows from (<ref>) that
_EDV∧_QEDV      =(DV,E-V,QE[QE,E])∧(DV,QE+V,E[QE,E])

Moreover, noting that $d^2=0,$  we compute
d (θ^2∇QE,QEV,QEdV,E)+V,QEdV,E∧d(θ^2∇QE,QE)

=d (θ^2∇QE,QEV,QE)∧dV,E+V,QEdV,E∧d(θ^2∇QE,QE)

=[V,QEd (θ^2∇QE,QE)+θ^2∇QE,QEdV,QE]∧dV,E

1/2 d(θ^2∇QE,QE|V|^2[QE,E])-1/2|V|^2 d(θ^2∇QE,QE[QE,E])

=1/2|V|^2 d(θ^2∇QE,QE[QE,E])+1/2d(|V|^2)∧(θ^2∇QE,QE[QE,E])

-1/2|V|^2 d(θ^2∇QE,QE[QE,E])

$$\div_gQ\nabla\n V=\<DV,Q\nabla\n\>,\quad\div_gV=\<DV,\id\>,$$
inserting (<ref>), (<ref>), (<ref>), and (<ref>) into (<ref>),  we obtain

=θ^2Q^*Π(_EDV,_EDV)+d (θ^2∇QE,QEV,QEdV,E),

+V,QEdV,E∧d(θ^2∇QE,QE),  +1/2d(θ^2∇QE,QE|V|^2[QE,E]),

≥θ^2Q^*Π(_EDV,_EDV)+d (θ^2∇QE,QEV,QEdV,E),

+1/2d(θ^2∇QE,QE|V|^2[QE,E]),  -C(|V||θDV|+|V|^2).We integrate the above inequality over $\Om$ to have

≥∫_θ^2Q^*Π(_EDV,_EDV)dg+(d (θ^2∇QE,QEV,QEdV,E),  )_^2(,Λ^2)

+1/2(d(θ^2∇QE,QE|V|^2[QE,E]),  )_^2(,Λ^2)-C∫_(|V||θDV|+|V|^2)dg

=∫_θ^2Q^*Π(_EDV,_EDV)dg+(θ^2∇QE,QEV,QEdV,E,  δ)_^2(,Λ^2)

+1/2(θ^2∇QE,QE|V|^2[QE,E],  δ)_^2(,Λ^2)-C∫_(|V||θDV|+|V|^2)dg

+∫_θ^2∇QE,QEν∧(V,QEdV,E+1/2|V|^2[QE,E],  d,where the following formula has been used

$$(d Y,T)_{\LL^2(\Om,\Lambda^2)}=(Y,\delta T)_{\LL^2(\Om,T)}+\int_\Ga\<\nu\wedge Y,T\>d\Ga\qfq Y\in TM,\quad T\in\Lambda^2M,$$
see <cit.>.
Thus (<ref>) follows.

Proof of Theorem <ref>.   We shall prove the case of $m=1.$ The cases of $m\geq2$ can be treated by repeating a similar argument.

Let $F=e^{-\g\kappa}(f_1QX+f_2\nabla\n X)$ be given with $f_1,$ $f_2\in\WW^{1,2}(\Om_{-\varepsilon})$ and let $V\in\LL^2(\Om_{-\varepsilon},T)$ be the solution to problem (<ref>). By Proposition <ref>,

$$D_\tau V\in\LL^2(\Om_{-\varepsilon},T),\quad \tau=\frac{\a_t}{|\a_t|},$$
since $\Phi=-e^\var\frac{\<D\psi,Q\a_t\>}{|\a_t|^2}\a_t.$ Noting that $\div_gQ\nabla\n=0$ and $Q\tau,$ $\tau$ forms an orthonormal frame on $\overline{\Om}_{-\varepsilon}$ with positive orientation, from (<ref>) and (<ref>), we have

+f_1+ηV,QXx∈_-ε,.which yield, by (<ref>),

$$\|D_{Q\tau}V\|^2_{\LL^2(\Om_{-\varepsilon},T)}\leq C(\|D_\tau V\|^2_{\LL^2(\Om_{-\varepsilon},T)}+\|V\|^2_{\LL^2(\Om_{-\varepsilon},T)}+\|F\|^2_{\LL^2(\Om_{-\varepsilon},T)}).$$

Thus $V\in\WW^{1,2}(\Om_{-\varepsilon},T).$  In the case of $m=1$ (<ref>) follows    from (<ref>) and (<ref>). $\Box$

Proof of Theorem <ref>.   We shall prove the case of $m=1.$ The cases of $m\geq2$ can be treated by repeating a similar argument. By Theorem <ref>, we may assume that $F\in\CC^1(\Om_{-\varepsilon},T).$ We take two cut-off functions $\theta_1,$ $\theta_2\in\CC^1(-\varepsilon,b_1)$ as follows:

$$ \theta_1(s)=1\qfq s\in(\varepsilon/3,b_1),\quad \theta_1(s)=0\qfq s\in(-\varepsilon,\varepsilon/4);$$

$$ \theta_2(s)=1\qfq s\in(-\varepsilon,\epsilon/3),\quad \theta_2(s)=0\qfq s\in(\varepsilon/2,b_1).$$

Let $\si_0>0$ be given such that

$$Q^*\Pi\geq3\si_0\id\qfq x\in S^+\setminus S_{\varepsilon/3}.$$

We take $\si=\si_0,$ $\theta=\theta_1,$ $E=\a_t/|\a_t|,$ and $E=Q\a_t/|\a_t|$ in (<ref>), respectively. Then we obtain

Then  $V_2$ satisfies problem

V_2,QX|__-ε=p,  V_2,ν|__ε/2=0..

