Uniqueness of solution of an inverse source problem for ultrahyperbolic equations

In this article, we consider an inverse problem for an ultrahyperbolic equation. One of our motivations to deal with this equation is its interesting structure from the point of view of the theory of partial differential equations. For instance, depending on the specific form of initial conditions, solutions possess both hyperbolic and elliptic properties (see [15]). Another motivation is recent discussions on the possibility of physics in multiple time dimensions, (e.g. [4, 20, 21]). Namely, in some superstring theories which attempt to unify the general theory of relativity and the quantum mechanics, extra dimensions are required for the consistency of theory. When the presence of more than one temporal dimension is considered, the mathematical model occurs as an ultrahyperbolic equation (e.g. [9]). More precisely, the paper [9] asserts that the equation in a form of

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \sum_{i=1}^{n}\partial _{x_{i}}^{2}u(x_{1},...,x_{n},y)-\partial _{y}^{2}u(x_{1},...,x_{n},y)=F(x_{1},...,x_{n},y) \nonumber \end{align*}$$

is of central physical importance, which describes the dynamical evolution of many physical quantities of classical and quantum field theories including the components of the electromagnetic fields in the case of a single time dimension, while the equation in a form of

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \sum_{i=1}^{n}\partial _{x_{i}}^{2}u(x_{1},...,x_{n},y_{1},...,y_{m})-\sum_{j=1}^{m}\partial _{y_{j}}^{2}u(x_{1},...,x_{n},y_{1},...,y_{m})=F(x_{1},...,x_{n},y_{1},...,y_{m}) \nonumber \end{align*}$$

is fundamental where $x\in \mathbb{R}^{n}$ and $y\in \mathbb{R}^{m}$ are, respectively, space-like and time-like variables.

Throughout the paper, we set

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle x &=(x_{1},^{\prime }x)\in \mathbb{R}^{n},\quad ^{\prime }x=(x_{2},...,x_{n})\in \mathbb{R}^{n-1}, \nonumber \\ y &=\left(y^{\prime },y_{m}\right) \in \mathbb{R}^{m},\quad y^{\prime }=\left(y_{1},...,y_{m-1}\right) \in \mathbb{R}^{m-1}, \nonumber \end{align*}$$

and use the following notation

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \partial _{x_{i}} &=\frac{\partial }{\partial x_{i}},{{\rm ~}}\partial _{y_{j}}=\frac{\partial }{\partial y_{j}},{{\rm ~}}\nabla _{^{\prime }x}=\left(\partial _{x_{2}},\partial _{x_{3}},\cdots ,\partial _{x_{n}}\right) ,{{\rm ~}}\partial _{x_{i}}^{2}=\frac{\partial ^{2}}{\partial x_{i}^{2}}, \nonumber \\ \nabla _{y} &=\left(\partial _{y_{1}},\partial _{y_{2}},\cdots ,\partial _{y_{m}}\right) ,{{\rm ~}}\partial _{x_{i}}\partial _{x_{j}}=\frac{\partial ^{2}}{\partial x_{i}\partial x_{j}},{{\rm ~}}\Delta _{x}=\sum\limits_{i=1}^{n}\partial _{x_{i}}^{2},{{\rm ~}}\Delta _{^{\prime }x}=\sum\limits_{i=2}^{n}\partial _{x_{i}}^{2}. \nonumber \end{align*}$$

Moreover for short, we write

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle u_{x_{i}}=\frac{\partial u}{\partial x_{i}},\quad u_{x_{i}x_{j}}=\frac{\partial ^{2}u}{\partial x_{i}\partial x_{j}},\,\,{{\rm etc.}} \nonumber \end{align*}$$

Henceforth for $n,m\geqslant 2$, let $D\subset \mathbb{R}^{n}$ be a bounded doman and $G:=G^{\prime }\times I\subset \mathbb{R}^{m}$ where I is an open bounded interval and $G^{\prime }\subset \mathbb{R}^{m-1}$ is a bounded domain. We set

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \Omega =D\times G \nonumber \end{align*}$$

and we assume that $\Omega$ is supported by the plane x₁  =  0, that is, for any $(x,y)\in \Omega$ there exists a cone which is contained in $\Omega$ and includes the point $(x,y)$, and whose base lies on the plane x₁  =  0.

Similarly to [9, 20, 21], we here consider an ultrahyperbolic equation in $u(x,y):=u(x_{1},...,x_{n},y_{1},...,y_{m})$, which is associated with general geometry in the time variables y :

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle & Lu(x,y)\equiv \sum_{i=1}^{n}\partial _{x_{i}}^{2}u(x,y)- \sum_{i,j=1}^{m}a_{ij}(x,y_{1},...,y_{m-1})u_{y_{i}y_{j}}(x,y) \nonumber \\ +& \sum_{i=1}^{n}a_{i}(x,y)\partial _{x_{i}}u(x,y)+\sum\limits_{j=1}^{m}b_{j}(x,y)\partial _{y_{j}}u(x,y)+a_{0}(x,y)u(x,y) \nonumber \end{align*}$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle =f(x,y)g(x,y_{1},...,y_{m-1}) \label{1} \nonumber \end{align} \tag{ 1 }$$

in the domain $\Omega =D\times G$.

Throughout we assume that we can choose constant M  >  0 such that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle a_{ij}\in C^{2}(\overline{D\times G^{\prime }}),\quad \left\vert \left\vert a_{ij}\right\vert \right\vert _{C^{2}\overline{\Omega })}\leqslant M, \label{2} \nonumber \end{align} \tag{ 2 }$$

and $a_{k},b_{j}\in C\left(\overline{\Omega }\right)$, $i,j=1,...,m;k=0,1,...,n$ and $f\in C^{2}\left(\overline{\Omega }\right)$. We note that in this article, we do not assume the ellipticity for $\sum\nolimits_{i,j=1}^{m}a_{ij}\partial _{y_{i}}\partial _{y_{j}}.$

The purpose of this article is to investigate the uniqueness of solution of the following problem:

Inverse source problem.

For given $f(x,y)$ and $u_{0}(x,y_{1},...,y_{m-1})$, find a pair of functions $(u(x,y),g(x,y_{1},...,y_{m-1}))$ in $\Omega$ satisfying equation (1), Cauchy data

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle u\left(0,x_{2},...,x_{n},y\right) =\partial _{x_{1}}u(0,x_{2},...,x_{n},y)=0 \label{3} \nonumber \end{align} \tag{ 3 }$$

and the additional information

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle u\left(x,y_{1},...,y_{m-1},0\right) =u_{0}\left(x,y_{1},...,y_{m-1}\right) . \label{4} \nonumber \end{align} \tag{ 4 }$$

This is an inverse problem of determining a factor g which is independent of one component y _m of the time-like variable of the right-hand side of (1). When we admit the multiple time dimensions related to a superstring theory (e.g. [9, 20, 21]), as a governing equation, we introduce an ultrahyperbolic equation (1) and then we should discuss an original cause initiating dynamical evolution of physical quantities. In our case, the right-hand side $f(x,y)g(x,y_{1},...,y_{m-1})$ in (1) is assumed to be such a cause and our inverse problem is the determination of a factor g.

Our main result is stated in theorem 1 and asserts that our data can determine the factor g uniquely.

Theorem 1. We assume that $f(x,y^{\prime },0)\neq 0$ and there exists a constant $\alpha _{1}>0$ such that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle -\overset{m}{\underset{i,j=1}{\sum }}(\partial _{x_{1}}a_{ij})(x,y^{\prime })\xi _{i}\xi _{j}\geqslant \alpha _{1}\sum_{i=1}^{m}|\xi _{i}|^{2},\quad \xi \in \mathbb{R}^{m},\,(x,y^{\prime })\in D\times G^{\prime }. \label{5} \nonumber \end{align} \tag{ 5 }$$

Then the inverse source problem has at most one solution $(u,g)$ such that $(u,g)\in C^{2}\left(\overline{\Omega }\right) \times$ $C\left(D\times G^{\prime }\right)$.