Noting that $\supp D\theta_2\subset(\varepsilon/3,\varepsilon/2),$ applying Theorem <ref> to problem (<ref>) yields, by (<ref>),
θ_2 V^2_^1,2(Σ_-ε^ε/2,T)      ≤C(F^2_^1,2( Σ_-ε^ε/2,T)+V^2_^1,2( Σ_-ε^ε/2,T)+ p^2_^1,2(_-ε))

+p^2_^1,2(_-ε)).Thus (<ref>) follows from (<ref>) and (<ref>). $\Box$

§ $\WW^{M,2}$ SOLUTIONS OF A COUPLING TENSOR SYSTEM OF MIXED TYPE

For given $(F,f)\in\LL^2(\Om_{-\varepsilon},T)$ arbitrarily, we directly seek some appropriate conditions on the boundary data
$(p,q)\in\LL^2(\Ga_{-\varepsilon})\times\LL^2(\Ga_{b_1})$ such that problem

V,QX|__-ε=p,  V,ν|__b_1=q,  ∫__-εvdg=0,.where $\rho=-\tr_g\Pi,$
admits a solution $(V,v)$ on $\Om_{-\varepsilon}.$

For convenience, we denote
V=ηe^-κ(V,QXQX+V,∇X∇X)V∈^2(_-ε,T),where the function $\eta$ and the vector field $X$ are given in (<ref>) and (<ref>), respectively.

We consider a uniqueness problem.

Let $S$ be of $\CC^5.$ Problem

admits a unique zero solution, where $\L_0^*$ is given in $(\ref{2.13}).$

$$w_1=e^{-\g\kappa}\<W,QX\>,\quad w_2=e^{-\g\kappa}\<W,\nabla\n X\>.$$

From (<ref>), we have

$$\L_0^*W=\nabla\n QDw_1-Dw_2-\cc W,$$
that is,
∇QDw_1-Dw_2=0x∈_-ε.Thus we obtain

$$\div_gQ\nabla\n QDw_1=0,\quad \div_g(\nabla\n)^{-1}Dw_2=0\qfq x\in S_+.$$

In addition $W|_{\Ga_{b_1}}=0$ implies

$$Dw_i=\<Dw_i,\nu\>\nu\qfq x\in\Ga_{b_1}.$$
It follows from (<ref>) that

$$\<Dw_1,\nu\>\nabla\n Q\nu=\<Dw_2,\nu\>\nu\qfq x\in\Ga_{b_1},$$
this is,

$$\<Dw_1,\nu\>=\<Dw_2,\nu\>=0\qfq x\in\Ga_{b_1},$$
since $\Pi(Q\nu,Q\nu)>0$ for $x\in\Ga_{b_1}.$ Noting that $-\div_gQ\nabla\n QDw_1$ and $\div_g(\nabla\n)^{-1}Dw_2$ are elliptic in $S_+,$ the uniqueness theorem in [9] implies that

Next, we need to prove that when given $\varepsilon>0$ is small,

Let $X_1$ and $X_2$ be given in (<ref>). Set

$$H=\frac1{\kappa^2}X_2\qfq x\in\Sigma_{-\varepsilon}^0.$$
From (<ref>), we have
QDw_2,D(Hw_1)      =Q∇QDw_1,D(Hw_1)=D_Q∇QDw_1H,Dw_1+D_HDw_1,Q∇QDw_1

=2D_Q∇QDw_1H,Dw_1+QDw_1,D_H(∇) QDw_1)

+QDw_1,∇QDw_1÷_gH-÷_g(QDw_1,∇QDw_1H),that is,

$$\Upsilon(w_1)=2\<D_{Q\nabla\n QDw_1}H,Dw_1\>+\<QDw_1,D_H(\nabla\n) QDw_1)\>+\<QDw_1,\nabla\n QDw_1\>\div_gH.$$

Moreover, using $H=\kappa^{-2}X_2,$ we compute
κ^3D_Q∇QDw_1H,Dw_1      =-2Q∇QDw_1,DκX_2(w_1)+κD_Q∇QDw_1X_2,Dw_1

=_2Ø(κ)[X_1(w_1)]^2+2_1X_2(κ)[X_2(w_1)]^2+Ø(κ)X_1(w_1)X_2(w_2),where the formula $\<D_{X_2}X_2,X_2\>=0$ is used,
κ^3QDw_1,D_H(∇) QDw_1)      =κQDw_1,D_X_2(∇) QDw_1)=κX_2(_2)[X_1(w_1)]^2

+Ø(κ){[X_2(w_1)]^2+X_1(w_1)X_2(w_1)},where the formula $D_{X_2}(\nabla\n)(X_2,X_2)=X_2(\lam_2)$ is used, and
κ^3QDw_1,∇QDw_1      ÷_gH=κ{_2[X_1(w_1)]^2+_1[X_2(w_1)]^2}÷_gX_2

Since $\kappa X_2(\lam_2)=\lam_2[X_2(\kappa)+\O(\kappa)],$ we obtain
κ^3Υ(w_1)      =-_2[X_2(κ)+Ø(κ)][X_1(w_1)]^2+[2_1X_2(κ)+Ø(κ)][X_2(w_1)]^2

+Ø(κ)X_1(w_1)X_2(w_2),which yields
-Υ(w_1)≥c/κ^2|Dw_1|^2x∈Σ_-ε^0, when $c>0$ and $\varepsilon>0$ are small enough, since $X_2(\kappa)>0$ by (<ref>).

Since $S$ is of $\CC^5,$ by Theorem <ref> $W\in\WW^{5,2}(\Om_{\varepsilon})$ and thus

$$w_1,\quad w_2\in\CC^3(\Om_{-\varepsilon},T).$$
From (<ref>) we have

$$\lim_{s\rw0^-}\frac{Dw_1}{\kappa^2}(\a(t,s))=\lim_{s\rw0^-}\frac{Dw_2}{\kappa^2}=0\qfq t\in[0,a).$$
lim_s0^-[2H(w_1)QDw_2+QDw_1,∇QDw_1H]((t,s))=0t∈[0,a).Moreover, since $\div_gQDw_2=0,$ we have
QDw_2,D(Hw_1)=÷_g(Hw_1QDw_2).For given $x\in\Ga_{-\varepsilon}$ and by (<ref>), we obtain

=X_2,ν{_2[X_1(w_1)]^2-_1[X_2(w_1)]^2}-2_2X_1,νX_1(w_1)X_2(w_1),that implies, by (<ref>),
κ^2[2H(w_1)QDw_2,ν+QDw_1,∇QDw_1H,ν]≥c_1|Dw_1|^2x∈_-ε,when given $\varepsilon>0$ is small enough.

For given $-\varepsilon<s<0,$ we integrate $\Upsilon(w_1)$ over $\Sigma_{-\varepsilon}^s$ to have, by (<ref>), (<ref>) and (<ref>),

$$ c\int_{\Sigma_{-\varepsilon}^s}\frac{|Dw_1|^2}{\kappa^2}dg\leq-\int_{\Ga_{-\varepsilon}\cup\Ga_s}[2H(w_1)\<QDw_2,\nu\>+\<QDw_1,\nabla\n QDw_1\>\<H,\nu\>]d\Ga,$$

from which we obtain, (<ref>) and (<ref>),

$$c\int_{\Sigma_{-\varepsilon}^0}\frac{|Dw_1|^2}{\kappa^2}dg\leq-\int_{\Ga_{-\varepsilon}}[2H(w_1)\<QDw_2,\nu\>+\<QDw_1,\nabla\n QDw_1\>\<H,\nu\>]d\Ga\leq0,$$
that is, (<ref>) is true.

_0={ W∈^2(_-ε,T) | Ł_0^*W+W=0, W,QX|__b_1=W,ν|__-ε=0 }.Set

where $\pounds_1$ is given in (<ref>).
Consider the direct sum composition

$$\LL^2(\Ga_{b_1})=\Ga(\W_0)\oplus\Ga^\bot(\W_0)\quad\mbox{in the norm of $\LL^2(\Ga_{b_1}).$}$$

V,ν|__b_1=q,  V,QX|__-ε=0
admits a solution $V\in\LL^2(\Om_{-\varepsilon},T)$ if and only if

Let $V$  satisfy  (<ref>) and $W\in \W_0.$ From (<ref>), we have

+∫__b_1∪_-ε(_1V,νW,ν-_2V,QXW,QX)d,which yield

Conversely, for given $q\in\Ga^\bot(\W_0),$ by Theorem <ref> there exists a unique solution $V_0\in\LL^2(\Om_{-\varepsilon},T)$ to problem

\<V_0,\nu\>|_{\Ga_{b_1}}=q,\quad\<V_0,QX\>|_{\Ga_{-\varepsilon}}=0.
\end{cases}$$

For given $Z\in\LL^2(\Om_{-\varepsilon},T),$ by Theorem <ref> problem

$$\begin{cases}\L_0V=-\cc Z,\\
\<V,\nu\>|_{\Ga_{b_1}}=\<V,QX\>|_{\Ga_{-\varepsilon}}=0
\end{cases}$$
admits a unique solution $V\in\LL^2(\Om_{-\varepsilon},T).$ By Theorem <ref>

We define a linear operator $\B:$ $\LL^2(\Om_{-\varepsilon},T)\rw\LL^2(\Om_{-\varepsilon},T)$ by

$$\B Z=V.$$
Then $\B$ is compact on $\LL^2(\Om_{-\varepsilon},T).$ By Fredholm's theorem, problem
(𝕀-)V=V_0admits a solution in $\LL^2(\Om_{-\varepsilon},T)$ if and only if
(V_0,U)_^2(_-ε,T)=0    U∈(𝕀-^*).In addition, it is easy to check that

For $U\in\Z(\id-\B^*),$ let $W=(\L_0^*)^{-1}U.$ Then

$$\L_0^*W+\cc W=0,$$
that is, $W\in\Ga(\W_0).$ Thus (<ref>) is equivalent to
0      =(V_0,Ł_0^*W)_^2(_-ε,T)=(Ł_0V_0,W)_^2(_-ε,T)-∫__b_1_1qW,νd

=-∫__b_1_1qW,νd,that is, $q\in\Ga^\bot(\W_0).$

Then there exists a solution $V\in\LL^2(\Om_{-\varepsilon},T)$ to problem (<ref>) that is also a solution to problem (<ref>).

By a similar argument and the duality, we have the following.

W,ν|__-ε=0,  W,QX|__b_1=p
admits a solution $W\in\LL^2(\Om_{-\varepsilon},T)$ if and only if

$$\int_{\Ga_{-\varepsilon}}\pounds_1p\<V,\nu\>d\Ga=0\qfq V\in\V_0.$$

Let $S$ be of $\CC^{m+5}.$ There exists a $H_0\in\WW^{m+5,2}(\Om_{-\varepsilon},T)$ satisfying

H_0,QX|__-ε=0,  ∫__-ε∇H_0,Qνd=1.
Step 1.   We consider a similar problem on a bigger region as

where $b_2>b_1$ such that

$$\kappa(\a(t,s))>0\qfq (t,s)\in[0,a)\times[b_1,b_2].$$

We claim that there is a solution   $H_0\in\LL^2(\Om_{-\varepsilon},T)$ to problem