For the uniqueness in the inverse problem, we need to assume (5) and in general we cannot expect the uniqueness for a_ij satisfying ellipticity condition and (2) even for the case m  =  n  =  1. We can interpret condition (5) in the case of m  =  1 that wave propagation speed increases at a neighborhood of the plane x₁  =  0, which implies the absence of waveguides in a medium with x₁  >  0 and as a related work, see Amirov and Yamamoto [3].

Inverse problems for ultrahyperbolic equations were studied in [1, 2, 6, 11], where the key method is based on weighted a priori estimates and was firstly developed by Bukhgeim and Klibanov [6]. A uniqueness theorem for ultrahyperbolic equations is given by [6] for a bounded domain with Dirichlet and Neumann type condition on a part of the boundary. In [1] and [2], uniqueness is investigated in an unbounded domain with an additional information for the solution of direct problem at y   =  0. In [11], Hölder stability estimates were obtained in a bounded domain by some lateral boundary data. A major difference of our work from the existing results is that, in the inverse problem, additional information is given only at y _m  =  0.

As for the direct problem (1) and (3) with given fg, it is known that the problem of determination of the function u from relations (1) and (3) is ill-posed in the Hadamard sense (see [16], chapter IV). By using the mean-value theorem of Asgeirsson (see e.g. [13]), it was shown by [8] that the existence of solutions fails if the initial conditions are not properly prescribed. We refer to [7, 10, 17–19], as for the uniqueness results for various Cauchy, Dirichlet and Neumann problems for ultrahyperbolic equations. Finally, in [9] it is proved that under a nonlocal constraint, the initial value problem is well-posed for initial data given on a codimension-one hypersurface.

This paper consists of four sections and one appendix. In section 2, we state proposition 1 which is the key Carleman estimate for the proof of theorem 1 and the proof of the proposition is given in section 4. In section 3, we prove theorem 1 on the basis of proposition 1. In appendix, we prove two lemmata which are used for the proofs of proposition 1 and theorem 1.

In order to prove theorem 1, the key tool is an Carleman type inequality which will be presented in proposition 1 below.

We now introduce a Carleman weight function in the form of $\newcommand{\e}{{\rm e}} \chi =\exp \left(\lambda \psi ^{-\nu }\right)$ where

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \psi (x,y)=\delta x_{1}+\frac{1}{2}\sum_{i=2}^{n}(x_{i}-x_{i}^{0})^{2}+\frac{1}{2}\sum_{i=1}^{m-1}(y_{i}-y_{i}^{0})^{2}+\frac{1}{2}y_{m}^{2}+\alpha _{0} {{\rm ,}} \nonumber \end{align*}$$

where $\alpha _{0}>0,$ $\delta \geqslant 4,$ $\left(x^{0},y^{0}\right) \in \overline{\Omega },$ and $\lambda ,$ $\nu$ are large positive parameters.

We define the domain $\Omega _{\gamma }$ as a level set by $\psi$. More precisely, we set

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \Omega _{\gamma }=\left\{(x,y);{{\rm ~}}x_{1}>0,{{\rm ~}}\psi (x,y)<\gamma +\alpha _{0}<1\right\} {{\rm ,~}}0<\gamma <1. \nonumber \end{align*}$$

Then we see

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \partial \Omega _{\gamma }=\Gamma _{1}\cup \Gamma _{2}, \quad \Gamma _{1}=\left\{x_{1}=0\right\} \cap \overline{\Omega }_{\gamma },{{\rm ~}} \Gamma _{2}=\{(x,y); \psi (x,y)=\gamma +\alpha _{0}\}\cap \overline{\Omega } _{\gamma }. \nonumber \end{align*}$$

First, we reduce equation (1) to a more suitable form by introducing a new variable $\newcommand{\e}{{\rm e}} \widetilde{x}_{1}=\sqrt{2x_{1}}-\eta _{0}$, that is, $\newcommand{\e}{{\rm e}} x_{1}=\frac{1 }{2}\left(\widetilde{x}_{1}+\eta _{0}\right) ^{2}$, where $\newcommand{\e}{{\rm e}} 2\eta _{0}=\min \left\{\alpha _{0},\gamma \right\}$. Without loss of generality, we assume that $\newcommand{\e}{{\rm e}} \sqrt{2x_{1}}-\eta _{0}>0,$ i.e. $\newcommand{\e}{{\rm e}} x_{1}>\frac{\eta _{0}^{2}}{2}$, and so we have $\widetilde{x}_{1}>0$. Then, for the new function $\newcommand{\e}{{\rm e}} \widetilde{u} \left(\widetilde{x}_{1},^{\prime }x,y\right) \equiv u\left(\frac{1}{2} \left(\widetilde{x}_{1}+\eta _{0}\right) ^{2},^{\prime }x,y\right)$, by using the relations

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle u_{x_{1}}=\widetilde{u}_{\widetilde{x}_{1}}\frac{{\rm d}\widetilde{x}_{1}}{{\rm d}x_{1}}= \widetilde{u}_{\widetilde{x}_{1}}\frac{1}{\widetilde{x}_{1}+\eta _{0}};{{\rm ~}}u_{x_{1}x_{1}}=\widetilde{u}_{\widetilde{x}_{1}\widetilde{x}_{1}}\left(\widetilde{x}_{1}+\eta _{0}\right) ^{-2}-\widetilde{u}_{\widetilde{x} _{1}}\left(\widetilde{x}_{1}+\eta _{0}\right) ^{-3}, \nonumber \end{align*}$$

we have

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \left(\widetilde{x}_{1}+\eta _{0}\right) ^{-2}\widetilde{u}_{\widetilde{x} _{1}\widetilde{x}_{1}}+\Delta _{^{\prime }x}\widetilde{u}-\sum_{i,j=1}^{m} \widetilde{a}_{ij}\widetilde{u}_{y_{i}y_{j}}+\sum_{i=2}^{n}\widetilde{a}_{i} \widetilde{u}_{x_{i}}+\sum_{j=1}^{m}\widetilde{b}_{j}\widetilde{u}_{y_{j}}+ \widetilde{a}_{1}\widetilde{u}_{\widetilde{x}_{1}}+\widetilde{a}_{0} \widetilde{u}=\widetilde{f}\widetilde{g}, \nonumber \end{align*}$$

where $\widetilde{a}_{ij},$ $\widetilde{a}_{k},$ $\widetilde{b}_{j}$, $i,j=1,...,m$, $k=0,1,2,...,n,$ are functions in $\widetilde{x}_{1}$, $^{\prime }x$ and $y^{\prime }$ or y , and given by $a_{ij},a_{k},b_{j}$; for example, $\newcommand{\e}{{\rm e}} \widetilde{a}_{1}=a_{1}(\frac{1}{2}\left(\widetilde{x} _{1}+\eta _{0}\right) ^{2},^{\prime }x,y)-\left(\widetilde{x}_{1}+\eta _{0}\right) ^{-3}$, and $\newcommand{\e}{{\rm e}} \widetilde{f}=f(\frac{1}{2}\left(\widetilde{x} _{1}+\eta _{0}\right) ^{2},^{\prime }x,y)$ and $\newcommand{\e}{{\rm e}} \widetilde{g}=g(\frac{1}{2} \left(\widetilde{x}_{1}+\eta _{0}\right) ^{2},^{\prime }x,y^{\prime }).$

For the sake of simplicity, let us denote $\widetilde{u},$ $\widetilde{a} _{ij},$ $\widetilde{x}_{1},$ $\widetilde{a}_{k},$ $\widetilde{b}_{j},$ $\widetilde{f}$, $\widetilde{g}$ by $u,$ a_ij, x₁, a_k, b_j, $f,$ g respectively, where $i,j=1,...,m;$ $k=0,1,2,...,n.$ Then we can write

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \left(x_{1}+\eta _{0}\right) ^{-1}u_{x_{1}x_{1}}+\left(x_{1}+\eta _{0}\right) \left(\Delta _{^{\prime }x}u-\sum_{i,j=1}^{m}a_{ij}u_{y_{i}y_{j}} \right) \nonumber \end{align*}$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle + (x_{1}+\eta _{0}) \left(\sum_{i=1}^{n}a_{i}u_{x_{i}}+\sum_{j=1}^{m}b_{j}u_{y_{j}}+a_{0}u \right) = (x_{1}+\eta _{0})\,fg.\label{6} \nonumber \end{align} \tag{ 6 }$$

We set

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle L_{0}u\equiv \left(x_{1}+\eta _{0}\right) ^{-1}u_{x_{1}x_{1}}+(x_{1}+\eta _{0})\left(\Delta _{^{\prime }x}u-\sum_{i,j=1}^{m}a_{ij}u_{y_{i}y_{j}}\right) . \nonumber \end{align*}$$

Henceforth C  >  0 denotes generic constants which are independent of $\lambda$ and $\nu .$

Then we can have the following, which is the key for the proof of theorem 1.

Proposition 1. (Carleman estimate for an ultrahyperbolic equation) Let condition (5) be satisfied and $\gamma >0$ be so small that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{7} 0<\gamma <\min \left\{\frac{1}{2},\quad \frac{4}{3}(Mm\left(Mm\varepsilon _{0}^{-1}+Mm+m+1\right))^{-1/2}\right\}, \nonumber \end{align} \tag{ 7 }$$

where $0<\varepsilon _{0}<\frac{\alpha _{1}}{4m}$. Then there exists a constant $\delta _{\ast }=\delta _{\ast }(\alpha _{1},M,n,m)>0$ such that for any $\delta \ge \delta _{\ast }$ and $\nu >1,$ there exists a positive number $\lambda _{\ast }=\lambda _{\ast }(\delta ,\nu)$ such that for any $\lambda \ge \lambda _{\ast }$, the following estimate holds

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle &\psi ^{\nu +1}\left(L_{0}u\right) ^{2}\chi ^{2}-2\lambda \nu \beta _{0}u\left(x_{1}+\eta _{0}\right) \left(L_{0}u\right) \chi ^{2} \nonumber \\ &\geqslant 2\lambda \nu \delta \left(x_{1}+\eta _{0}\right) ^{-3}u_{x_{1}}^{2}\chi ^{2}+2\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}\left\vert \nabla _{^{\prime }x}u\right\vert ^{2}\chi ^{2} \nonumber \\ &+2\lambda \nu \left(x_{1}+\eta _{0}\right) \left\vert \nabla _{y}u\right\vert ^{2}\chi ^{2}+\lambda ^{3}\nu ^{4}\delta ^{4}\psi ^{-2\nu -3}u^{2}\chi ^{2}+\nabla _{x,y}\cdot U\quad {{\rm in~}} \Omega_{\gamma} \nonumber \end{align*}$$

for all $u=u(x,y) \in C^{2}\left(\overline{\Omega }_{\gamma }\right)$. Here $\beta _{0}=n+2+Mm((1+3\sqrt{2\gamma })m+1)$ and the vector-valued function U satisfies the following inequality:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{8} \left\vert U\right\vert \leqslant C\lambda ^{2}\nu ^{2}\psi ^{-2\nu -2}\left(x_{1}+\eta _{0}\right) ^{-2}(\left\vert \nabla _{x}u\right\vert ^{2}+\left\vert \nabla _{y}u\right\vert ^{2}+\lambda \nu u^{2})\chi ^{2}. \nonumber \end{align} \tag{ 8 }$$

The Carleman estimate proposition 1 is an estimate which is uniform in large parameter $\lambda>0$, that is, the constant C  >  0 is independent of $\lambda$ provided that $\lambda$ is sufficiently large. This uniformity is essential for the application to the inverse problem.

As for Carleman estimates for ultrahyperbolic equation (1) with constant principal parts a_ii  =  1 and a_ij  =  0 for $i\neq j$, see section 2 of chapter 4 of [1, 2] and section 4 of chapter IV of [16]. As for a Carleman estimate for an ultrahyperbolic equation with variable a_ij, see also Romanov [19] which assumes the ellipticity for a_ij unlike our paper.

We note that reflecting a similar property to the hyperbolicity, the ultrahyperbolic equations require some condition (5) to the principal coefficients a_ij for proving a Carleman estimate.

As general but very limited references on Carleman estimates, we refer, for example, to Bellassoued and Yamamoto [5], Hörmander [12], and Klibanov and Timonov [14]. Our Carleman estimates are different from more common Carleman estimates in e.g. [5, 12] in the following senses:

• Our Carleman estimate asserts a pointwise inequality, while the conventional Carleman estimates are proved in terms of weighted L² -integrals of solutions u and boundary data.
• In conventional Carleman estimates, the weight function has a form ${\rm e}^{2\lambda\varphi(x,y)}$ with suitable $\varphi$, and the estimate holds uniformly in all large $\lambda > 0$.

One can prove a conventional type of Carleman estimate for (1), and we can apply both types similarly to inverse problems and establish the uniqueness for the inverse problem. We note that our weight function is the same as in [1, 2], while [16] applies a weight in the form of ${\rm e}^{2\lambda\varphi(x,y)}$.

Once that a relevant Carleman estimate proposition 1 is established, we can apply it for the proof of the uniqueness of the inverse problem by the method originating from Bukhgeim and Klibanov [6]. As for such a method, see also [3, 5, 14]. In this section, postponing the proof of proposition 1 to section 4, we first complete the proof of theorem 1.

Let $(u,g)$ be a solution to (1), (3) and (4) with $\newcommand{\e}{{\rm e}} u_{0}\equiv 0$ in $\Omega _{\gamma }$. Since $f\left(x,y^{\prime },0\right) \neq 0$ and $f\in C^{2}\left(\overline{\Omega }\right)$, there exists a number $0<\gamma <1$ such that $f(x,y)\neq 0$ also in $\Omega _{\gamma }.$ We assume that $\gamma$, which was introduced before, satisfies this condition. We define a new unknown function $v=\dfrac{u}{f}$ in $\Omega _{\gamma }.$ Then dividing equation (6) by $f(x,y)$ and taking into account relations (3)–(4), we obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle &\left(x_{1}+\eta _{0}\right) ^{-1}v_{x_{1}x_{1}}+\left(x_{1}+\eta _{0}\right) \left(\Delta _{^{\prime }x}v-\overset{m}{\underset{i,j=1}{\sum } }a_{ij}v_{y_{i}y_{j}}\right) \nonumber \\ &+\left(x_{1}+\eta _{0}\right) \left(\overset{n}{\underset{i=1}{\sum }} \overline{a}_{i}v_{x_{i}}+\overset{m}{\underset{j=1}{\sum }}\overline{b} _{j}v_{y_{j}}+\overline{a}_{0}v\right) =\left(x_{1}+\eta _{0}\right) g,\label{9} \nonumber \end{align} \tag{ 9 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{10} v\left(0,^{\prime }x,y\right) =v_{x_{1}}\left(0,^{\prime }x,y\right) =0, \nonumber \end{align} \tag{ 10 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{11} v\left(x,y^{\prime },0\right) =0, \nonumber \end{align} \tag{ 11 }$$

where $\overline{a}_{i}$, $i=0,1,...,n$, $\overline{b}_{j}$, $j=1,...,m$ depend on the coefficients of equation (1), the function $f(x,y)$ and its derivatives.