H_0,ν|__b_2=q,  H_0,QX|__-ε=0
such that

$$\int_{\Ga_{-\varepsilon}}\<\nabla\n H_0,Q\nu\>d\Ga=1.$$

By contradiction. We suppose that for all $q\in\Ga^\bot(\W_0)$ solutions $V$ to problem (<ref>) satisfy

$$p_0=\frac{e^{\g\kappa}\Pi(X,Q\nu)}{\Pi(X,X)}\qfq x\in\Ga_{-\varepsilon}.$$
From (<ref>), $p_0\not=0$ for $x\in\Ga_{-\varepsilon}.$ By (<ref>)

$$\<X,\nu\>V=\<V,QX\>Q\nu+\<V,\nu\>X\qfq x\in\Ga_{-\varepsilon},\quad V\in\V_0.$$

Then $V\in\V_0$ and (<ref>)  imply that

=\int_{\Ga_{-\varepsilon}}\<\nabla\n V,Q\nu\>d\Ga=0$$
for all $V\in\V_0,$ where $\pounds_1$ is given in (<ref>). By Lemma <ref> problem (<ref>) admits a solution $W_0\in\LL^2(\Om_{-\varepsilon},T)$ with
$p=p_0,$ where $\Om_{-\varepsilon}$ and $b_1$ are replaced with $\hat\Om_{-\varepsilon}$ and $b_2,$ respectively.

Now we claim that
∫__+_1W_0,νqd=0    q∈^(_0).By Lemma <ref>, for each $q\in\Ga^\bot(\W_0),$ problem
(<ref>) admits a solution $V.$ Then (<ref>) follows  from the formula (<ref>).

The formula (<ref>) implies that

that is, there exists $W_1\in\W_0$ such that

$$\<W_1,\nu\>=\<W_0,\nu\>\quad x\in\Ga_{b_2}.$$
Since $\<W_1,QX\>=\<W_0,QX\>=0$ for $x\in\Ga_{b_2},$ we have

$$W_0=W_1\qfq x\in\Ga_{b_2}.$$
By Proposition <ref>, $W_1=W_0$ for all $x\in\hat\Om_{-\varepsilon}.$ This is a contradiction since $\<W_1,\nu\>|_{\Ga_{-\varepsilon}}
=\<W_0,\nu\>|_{\Ga_{-\varepsilon}}=p_0\not=0.$ The proof is complete.

Step 2.   Next, we show that $H_0\in\WW^{1,2}(\Om_{-\varepsilon},T).$ Let $\var\in\CC^{m+5}(\hat\Om_{-\varepsilon})$ be a cutoff function satisfying

$$\var(x)=1\qfq x\in\Om_{-\varepsilon};\quad\var(x)=0\qfq x\in\Ga_{b_2}.$$

$$H_1=\var H_0.$$
Then $H_1\in\LL^2(\hat\Om_{-\varepsilon},T)$ solves problem

$$\begin{cases}\L_0H_1+\cc H_1=\<Q\nabla\n H_0,D\var\>QX+\<H_0,D\var\>\nabla\n X\qfq x\in\hat\Om_{-\varepsilon},\\
\<H_1,\nu\>|_{\Ga_{b_2}}=\<H_1,QX\>|_{\Ga_{-\varepsilon}}=0.\end{cases}$$
Ii follows from Theorem <ref> that $H_1\in\WW^{m+5,2}(\hat\Om_{-\varepsilon},T).$ The proof is complete.

with the norm

For given $V\in\LL^2(\Om_{-\varepsilon},T),$ we solve problem

ϱ|__-ε=ϱ|__b_1=0,and define an operator $\varrho_{\cdot}:$ $\LL^2(\Om_{-\varepsilon},T)\rw\WW^{2,2}(\Om_{-\varepsilon})$ by

For given $(V,v)\in\LL^2(\Om_{-\varepsilon},T)\times\dot{\WW}^{1,2}(\Om_{-\varepsilon}),$ we solve problem

W,ν|__b_1=W,QX|__-ε=0,by Theorem <ref>, to have $W\in\LL^2(\Om_{-\varepsilon},T),$ where $\rho=-\tr_g\Pi.$ Moreover, by Theorem <ref>

Then we have

$$\div_gQ\nabla\n W=\eta\<W-V,QX\>=\Delta_g\varrho_{W-V}.$$
Thus we obtain
÷_gQ[∇W+QDϱ_W-V+ϕ∇H_0]=0x∈_-ε,where $H_0\in\WW^{1,2}(\Om_{-\varepsilon},T)$ is given in Proposition <ref> and

$$\phi=-\int_{\Ga_{-\varepsilon}}\<\nabla\n W+QD\varrho_{W-V},Q\nu\>d\Ga.$$

Since (<ref>) means that $d[\nabla\n W+QD\varrho_{W-V}+\phi\nabla\n H_0]=0$ where $d$ is the exterior derivative, by the de Rham cohomology group theorem
there exists a unique $w\in\dot{\WW}^{1,2}(\Om_{-\varepsilon})$ such that
Dw=∇W+QDϱ_W-V+ϕ∇H_0.Clearly, $w\in\WW^{2,2}(\Om_{-\varepsilon}).$ We define a linear operator
$\A:$ $\LL^2(\Om_{-\varepsilon},T)\times\dot{\WW}^{1,2}(\Om_{-\varepsilon})\rw\LL^2(\Om_{-\varepsilon},T)\times\dot{\WW}^{1,2}(\Om_{-\varepsilon})$ by
(V,v)=(W,w).Then $\A$ is  compact on $\LL^2(\Om_{-\varepsilon},T)\times\dot{\WW}^{1,2}(\Om_{-\varepsilon}).$

We consider the structure of

Clearly, $\dim\Z(\id-\A^*)<\infty.$ Suppose that $W_0$  satisfies problem

Ł^*_0W_0=ηϱ_0 QX,

W_0,ν|__-ε=e^κΠ(X,Qν)/Π(X,X),  W_0,QX|__b_1=0,

ϱ_0|__b_1=0,  ϱ_0|__-ε=1.