Differentiating equation (9) with respect to y _m, setting $z=\partial _{y_{m}}v$ and using (11), we obtain the integro-differential equation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle &\left(x_{1}+\eta _{0}\right) ^{-1}z_{x_{1}x_{1}}+(x_{1}+\eta _{0})\left(\Delta _{^{\prime }x}z-\sum_{i,j=1}^{m}a_{ij}z_{y_{i}y_{j}}\right) \nonumber \\ &+(x_{1}+\eta _{0})\left(\sum\limits_{i=1}^{n}\overline{a} _{i}z_{x_{i}}+\sum_{j=1}^{m}\overline{b}_{j}z_{y_{j}}+\overline{a} _{0}z\right) \nonumber \\ &+(x_{1}+\eta _{0})\left(\sum_{i=1}^{n}\frac{\partial \overline{a}_{i}}{\partial y_{m}}I_{x_{i}}z+\sum_{j=1}^{m}\frac{\partial \overline{b}_{j}}{\partial y_{m}}I_{y_{j}}z+\frac{\partial \overline{a}_{0}}{\partial y_{m}} Iz\right) =0\label{12} \nonumber \end{align} \tag{ 12 }$$

with the Cauchy data

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{13} z\left(0,^{\prime }x,y\right) =z_{x_{1}}\left(0,^{\prime }x,y\right) =0, \nonumber \end{align} \tag{ 13 }$$

where

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle Iz=\left\{\begin{array}{@{}c@{}} \int_{0}^{y_{m}}z(x,y^{\prime },\tau){\rm d}\tau ,{{\rm ~}}y_{m}\geqslant 0 \nonumber \\ \int_{y_{m}}^{0}z(x,y^{\prime },\tau){\rm d}\tau ,{{\rm ~}}y_{m}<0 \end{array} \right. , \nonumber \end{align*}$$

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle I_{x_{i}}z=\left\{\begin{array}{@{}c@{}} \frac{\partial }{\partial x_{i}}\int_{0}^{y_{m}}z(x,y^{\prime },\tau){\rm d}\tau , {{\rm ~}}y_{m}\geqslant 0 \nonumber \\ \frac{\partial }{\partial x_{i}}\int_{y_{m}}^{0}z(x,y^{\prime },\tau){\rm d}\tau , {{\rm ~}}y_{m}<0 \end{array} \right. ,{{\rm ~}}i=1,..,n; \nonumber \end{align*}$$

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle I_{y_{j}}z=\left\{\begin{array}{@{}c@{}} \frac{\partial }{\partial y_{j}}\int_{0}^{y_{m}}z(x,y^{\prime },\tau){\rm d}\tau , {{\rm ~}}y_{m}\geqslant 0 \nonumber \\ \frac{\partial }{\partial y_{j}}\int_{y_{m}}^{0}z(x,y^{\prime },\tau){\rm d}\tau , {{\rm ~}}y_{m}<0 \end{array} \right. ,{{\rm ~}}j=1,...,m. \nonumber \end{align*}$$

We now prove that, if $z(x,y)$ satisfies (12) and (13), then $z(x,y) =0$ in $\Omega _{\gamma }$.

From (12), we obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left(L_{0}z\right) ^{2} =&~(x_{1}+\eta _{0})^{2}\left(\sum\limits_{i=1}^{n}\left(\overline{a}_{i}z_{x_{i}}+\frac{\partial \overline{a}_{i}}{\partial y_{m}}I_{x_{i}}z\right) +\sum_{j=1}^{m}\left(\overline{b}_{j}z_{y_{j}}+\frac{\partial \overline{b}_{j}}{\partial y_{m}} I_{y_{j}}z\right) \right. \nonumber \\ &+\left. \overline{a}_{0}z+\frac{\partial \overline{a}_{0}}{\partial y_{m}} Iz\right) ^{2} \nonumber \\ &\leqslant 6M_{0}\max \left\{n,m\right\} \left(x_{1}+\eta _{0}\right) ^{2}\left(\left\vert \nabla _{x}z\right\vert ^{2}+\sum\limits_{i=1}^{n}(I_{x_{i}}z)^{2}+|\nabla _{y}z|^{2}\right. \nonumber \\ &+\left. \sum_{j=1}^{m}(I_{y_{j}}z)^{2}+z^{2}+(Iz)^{2}\right) ,\label{14} \nonumber \end{align} \tag{ 14 }$$

where the constant M₀  >  0 depends on the coefficients a_ij, a_k, b_j, $i,j=1,...,m, k=0,1,...,n$ and $\Vert\,f\Vert _{C^{2}(\overline{\Omega }_{\gamma })}$. Here we shall use the following lemma, whose proof is given in appendix.

Lemma 1. The following relations hold:

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \int_{\Omega _{\gamma }}(Iz) ^{2}\chi ^{2}{\rm d}\Omega _{\gamma } &\leqslant \gamma \int_{\Omega _{\gamma }}z^{2}\chi ^{2}{\rm d}\Omega _{\gamma }, \nonumber \\ \int_{\Omega _{\gamma }}\left(I_{x_{i}}z\right) ^{2}\chi ^{2}{\rm d}\Omega _{\gamma } &\leqslant \gamma \int_{\Omega _{\gamma }}z_{x_{i}}^{2}\chi ^{2}{\rm d}\Omega _{\gamma }, \nonumber \\ \int_{\Omega _{\gamma }}\left(I_{y_{j}}z\right) ^{2}\chi ^{2}{\rm d}\Omega _{\gamma } &\leqslant \gamma \int_{\Omega _{\gamma }}z_{y_{j}}^{2}\chi ^{2}{\rm d}\Omega _{\gamma }, \nonumber \end{align*}$$

where $i=1,...,n;$ $j=1,...,m.$

By lemma 1, we can write inequality (14) in the following form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{15} \left(L_{0}z\right) ^{2}\leqslant 6M_{0}\max \left\{n,m\right\} \left(x_{1}+\eta _{0}\right) ^{2}(1+\gamma)(\left\vert \nabla _{x}z\right\vert ^{2}+|\nabla _{y}z|^{2}+z^{2}). \nonumber \end{align} \tag{ 15 }$$

On the other hand, by proposition 1, we can write

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle &\left(L_{0}z\right) ^{2}\chi ^{2}+(x_{1}+\eta _{0})^{2}(L_{0}z)^{2}\chi ^{2}+\lambda ^{2}\nu ^{2}\beta _{0}^{2}z^{2}\chi ^{2} \nonumber \\ &\geqslant \psi ^{\nu +1}\left(L_{0}z\right) ^{2}\chi ^{2}-2\lambda \nu \beta _{0}z(x_{1}+\eta _{0})(L_{0}z)\chi ^{2} \nonumber \\ &\geqslant 2\lambda \nu \delta \left(x_{1}+\eta _{0}\right) ^{-3}z_{x_{1}}^{2}\chi ^{2}+2\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}\left\vert \nabla _{^{\prime }x}z\right\vert ^{2}\chi ^{2} \nonumber \\ &+2\lambda \nu \left(x_{1}+\eta _{0}\right) \left\vert \nabla _{y}z\right\vert ^{2}\chi ^{2}+\lambda ^{3}\nu ^{4}\delta ^{4}(x_{1}+\eta _{0})^{-2}\psi ^{-2\nu -3}z^{2}\chi ^{2} \nonumber \\ &+\nabla _{x,y}\cdot U\label{16} \nonumber \end{align} \tag{ 16 }$$

for $\delta \ge \delta _{\ast },$ $\nu >1,$ $\lambda \ge \lambda _{\ast }$. From (15) and (16), it follows that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle &6M_{0}\max \left\{n,m\right\} \left(x_{1}+\eta _{0}\right) ^{2}(1+\gamma)(1+\left(x_{1}+\eta _{0}\right) ^{2}) \nonumber \\ &\times (\left\vert \nabla _{x}z\right\vert ^{2}+\left\vert \nabla _{y}z\right\vert ^{2}+z^{2})\chi ^{2}+\lambda ^{2}\nu ^{2}\beta _{0}^{2}z^{2}\chi ^{2} \nonumber \\ &\geqslant 2\lambda \nu \delta \left(x_{1}+\eta _{0}\right) ^{-3}z_{x_{1}}^{2}\chi ^{2}+2\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}\left\vert \nabla _{^{\prime }x}z\right\vert ^{2}\chi ^{2} \nonumber \\ &+2\lambda \nu \left(x_{1}+\eta _{0}\right) \left\vert \nabla _{y}z\right\vert ^{2}\chi ^{2}+\lambda ^{3}\nu ^{4}\delta ^{4}(x_{1}+\eta _{0})^{-2}\psi ^{-2\nu -3}z^{2}\chi ^{2} \nonumber \\ &+\nabla _{x,y}\cdot U.\label{17} \nonumber \end{align} \tag{ 17 }$$

Integrating inequality (17) in $\Omega _{\gamma }$ and using the fact that $\newcommand{\e}{{\rm e}} (x_{1}+\eta _{0})<\frac{3}{4}\gamma <1$, we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle &\int_{\Omega _{\gamma }}(\lambda ^{3}\nu ^{3}\psi ^{-2\nu -3}z^{2}+\lambda \nu (z_{x_{1}}^{2}+\left(x_{1}+\eta _{0}\right) ^{2}\left\vert \nabla _{^{\prime }x}z\right\vert ^{2}+\left(x_{1}+\eta _{0}\right) \left\vert \nabla _{y}z\right\vert ^{2}))\chi ^{2}{\rm d}\Omega _{\gamma } \nonumber \\ &\leqslant -\int_{\Omega _{\gamma }}\nabla _{x,y}\cdot U{\rm d}\Omega _{\gamma }\label{18} \nonumber \end{align} \tag{ 18 }$$

for $\lambda \geqslant 12M_{0}\max \left\{n,m\right\} (1+\gamma)>1$ and $\nu \geqslant \delta ^{-4}(1+\beta _{0}^{2})$. Taking into account condition (3) and inequality (8), from (18) we can write

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle &\lambda ^{3}\nu ^{3}\int_{\Omega _{\gamma }}\psi ^{-2\nu -3}z^{2}\chi ^{2}{\rm d}\Omega _{\gamma }\leqslant C\lambda ^{2}\nu ^{2}(\gamma +\alpha _{0})^{-2\nu -2}\exp (2\lambda (\gamma +\alpha _{0})^{-\nu }) \nonumber \\ &\times \int_{\Gamma _{2}}\left(x_{1}+\eta _{0}\right) ^{-2}(\left\vert \nabla _{x}z\right\vert ^{2}+\left\vert \nabla _{y}z\right\vert ^{2}+\lambda \nu z^{2}){\rm d}\Gamma _{2}\label{19}. \nonumber \end{align} \tag{ 19 }$$

Now let us choose small $\gamma _{0}\in (0,\gamma).$ Then we see that $\Omega _{\gamma _{0}}\subset \Omega _{\gamma }$ and

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \chi ^{2}(x,y)\geqslant \exp (2\lambda (\gamma _{0}+\alpha _{0})^{-\nu }) \nonumber \end{align*}$$

in $\Omega _{\gamma _{0}}.$ Hence inequality (19) implies that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle &\lambda ^{3}\nu ^{3}(\gamma _{0}+\alpha _{0})^{-2\nu -3}\exp (2\lambda (\gamma _{0}+\alpha _{0})^{-\nu })\int_{\Omega _{\gamma _{0}}}z^{2}{\rm d}\Omega _{\gamma _{0}} \nonumber \\ &\leqslant C\lambda ^{3}\nu ^{3}(\gamma +\alpha _{0})^{-2\nu -2}\exp (2\lambda (\gamma +\alpha _{0})^{-\nu }) \nonumber \\ &\times \int_{\Gamma _{2}}\left(x_{1}+\eta _{0}\right) ^{-2}(\left\vert \nabla _{x}z\right\vert ^{2}+\left\vert \nabla _{y}z\right\vert ^{2}+z^{2}){\rm d}\Gamma _{2}.\label{20} \nonumber \end{align} \tag{ 20 }$$

Since $\gamma _{0}+\alpha _{0}<\gamma +\alpha _{0}<1,$ and

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \underset{\lambda \rightarrow \infty }{\lim }\left\{\exp \left (-2\lambda ((\gamma _{0}+\alpha _{0})^{-\nu }-(\gamma +\alpha _{0})^{-\nu })\right) \right\} =0, \nonumber \end{align*}$$

passing to the limit as $\lambda \rightarrow \infty$ in (20), we conclude that

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \int_{\Omega _{\gamma _{0}}}z^{2}{\rm d}\Omega _{\gamma _{0}}\leqslant 0, \nonumber \end{align*}$$

which means that $z(x,y)=0$ in $\Omega _{\gamma _{0}}$. Since $\gamma _{0}\in (0,\gamma)$ is an arbitrary number, $z(x,y)=0$ in $\Omega _{\gamma }.$

Varying the point $x^{0}=\left(0,x_{2}^{0},x_{3}^{0},...,x_{n}^{0}\right)$ of the plane x₁  =  0, we establish that z  =  0 on $\widetilde{\Omega} _{\gamma }=\left\{(x,y)\in \Omega ;{{\rm ~}}0\leqslant \delta x_{1}\leqslant \gamma \right\} ,$ that is, $\partial_{y_{m}}v=0$ on $\widetilde{\Omega}_{\gamma }$. Then from equation (9) and by condition (11) we conclude that $g\left(x,y^{\prime }\right) =0$ on $\widetilde{\Omega}_{\gamma }^{\prime }=\left\{(x,y^{\prime })\in D\times G^{\prime };{{\rm ~}}0\leqslant \delta x_{1}\leqslant \gamma \right\}$, where $\widetilde{\Omega}_{\gamma }^{\prime }$ is the projection of $\widetilde{\Omega}_{\gamma }$ in $\mathbb{R} ^{n+m-1}$. Repeating the same argument leads to z  =  0 in $\widetilde{\Omega}_{2\gamma }$ and $g\left(x,y^{\prime }\right) =0$ on $\widetilde{\Omega}_{2\gamma }^{\prime }$. Thus, continuing in this way we complete the proof.

In the proof of proposition 1, we shall use two lemmata. The first of them is lemma 2, the proof of which is technical and lengthy, and we postpone it to appendix.