Let $H_0$ be given in Proposition $\ref{p5.2}.$ Then $(U,u)\in\Z(\id-\A^*)$ if and only if $(U,u)$ satisfies

Du,ν|__-ε=0,  ∫__-εudg=0,
where $c(u)=(\nabla\n H_0,Du)_{\LL^2(\Om_{-\varepsilon},T)}.$

Let $W$ solve problem (<ref>). It follows from the formula $\<X,\nu\>W=\<W,QX\>Q\nu+\<W,\nu\>X$ that

$$\<\nabla\n W,Q\nu\>=\frac{\Pi(X,Q\nu)}{\<X,\nu\>}\<W,\nu\>\qfq x\in\Ga_{-\varepsilon}.$$
By (<ref>) we obtain
∫__-ε∇W,Qνd      =∫__-ε_1W,νW_0,νd

=(W_0,Ł_0W)_^2(_-ε,T)-(W,Ł_0^*W_0)_^2(_-ε,T).In addition, from (<ref>) we have
∫__-εDϱ_W-V,νd=∫__-ε÷_gϱ_0Dϱ_W-Vdg=∫__-εϱ_0ηW-V,QXdg.It follows from (<ref>)-(<ref>) that
ϕ      =∫__-ε[ηϱ_0V,QX+V, W_0-e^-κρvW_0,∇X]dg

For given $(V,v),$ noting that

using (<ref>), (<ref>), and (<ref>), we have

=(V,  -(Ł_0^*)^-1(U+∇Du)+c(u)(W_0+ηϱ_0QX))_^2(_-ε,T)

+(v,  e^-κρ(Ł_0^*)^-1(U+∇Du),∇X-e^-κc(u)ρW_0,∇X)_^2(_-ε),which yield (<ref>).

By Theorem <ref> and a similar argument as in the proof of Proposition <ref>, we have the following. The details are omitted.
Let $S$ be of $\CC^5.$  Let $\U$ consist of all $(Z,z)\in\LL^2(\Om_{-\varepsilon},T)\times\dot{\WW}^{1,2}(\Om_{-\varepsilon})$ satisfying

Dz,ν|__-ε∪_b_1=0,  ∫__-εzdg=0.
Then $\dim\U<\infty$ and

with the norm

By a similar argument as for (<ref>), we have the following.
Let $H_0$ be given in Proposition $\ref{p5.2}.$ For given $(F,f,p,q)\in\H_0,$ problem

Dv=∇(V+(̱F,f,p,q) H_0)+QDϱ_V+F,

÷_g(V+(̱F,f,p,q) H_0)=ηV,∇X+f,

∫__-εvdg=0,  V,QX|__-ε=p,  V,ν|__b_1=q
admits a unique solution $(V,v)\in\LL^2(\Om_{-\varepsilon},T)\times\dot{\WW}^{1,2}(\Om_{-\varepsilon}),$ where $\varrho_V$ is given by $(\ref{5.27})$ and

$$\b(F,f,p,q)=-\int_{\Ga_{-\varepsilon}}\<\nabla\n V+QD\varrho_{V}+F,Q\nu\>d\Ga.$$

Consider problem

V,QX|__-ε=p,  V,ν|__b_1=q,  ∫__-εvdg=0,.where $\rho=-\tr_g\Pi.$ Let  $\N_0$ consist of  solutions to

$$\left\{\begin{array}{l}Dv=\nabla\n V\qfq x\in\Om_{-\varepsilon},\\
\div_gV=\rho v\qfq x\in\Om_{-\varepsilon},\\
\<V,QX\>|_{\Ga_{-\varepsilon}}=\<V,\nu\>|_{\Ga_{b_1}}=\int_{\Om_{-\varepsilon}}vdg=0.\end{array}\right.$$
By Theorems <ref> and <ref>

if $S$ is of $\CC^{m+5}.$
^m(_-ε,T)={ F∈^m,2(_-ε,T) | ÷_gQF∈^m,2(_-ε) },

Let $S$ be of $\CC^{m+5}.$  Let $(F,f,p)\in\W^0(\Om_{-\varepsilon},T)\times\LL^2(\Om_{-\varepsilon})\times\LL^2(\Ga_{-\varepsilon})$ be given. Then there exists

such that problem $(\ref{n5.33})$ admits a unique  solution $\N_0^\bot\cap[\LL^2(\Om_{-\varepsilon},T)\times\WW^{1,2}(\Om_{-\varepsilon})].$ Moreover, if
$(F,f,p)\in\W^m(\Om_{-\varepsilon},T)\times\WW^{m,2}(\Om_{-\varepsilon})\times\WW^{m,2}(\Ga_{-\varepsilon}),$ then

Step 1.   Let $(F,f,p,q)\in\H_0$ be given. We solve problem (<ref>) to have the solution $(V_0,v_0)\in\LL^2(\Om_{-\varepsilon},T)\times\dot{\WW}^{1,2}(\Om_{-\varepsilon}).$ Then

$$\L_0V_0=\G(F,f),\quad \<V_0,QX\>|_{\Ga_{-\varepsilon}}=p,\quad \<V_0,\nu\>|_{\Ga_{b_1}}=q,$$
where  $\G(F,f)=e^{-\g\kappa}[(-\div_gQF)QX+f\nabla\n X].$

We compute $\b(F,f,p,q)$ as follows. Let $W_0$ be given in (<ref>). Noting that $\<X,\nu\>V_0=\<V_0,QX\>Q\nu+\<V_0,\nu\>X,$ we have, by (<ref>) and (<ref>),
∫__-ε∇V_0,Qνd      =∫__-ε1/X,ν[pΠ(Qν, Qν)+Π(X,Qν)V_0,ν]d

Noting that

we obtain
-(̱F,f,p,q)      =(p,_2(e^κ+W_0,QX))_^2(_-ε)-(q,_1W_0,ν)_^2(_b_1)

Step 2.   Consider problem
(𝕀-)(V,v)=(V_0, v_0)^2(_-ε,T)×^1,2(_-ε),where $\A$ is defined by (<ref>).
By the Fredholm theorem  problem (<ref>) admits a solution in $\LL^2(\Om_{-\varepsilon},T)\times\dot{\WW}^{1,2}(\Om_{-\varepsilon})$ if and only if
((V_0, v_0),(U,u))_^2(_-ε,T)×^1,2(_-ε)=0(U,u)∈(𝕀-^*).