Lemma 2. Under the hypothesis of proposition 1, there exists a constant $\delta _{0}\,=$ $\delta _{0}(\alpha _{1},M,n,m)>0$ such that for any $\delta \ge \delta _{0}$ and $\nu >1$, there exists a positive number $\lambda _{0}=\lambda _{0}(\delta ,\nu)$ such that for all $\lambda \ge \lambda _{0}$ and for all $u\in C^{2}(\overline{\Omega }_{\gamma }),$ the following inequality is valid

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \psi ^{\nu +1}\left(L_{0}u\right) ^{2}\chi ^{2} &\geqslant 2\lambda \nu \delta \left(x_{1}+\eta _{0}\right) ^{-3}u_{x_{1}}^{2}\chi ^{2}-2\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}(\beta _{0}-1)\left\vert \nabla _{^{\prime }x}u\right\vert ^{2}\chi ^{2} \nonumber \\ &+\lambda \nu \delta \alpha _{1}\left(x_{1}+\eta _{0}\right) \left\vert \nabla _{y}u\right\vert ^{2}\chi ^{2}+2\lambda ^{3}\nu ^{4}\delta ^{4}\left(x_{1}+\eta _{0}\right) ^{-2}\psi ^{-2\nu -3}u^{2}\chi ^{2} \nonumber \\ &+\sigma _{1}(\lambda ,\nu)u^{2}\chi ^{2}+\nabla _{x,y}\cdot U_{1},\label{21} \nonumber \end{align} \tag{ 21 }$$

where $\sigma _{1}(\lambda ,\nu)$ is a polynomial in $\lambda$ and $\nu$ of degree 6, and the term $\nabla _{x,y}\cdot U_{1}$ is given in the proof explicitly.

In (21), the sign of the term $\left\vert \nabla _{^{\prime }x}u \right\vert ^{2}$ is minus and therefore we need to perform another estimation:

Lemma 3. The following equality holds for any function $u\in C^{2}(\overline{\Omega })$:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle -(x_{1}+\eta _{0})u(L_{0}u)\chi ^{2} &=u_{x_{1}}^{2}\chi ^{2}+\chi ^{2}\left(x_{1}+\eta _{0}\right) ^{2}\left(|\nabla _{^{\prime }x}u|^{2}-\sum_{i,j=1}^{m}a_{ij}u_{y_{i}}u_{y_{j}}\right) \nonumber \\ & \quad +\sigma _{2}(\lambda ,\nu)u^{2}\chi ^{2}+\nabla _{x,y}\cdot U_{2}, \label{22} \nonumber \end{align} \tag{ 22 }$$

where $\sigma _{2}(\lambda ,\nu)$ is a polynomial in $\lambda$ and $\nu$ of degree 4. Moreover, the term $\nabla _{x,y}\cdot U_{2}$ is given as follows

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \nabla _{x,y}\cdot U_{2} =&~-((uu_{x_{1}}+\lambda \nu \delta \psi ^{-\nu -1}u^{2})\chi ^{2})_{x_{1}} \nonumber \\ &-\left(x_{1}+\eta _{0}\right) ^{2}\sum_{i=2}^{n}\left((uu_{x_{i}}+\lambda \nu \psi _{x_{i}}\psi ^{-\nu -1}u^{2})\chi ^{2}\right) _{x_{i}} \nonumber \\ &+\left(x_{1}+\eta _{0}\right) ^{2}\sum_{i,j=1}^{m}\left(\left(a_{ij}\left(u_{y_{j}}u+\lambda \nu \psi _{y_{j}}\psi ^{-\nu -1}u^{2}\right) -\frac{1}{2}\left(a_{ij}\right) _{y_{j}}u^{2}\right) \chi ^{2}\right) _{y_{i}}. \nonumber \end{align*}$$

Lemma 3 can be proved easily by direct calculations and we omit the proof here.

We now proceed to the completion of the proof of proposition 1. We multiply equality (22) by $2\lambda \nu \beta _{0}$ and add to inequality (21) to have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle &\psi ^{\nu +1}\left(L_{0}u\right) ^{2}\chi ^{2}-2\lambda \nu \beta _{0}(x_{1}+\eta _{0})u\left(L_{0}u\right) \chi ^{2} \nonumber \\ &\geqslant 2\lambda \nu \delta \left(x_{1}+\eta _{0}\right) ^{-3}u_{x_{1}}^{2}\chi ^{2}+2\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}\left\vert \nabla _{^{\prime }x}u\right\vert ^{2}\chi ^{2} \nonumber \\ &+\lambda \nu \delta \alpha _{1}\left(x_{1}+\eta _{0}\right) \left\vert \nabla _{y}u\right\vert ^{2}\chi ^{2}-2\lambda \nu \beta _{0}\left(x_{1}+\eta _{0}\right) ^{2}\chi ^{2}\sum_{i,j=1}^{m}a_{ij}u_{y_{i}}u_{y_{j}} \nonumber \\ &+2\lambda ^{3}\nu ^{4}\delta ^{4}\left(x_{1}+\eta _{0}\right) ^{-2}\psi ^{-2\nu -3}u^{2}\chi ^{2}+\sigma _{3}(\lambda ,\nu)u^{2}\chi ^{2} \nonumber \\ &+\nabla _{x,y}\cdot U_{1}+2\lambda \nu \beta _{0}\nabla _{x,y}\cdot U_{2} {}\label{23} \nonumber \end{align} \tag{ 23 }$$

for $\delta >\delta _{0}$, $\nu >1,\lambda >\lambda _{0},$ where $\sigma _{3}(\lambda ,\nu)=\sigma _{1}(\lambda ,\nu)+2\lambda \nu \beta _{0}\sigma _{2}(\lambda ,\nu).$

Setting $\delta _{1}=\frac{2}{\alpha _{1}}\left(1+\frac{3}{4}m\beta _{0}\gamma M\right)$ and using the inequality

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle -\sum_{i,j=1}^{m}a_{ij}u _{y_{i}}u _{y_{j}}\geqslant -Mm\left\vert \nabla _{y}u \right\vert ^{2}, \nonumber \end{align*}$$

we can estimate the coefficient of $\left\vert \nabla _{y}u \right\vert ^{2}$ in (23):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle &\lambda \nu \delta \alpha _{1}\left(x_{1}+\eta _{0}\right) \left\vert \nabla _{y}u \right\vert ^{2}\chi ^{2}-2\lambda \nu \beta _{0}\left(x_{1}+\eta _{0}\right) ^{2}\sum_{i,j=1}^{m}a_{ij}u _{y_{i}}u _{y_{j}}\chi ^{2} \nonumber \\ &\geqslant \lambda \nu \left(x_{1}+\eta _{0}\right) (\delta \alpha _{1}-2\beta _{0}mM\left(x_{1}+\eta _{0}\right))\left\vert \nabla _{y}u \right\vert ^{2}\chi ^{2} \nonumber \\ &\geqslant 2\lambda \nu \left(x_{1}+\eta _{0}\right) \left\vert \nabla _{y}u \right\vert ^{2}\chi ^{2}\label{24} \nonumber \end{align} \tag{ 24 }$$

for $\delta \geqslant \delta _{1}.$

As for the coefficient of u², we can write $\sigma _{3}(\lambda ,\nu)$ in the form

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \sigma _{3}(\lambda ,\nu)=\lambda ^{3}\nu ^{3}\sigma _{31}+\lambda ^{2}\nu ^{2}\sigma _{32}+\lambda \nu \sigma _{33}, \nonumber \end{align*}$$

where the expressions $\sigma _{31},$ $\sigma _{32},$ $\sigma _{33}$ depend on $\delta$, $\nu ,$ a_ij and $\psi$. Since the functions a_ij and $\psi$ are bounded in the space $C^{2}\left(\overline{\Omega }_{\gamma }\right)$, it can be seen that the function $\widetilde{\sigma }_{31}= \frac{\sigma _{31}}{\delta ^{3}\nu \psi ^{-2\nu -3}}$ is bounded uniformly with respect to $(x,y) \in \overline{\Omega }_{\gamma }$ and $\delta :$ $\left\vert \widetilde{\sigma }_{31}\right\vert \leqslant M_{1}$, M₁  >  0. Then, noting that $\newcommand{\e}{{\rm e}} \left(x_{1}+\eta _{0}\right) ^{-2}\geqslant (\frac{3 }{4}\gamma){}^{-2}>1$ we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \lambda ^{3}\nu ^{4}\delta ^{4}\left(x_{1}+\eta _{0}\right) ^{-2}\psi ^{-2\nu -3}+\lambda ^{3}\nu ^{3}\sigma _{31} &>\lambda ^{3}\nu ^{4}\delta ^{4}\psi ^{-2\nu -3}\left(1+\frac{1}{\delta }\widetilde{\sigma }_{31}\right) \nonumber \\ &\geqslant \lambda ^{3}\nu ^{4}\delta ^{4}\psi ^{-2\nu -3}\left(1-\frac{1}{\delta }M_{1}\right) \nonumber \\ &\geqslant \frac{1}{2}\lambda ^{3}\nu ^{4}\delta ^{4}\psi ^{-2\nu -3}\label{25} \nonumber \end{align} \tag{ 25 }$$

for $\delta \geqslant \delta _{2}=2M_{1}$. It is obvious that the functions $\sigma _{32}$ and $\sigma _{33}$ are also bounded on $\Omega _{\gamma }$ for fixed $\delta \geqslant \delta _{2}$, $\nu >1$, that is, there exist constants M₂, M₃  >  0 such that $\left\vert \sigma _{32}\right\vert \leqslant M_{2},$ $\left\vert \sigma _{33}\right\vert \leqslant M_{3}.$ Hence, by using (25), we obtain

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle &\lambda ^{3}\nu ^{4}\delta ^{4}\left(x_{1}+\eta _{0}\right) ^{-2}\psi ^{-2\nu -3}+\lambda ^{3}\nu ^{3}\sigma _{31}+\lambda ^{2}\nu ^{2}\sigma _{32}+\lambda \nu \sigma _{33} \nonumber \\ &\geqslant \frac{1}{2}\lambda ^{3}\nu ^{4}\delta ^{4}\psi ^{-2\nu -3}-\lambda ^{2}\nu ^{2}M_{2}-\lambda \nu M_{3}\geqslant 0 \nonumber \end{align*}$$

for $\lambda \geqslant \lambda _{1}=\max \left\{M_{2},\sqrt{M_{3}}\right\} ,$ which yields

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{26} 2\lambda ^{3}\nu ^{4}\delta ^{4}\left(x_{1}+\eta _{0}\right) ^{-2}\psi ^{-2\nu -3}+\sigma _{3}(\lambda ,\delta)\geqslant \lambda ^{3}\nu ^{4}\delta ^{4}\psi ^{-2\nu -3}. \nonumber \end{align} \tag{ 26 }$$

Consequently, inequalities (23), (24) and (26) imply that

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle &\psi ^{\nu +1}\left(L_{0}u\right) ^{2}\chi ^{2}-2\lambda \nu \beta _{0}u\left(L_{0}u\right) \chi ^{2} \nonumber \\ &\geqslant 2\lambda \nu \delta \left(x_{1}+\eta _{0}\right) ^{-3}u_{x_{1}}^{2}\chi ^{2}+2\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}\chi ^{2}\left\vert \nabla _{^{\prime }x}u\right\vert ^{2}\chi ^{2} \nonumber \\ &+2\lambda \nu \left(x_{1}+\eta _{0}\right) \left\vert \nabla _{y}u\right\vert ^{2}\chi ^{2}+\lambda ^{3}\nu ^{4}\delta ^{4}\psi ^{-2\nu -3}u^{2}\chi ^{2}+\nabla _{x,y}\cdot U \nonumber \end{align*}$$

for $\delta \geqslant \delta _{\ast }=\max \left\{4,\delta _{0},\delta _{1},\delta _{2}\right\} ,$ $\nu >1$ and $\lambda \geqslant \lambda _{\ast }=\max \left\{\lambda _{0},\lambda _{1}\right\}$, where $\nabla _{x,y}\cdot U\,=$ $\nabla _{x,y}\cdot U_{1}+2\lambda \nu \beta _{0}\nabla _{x,y}\cdot U_{2}$. It is easy to see that the vector function U satisfies relation (8). Thus the proof of proposition 1 is completed.

The second author was supported by Grant-in-Aid for Scientific Research (S) 15H05740 of Japan Society for the Promotion of Science and by The National Natural Science Foundation of China (no. 11771270, 91730303). This work was prepared with the support of the "RUDN University Program 5-100".

Let K be the part of the domain $\Omega _{\gamma }$ produced by the plane y _m  =  0 and let $\sigma (x,y^{\prime })$ be the distance between the boundary point $(x,y)$ of $\Omega _{\gamma }$ and the point $(x,y^{\prime },0)\in K.$

Since $\chi ^{2}$ is monotone with respect to y _m, we have

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle &\int_{\Omega _{\gamma }}\chi ^{2}(Iz)^{2}{\rm d}\Omega _{\gamma }=\int_{K}\left(\int\nolimits_{0}^{\sigma (x,y^{\prime }) }\chi ^{2}(Iz)^{2}{\rm d}y_{m}+\int\nolimits_{-\sigma (x,y^{\prime }) }^{0}\chi ^{2}(Iz)^{2}{\rm d}y_{m}\right) {\rm d}x{\rm d}y^{\prime } \nonumber \\ &\leqslant \int_{K}\left(\sqrt{\gamma }\int\nolimits_{0}^{a} \int_{0}^{y_{m}}z^{2}(x,y^{\prime },\tau)\chi ^{2}{\rm d}\tau {\rm d}y_{m}+\sqrt{\gamma }\int\nolimits_{-a}^{0}\int_{y_{m}}^{0}z^{2}(x,y^{\prime },\tau)\chi ^{2}{\rm d}\tau {\rm d}y_{m}\right) {\rm d}x{\rm d}y^{\prime } \nonumber \\ &\leqslant \gamma \int_{K}(\int\nolimits_{0}^{a}z^{2}(x,y^{\prime },\tau)\chi ^{2}(x,y^{\prime },\tau){\rm d}\tau +\int\nolimits_{-a}^{0}z^{2}(x,y^{\prime },\tau)\chi ^{2}(x,y^{\prime },\tau){\rm d}\tau){\rm d}x{\rm d}y^{\prime } \nonumber \\ &=\gamma \int_{\Omega _{\gamma }}z^{2}\chi ^{2}{\rm d}\Omega _{\gamma } \nonumber \end{align*}$$

for $a=\sigma (x,y^{\prime })$. Here we used the relations

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \int\nolimits_{0}^{a}\chi ^{2}(x,y)\int_{0}^{y_{m}}z^{2}(x,y^{\prime },\tau){\rm d}\tau {\rm d}y_{m} &=\int\nolimits_{0}^{a}\int_{0}^{y_{m}}\chi ^{2}(x,y)z^{2}(x,y^{\prime },\tau){\rm d}\tau {\rm d}y_{m} \nonumber \\ &=\int\nolimits_{0}^{a}z^{2}(x,y^{\prime },\tau)\int_{\tau }^{a}\chi ^{2}{\rm d}y_{m}{\rm d}\tau \nonumber \\ &\leqslant \int\nolimits_{0}^{a}z^{2}(x,y^{\prime },\tau)\chi ^{2}(x,y^{\prime },\tau)(a-\tau){\rm d}\tau \nonumber \\ &\leqslant \sqrt{\gamma }\int\nolimits_{0}^{a}z^{2}(x,y^{\prime },\tau)\chi ^{2}(x,y^{\prime },\tau){\rm d}\tau \nonumber \end{align*}$$

and

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \int\nolimits_{-a}^{0}\chi ^{2}\int_{y_{m}}^{0}z^{2}(x,y^{\prime },\tau){\rm d}\tau {\rm d}y_{m} &=\int\nolimits_{-a}^{0}z^{2}(x,y^{\prime },\tau)\int_{-a}^{\tau }\chi ^{2}{\rm d}y_{m}{\rm d}\tau \nonumber \\ &\leqslant \sqrt{\gamma }\int\nolimits_{-a}^{0}z^{2}(x,y^{\prime },\tau)\chi ^{2}(x,y^{\prime },\tau){\rm d}\tau . \nonumber \end{align*}$$

Similarly we can prove the rest part.