Z=(Ł_0^*)^-1(U+∇Du),  z=u(U,u)∈(𝕀-^*).Then $(Z-c(z)W_0,z)$ satisfies (<ref>) and $(Z-c(z)W_0,z)\in\U.$
Using (<ref>), (<ref>) and (<ref>), we have

=I(U,u)-(q,_1Z-c(z)W_0,ν)_^2(_b_1),where I(U,u)      =((F,f),Z-c(z)W_0)_^2(_-ε,T)+(p,_2(e^κc(z)+Z-c(z)W_0,QX))_^2(_-ε)

$$\X=\{\,\<Z-c(z)W_0,\nu\>|_{\Ga_{b_1}}\,|\,Z=(\L_0^*)^{-1}(U+\nabla\n Du),\,\,z=u,\,\,(U,u)\in\Z(\id-\A^*)\,\}.$$
By Proposition <ref>

Let $\{\,\<Z_i-c(z_i)W_0,\nu\>\,\}_{i=1}^k$ be a basis of $\X$ such that

$$\Big(\<Z_i-c(z_i)W_0,\nu\>,\pounds_1\<Z_j-c(z_j)W_0,\nu\>\Big)_{\Ga_{b_1}}=\delta_{ij}\qfq1\leq i,\,j\leq k.$$

For given $(F,f,p)\in\W^0(\Om_{-\varepsilon},T)\times\LL^2(\Om_{-\varepsilon})\times\LL^2(\Ga_{-\varepsilon}),$ let

and $(U_i,z_i)\in\Z(\id-\A^*)$ such that

$$Z_i=(\L_0^*)^{-1}(U_i+\nabla\n Du_i),\,\,z_i=u_i\qfq1\leq i\leq k.$$
Then (<ref>) hold true. Thus problem (<ref>) admits a unique solution $(V,v)\in\LL^2(\Om_{-\varepsilon},T)\times\dot{\WW}^{1,2}(\Om_{-\varepsilon}),$ which implies that

$$(V+\b_0H_0,v)\in \LL^2(\Om_{-\varepsilon},T)\times\dot{\WW}^{1,2}(\Om_{-\varepsilon})$$
is a solution to problem (<ref>) with

$$\b_0=-\int_{\Ga_{-\varepsilon}}\<\nabla\n V+F,Q\nu\>d\Ga.$$

Finally, (<ref>) follows from Theorems <ref> and <ref>.

Let $W_0$ solve problem (<ref>). We define

(Z,z)  }.By Theorem Theorems <ref> and <ref>

if $S$ is of $\CC^{m+5}.$

Consider problem

V,QX|__-ε=p,  V,ν|__b_1=q..Let  $\N_1$ consist of  solutions to

if $S$ is of $\CC^{m+5}.$

Let $S$ be of $\CC^{m+5}.$ Problem $(\ref{5.41})$ admits a unique solution

$$(V,v)\bot\N_1\qiq \LL^2(\Om_{-\varepsilon},T)\times\WW^{1,2}(\Om_{-\varepsilon})$$
if and only if
(p,q)ℵ^2(_-ε)×^2(_b_1).Moreover, if $(p,q)\in\WW^{j,2}(\Ga_{-\varepsilon})\times\WW^{j,2}(\Ga_{b_1})$ with $(\ref{5.44}),$ then

$$(V,v)\in\WW^{j,2}(\Om_{-\varepsilon},T)\times\WW^{j+1,2}(\Om_{-\varepsilon})\qfq 0\leq j\leq m+4.$$

Let $(V,v)$ be a solution to problem (<ref>). Then $(V, v-\theta)$ solves problem (<ref>) with
(F,f,p,q)=(0,θρ,p,q),where $\theta=(v,1)_{\LL^2(\Om_{-\varepsilon})}/m(\Om_{-\varepsilon}).$ Let ${\cal P}$ be the orthogonal projection operator from $\LL^2(\Ga_{-\varepsilon})\times\LL^2(\Ga_{b_1})$ on to $\aleph.$ By (<ref>) problem (<ref>) admits a solution if and only

=-θ∫__-ερe^-κZ-c(z)W_0,∇Xdg=θ∫__-εΔ_gzdg=0,where $Z-c(z)W_0$ satisfies  (<ref>).
Thus the desired results follow.

§ PROOFS OF THE MAIN RESULTS

Proof of Theorem <ref>. Let

$$U\in T^2_{\sym}S.$$
Consider problem

÷_gV+v_gΠ=fx∈S,.where $(V,v)\in\LL^2(S,T)\times\WW^{1,2}(S)$ is the unknown and
F=Q[D(_g U)-÷_gU],  f=-_gU(Q∇·,·)x∈S.

For $y\in\WW^{1,2}(S,\R^3),$ let
2v=∇y(e_2,e_1)-∇y(e_1,e_2)x∈S,V=(∇)^-1(Dv-F)x∈S,where $e_1,$ $e_2$ is an orthonormal basis of $T_xS$ with positive orientation.

By <cit.>, there is a $y\in\WW^{1,2}(S,\R^3)$ to solve problem (<ref>) if and only if $(V,v),$ being given in (<ref>) and (<ref>), respectively, solves problem (<ref>). In that case, we have

∇_e_2y=[-v+U(e_1,e_2)]e_1+U(e_2,e_2)e_2-QV,e_2,.x∈S.Moreover, from <cit.>,   $(V,v)$ is a solution to problem (<ref>) if and only if $v$ solves problem
D^2v,Q^*Π+1/κX_0v+vκ_gΠ=κf+1/κX_0,F+DF,Q^*Πx∈S,  κ≠0where $f$ and $F$ are given in (<ref>).

Let $U\in\WW^{m+1,2}(S,T^2_{\sym})$ be given.  We will find a solution

to problem (<ref>) as follows.

Consider the region $\Om_{-\varepsilon}$ in Theorem <ref>. By (<ref>)

By Theorem <ref> there exists  $q\in\WW^{m+9/2,2}(\Ga_{b_1})$ such that problem

V,QX|__-ε=p,  V,ν|__b_1=q,  ∫__-εvdg=0,.where $\rho=-\tr_g\Pi,$
admits a solution $(U,u)$ satisfying

The above solution can be extended to the region $S_1=S/\Om_{-\varepsilon}$ below. Since $S_1$ is a noncharacteristic region, the boundary operators $\T_1$ and $T_2$ as in [13] can defined.
Let $x\in\Ga_{-\varepsilon}$ be given. $\mu\in T_xS_1$ with $|\mu|=1$ is said to be the noncharacteristic normal outside $S_1$ if there is a curve
$\zeta:$ $(0,\iota)\rw S$ such that

$$\zeta(0)=x,\quad \zeta'(0)=-\mu,\quad\Pi(\mu,Y)=0\qfq Y\in T_x\Ga_{-\varepsilon}.$$
Let $\mu$ be the the noncharacteristic normal field along $\Ga_{-\varepsilon}.$
We define boundary operators $\T_i:$ $T_xM\rw T_xM$ by
_iX=1/2[Y+(-1)^iχ(μ,Y)ω(Y)Q∇Y]Y∈T_xM,  i=1,  2,where
χ(μ,Y)=(μ,Y,),  ω(Y)=1/√(-κ)Π(Y,Y),and  $\sign$ is the sign function. By a similar argument for <cit.>

$$\begin{cases}\<D^2v,Q^*\Pi\>+\frac1\kappa X_0v+v\kappa\tr_g\Pi=\kappa f+\frac1\kappa\<X_0,F\>+\<DF,Q^*\Pi\>\qfq x\in S_1,\\
w=u,\quad \frac1{\sqrt{2}}\<Dw,(\T_1-\T_2)\a_t\>=\frac1{\sqrt{2}}\<Du,(\T_1-\T_2)\a_t\>\qfq x\in\Ga_{-\varepsilon}
\end{cases}$$

admits a unique solution $w\in\WW^{m+1,2}(S_1),$ where $u$ is the component of the solution $(U,u)$ to problem (<ref>). Let

$$W=(\nabla\n)^{-1}(Dw-F)\qfq x\in S_1.$$

We define

$$(V,v)=\begin{cases}(U,u)\qfq x\in\Om_{-\varepsilon},\\
(W,w)\qfq x\in S_1.\end{cases}$$
Then $(V,v)\in\WW^{m-1,2}(S,T)\times\WW^{m,2}(S)$ is a solution to (<ref>). Then by (<ref>) $y\in\WW^{m,2}(S,\R^3).$ Thus by <cit.>

$$\sym DW=U-\<y,\n\>\Pi\in\WW^{m,2}(S,T).$$
The proof is complete.$\Box$

Proof of Theorem 1.2   Let $y\in\WW^{2,2}(S,\R^3)$ be an  infinitesimal isometry. Let

$$ 2v=\nabla y(e_2,e_1)-\nabla y(e_1,e_2),\quad V=(\nabla\n)^{-1}Dv\qfq x\in S,$$
where $e_1,$ $e_2$ is an orthonormal basis of $T_xS$ with positive orientation. Then $(V,v)$ solves problem

÷_gV+v_gΠ=0x∈S..By (<ref>) with $U=0,$ $y\in\WW^{2,2}(S,\R^3)$ implies $(V,v)\in\WW^{1,2}(S,T)\times \WW^{2,2}(S).$

Consider the region $\Om_{-\varepsilon}$ as in Theorem <ref>.
By <cit.> $v|_{\Ga_{b_1}}\in\CC^{m+4,\a}(\Ga_{b_1})$ due to that $v$ solves an elliptic equation on $S_+.$ In addition, since $v$ solves a hyperbolic equation on $S_-,$
$v|_{\Ga_{-\varepsilon}}\in\WW^{2,2}(\Ga_{-\varepsilon})$ by [13]. Thus

$$V|_{\Ga_{b_1}}\in\CC^{m+1}(\Ga_{b_1}),\quad V|_{\Ga_{-\varepsilon}}\in\WW^{1,2}(\Ga_{-\varepsilon}).$$

$$p=\<V,QX\>|_{\Ga_{-\varepsilon}},\quad q=\<V,\nu\>|_{\Ga_{b_1}}.$$
We decompose as the direct sum

where $(V_0,v_0)\in\N_1$ and $\N_1$ is given by (<ref>). Then
and $(V_1,v_1)\in\LL^2(\Om_{-\varepsilon},T)\times\WW^{1,2}(\Om_{-\varepsilon})$ solves problem (<ref>).
By Theorem <ref>

$$\|(V_1,v_1)\|_{\WW^{1,2}(\Om_{-\varepsilon},T)\times\WW^{2,2}(\Om_{-\varepsilon})}\leq C\|(p,q)\|^2_{\WW^{1,2}(\Ga_{-\varepsilon})\times\WW^{1,2}(\Ga_{b_1})},$$
where $\aleph$ is given in (<ref>).

Next, for given $\iota>0$ arbitrarily, we take $(p_\iota,q_\iota)\in\WW^{m+1,2}(\Ga_{-\varepsilon})\times\WW^{m+1,2}(\Ga_{b_1})$ such that

Denote by $(\hat p_\iota,\hat q_\iota)$ by the orthogonal projection of $(p_\iota,q_\iota)$ from $\LL^2(\Ga_{-\varepsilon})\times\LL^2(\Ga_{b_1})$ to $\aleph^\bot$ in $\LL^2(\Ga_{-\varepsilon})\times\LL^2(\Ga_{b_1}).$ Since $\dim\aleph<\infty,$ we have

$$\|(p,q)-(\hat p_\iota,\hat q_\iota)\|_{\WW^{1,2}(\Ga_{-\varepsilon})\times\WW^{1,2}(\Ga_{b_1})}\leq C\iota.$$

By Theorem <ref> problem (<ref>) admits a unique solution $(V_\iota,v_\iota)\in\N_1^\bot$ in $\LL^2(\Om_{-\varepsilon})\times\WW^{1,2}(\Om_{-\varepsilon}),$ and

$$\|(V_\iota,v_\iota)\|_{\WW^{m+1,2}(\Om_{-\varepsilon},T)\times\WW^{m+2,2}(\Om_{-\varepsilon})}\leq C\|(\hat p_\iota,\hat q_\iota)\|_{\WW^{m+1,2}(\Ga_{-\varepsilon})\times\WW^{m+1,2}(\Ga_{b_1})}.$$

$$\|(V_1,v_1)-(V_\iota,v_\iota)\|_{\WW^{1,2}(\Ga_{-\varepsilon})\times\WW^{1,2}(\Ga_{b_1})}\leq C\iota.$$

Now we extend the domains of $(V_1,v_1),$  $(V_0,v_0),$ and $(V_\iota,v_\iota)$ from $\Om_{-\varepsilon}$ to $S,$ respectively, as in the proof of Theorem <ref>, with the same notations such that

$(V_1,v_1),$  $(V_0,v_0),$ and $(V_\iota,v_\iota)$ satisfy problem (<ref>) on the region $S,$ respectively, and

$$\|(V,v)-(V_\iota+V_0,v_\iota+v_0)\|_{\WW^{1,2}(S,T)\times\WW^{2,2}(S)}\leq C\iota.$$
The proof is complete by the formula (<ref>). $\Box$

Proof of Theorem <ref>   As in $[4]$ we conduct in $2\leq i\leq m.$  Let

be an $(i-1)$th order isometry of class $\CC^{2+4(m-i+1)}_B(S,\R^3),$ where $z_0=\id$ and $z_2=y$ for some $i\geq2.$ Then

$$\sum_{j=0}^k\na^Tz_j\na z_{k-j}=0\qfq 1\leq k\leq i-1.$$

Next, we shall find out $z_i\in\CC^{2+4(m-i)}_B(S,\R^3)$ such that

is an $i$th order isometry.
By Corollary  $\ref{c1.1}$ there exists  a solution $z_i\in\CC^{2+4(m-i)}_B(S,\R^3)$ to problem

$$\sym \nabla z_i=-\frac{1}{2}\sym\sum_{j=1}^{i-1}\na^Tz_j\na z_{i-j}$$
which satisfies
z_i_^2+4(m-i)_B(S,^3)      ≤C∑_j=1^i-1^Tz_jz_i-j_^2+4(m-i)+3_B(S,^3)

≤C∑_j=1^i-1z_j_^2+4(m-i+1)_B(S,^3)z_i-j_^2+4(m-i+1)_B(S,^3).The conduction completes. $\Box$

Appendix: A theorem of  Lax-Milgram's type

The following theorem is an improved version of the Lax-Milgram theorem.

Theorem A  Let $(\V,\<\cdot,\cdot\>_\V)$ be a Hilbert space. Suppose that $\W\subset\V$ is a linear subspace of $\V$ such that $(\W,\<\cdot,\cdot\>_\W)$ is an inner product space, which may not be complete. Let $\b:$ $\V\times\W\rw\R$ be a bilinear functional satisfying
|(̱v,w)|≤Cv_w_v∈,  w∈,(̱w,w)≥w^2_w∈.Then there is a closed linear subspace $\V'\subset\V$ such that for  given bounded linear functional $\F$ on $\V,$ there  exists a unique $v\in\V'$ satisfying

$$\F(w)=\b(v,w)\qfq w\in\W,$$

For given $w\in\W,$ from (<ref>) $\b(\cdot,w)$ is a bounded linear functional on $\V.$ The Riesz representation theorem implies that there exists a bounded linear operator $A:$ $\W\rw\V$ such that

$$\b(v,w)=\<v,A w\>_\V\qfq v\in\V,\,\,w\in\W.$$

Then $\|A\|\leq C.$ Thus

$$\<w,A w\>_\V=\b(w,w)\geq\si\|w\|^2_\V\qfq w\in\W,$$
which yield
A w_≥w_w∈.

Let $\V'$ be the closure of $R(A)$ in $\V.$ Then for given $y\in\V'$ there is a sequence $\{w_k\}_1^\infty\subset\W$ such that

$$\|y-Aw_k\|_\V\rw0\quad\mbox{as}\quad k\rw\infty.$$

By (<ref>) $\var$ is a bounded linear functional on $\V'.$ By the Riesz representation theorem there exists a unique $v\in\V'$ such that

$$\var(y)=\<v,y\>_\V\qfq y\in\V'.$$
In particular, letting $y=Aw$ for $w\in\W$ yields

$$\F(w)=\var(y)=\<v,Aw\>_\V=\b(v,w)\qfq w\in\W.$$
Finally, (<ref>) follows from (<ref>).

Compliance with Ethical Standards

Conflict of Interest: The author declares that there is no conflict of interest.

Ethical approval: This article does not contain any studies with human participants or animals performed by the authors.

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