The proof is based on the method described in section 4 of chapter IV in [16].

We introduce a new function

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle w =\chi u . \nonumber \end{align*}$$

Then using the relations

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle u _{x_{1}x_{1}}=\chi ^{-1}\left(w _{x_{1}x_{1}}+2\lambda \nu \psi ^{-\nu -1}\psi _{x_{1}}w _{x_{1}}+\delta ^{2}\phi _{1}w \right) \nonumber \end{align*}$$

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \Delta _{^{\prime }x}u =\chi ^{-1}\left(\Delta _{^{\prime }x}w +2\lambda \nu \psi ^{-\nu -1}(\nabla _{^{\prime }x}\psi ,\nabla _{^{\prime }x}w)+\phi _{2}w \right) , \nonumber \end{align*}$$

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \sum_{i,j=1}^{m}a_{ij}u _{y_{i}y_{j}}=\chi ^{-1}\left(\sum_{i,j=1}^{m}a_{ij}\left(w _{y_{i}y_{j}}+\lambda \nu \psi ^{-\nu -1}\psi _{y_{i}}w _{y_{j}}+\lambda \nu \psi ^{-\nu -1}\psi _{y_{j}}w _{y_{i}}\right) +\phi _{3}w \right) , \nonumber \end{align*}$$

we obtain

$$\begin{align*} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle &\psi ^{\nu +1}\left(L_{0}u \right) ^{2}\chi ^{2} \nonumber \\ &=\psi ^{\nu +1}\biggl\{\left(x_{1}+\eta _{0}\right) ^{-1}w _{x_{1}x_{1}}+\left(x_{1}+\eta _{0}\right) \left(\Delta _{^{\prime }x}w - \overset{m}{\underset{i,j=1}{\sum }}a_{ij}w _{y_{i}y_{j}}\right) \nonumber \\ &+w \left(\left(x_{1}+\eta _{0}\right) ^{-1}\delta ^{2}\phi _{1}+\left(x_{1}+\eta _{0}\right) \left(\phi _{2}-\phi _{3}\right))\right. \nonumber \\ &+2\lambda \nu \psi ^{-\nu -1}\left(\delta \left(x_{1}+\eta _{0}\right) ^{-1}w _{x_{1}}+\left(x_{1}+\eta _{0}\right) \left((\nabla _{^{\prime }x}\psi ,\nabla _{^{\prime }x}w)-\overset{m}{\underset{i,j=1}{\sum }} a_{ij}\psi _{y_{i}}w _{y_{j}}\right) \right) \biggr\}^{2} \nonumber \end{align*}$$

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle &\geqslant 4\lambda \nu \biggl\{\left(x_{1}+\eta _{0}\right) ^{-1}w _{x_{1}x_{1}}+\left(x_{1}+\eta _{0}\right) \left(\Delta _{^{\prime }x}w - \overset{m}{\underset{i,j=1}{\sum }}a_{ij}w _{y_{i}y_{j}}\right) \nonumber \\ &+w (\left(x_{1}+\eta _{0}\right) ^{-1}\delta ^{2}\phi _{1}+\left(x_{1}+\eta _{0}\right) \left(\phi _{2}-\phi _{3}\right))\biggr\} \nonumber \\ &\times \left(\delta \left(x_{1}+\eta _{0}\right) ^{-1}w _{x_{1}}+\left(x_{1}+\eta _{0}\right) \left((\nabla _{^{\prime }x}\psi ,\nabla _{^{\prime }x}w)-\overset{m}{\underset{i,j=1}{\sum }}a_{ij}\psi _{y_{i}}w _{y_{j}}\right) \right) \nonumber \\ &=:\sum\limits_{k=1}^{14}T_{k}{{\rm ,}}\label{27} \nonumber \end{align} \tag{ B.1 }$$

where we set

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \phi _{1} &=\lambda ^{2}\nu ^{2}\psi ^{-2\nu -2}-\lambda \nu (\nu +1)\psi ^{-\nu -2}, \nonumber \\ \phi _{2} &=\lambda ^{2}\nu ^{2}\phi _{21}-\lambda \nu \phi _{22},{{\rm ~}} \phi _{3}=\lambda ^{2}\nu ^{2}\phi _{31}-\lambda \nu \phi _{32} \nonumber \end{align*}$$

and

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \phi _{21} &=\psi ^{-2\nu -2}\left\vert \nabla _{^{\prime }x}\psi \right\vert ^{2},{{\rm ~}}\phi _{22}=\psi ^{-\nu -2}((\nu +1)\left\vert \nabla _{^{\prime }x}\psi \right\vert ^{2}-(n-1) \psi), \nonumber \\ \phi _{31} &=\psi ^{-2\nu -2}\sum_{i,j=1}^{m}a_{ij}\psi _{y_{i}}\psi _{y_{j}},{{\rm ~}}\phi _{32}=\psi ^{-\nu -2}\sum_{i,j=1}^{m}a_{ij}\left((\nu +1)\psi _{y_{i}}\psi _{y_{j}}-\psi \delta _{ij}\right) . \nonumber \end{align*}$$

Here we set $\delta_{ii}=1$ and $\delta_{ij} = 0$ if $i\ne j$. Noting that $\left\Vert a_{ij}\right\Vert _{C^{1}\left(\Omega \right) }\leqslant M$, $\left\vert \psi _{x_{i}}\right\vert \leqslant \sqrt{2\gamma }$, $2\leqslant i\leqslant n$ and $|\psi _{y_{k}}|\leqslant \sqrt{2\gamma }$, $1\leqslant k\leqslant m$ in $\Omega _{\gamma },$ we estimate the terms T_i, $1\leqslant i\leqslant 14$ as follows:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{28} T_{1}=4\lambda \nu \delta \left(x_{1}+\eta _{0}\right) ^{-2}w _{x_{1}}w _{x_{1}x_{1}}=d_{1}\left(w \right) +4\lambda \nu \delta \left(x_{1}+\eta _{0}\right) ^{-3}w _{x_{1}}^{2}, \nonumber \end{align} \tag{ B.2 }$$

where $\newcommand{\e}{{\rm e}} d_{1}\left(w \right) =2\lambda \nu \delta (\left(x_{1}+\eta _{0}\right) ^{-2}w _{x_{1}}^{2})_{x_{1}}$;

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{29} T_{2}=4\lambda \nu \delta w _{x_{1}}\Delta _{^{\prime }x}w =2\lambda \nu \delta \sum_{i=2}^{n}(2\left(w _{x_{1}}w _{x_{i}}\right) _{x_{i}}-\left(w _{x_{i}}^{2}\right) _{x_{1}})=:d_{2}\left(w \right) {{\rm ;}} \nonumber \end{align} \tag{ B.3 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle T_{3} &=-4\lambda \nu \delta w _{x_{1}}\overset{m}{\underset{i,j=1}{\sum }} a_{ij}w _{y_{i}y_{j}} \nonumber \\ &=d_{3}\left(w \right) -2\lambda \nu \delta \overset{m}{\underset{i,j=1}{\sum }}(-2\frac{\partial a_{ij}}{\partial y_{j}}w _{y_{i}}w _{x_{1}}+\frac{\partial a_{ij}}{\partial x_{1}}w _{y_{i}}w _{y_{j}}) \nonumber \\ &\geqslant d_{3}\left(w \right) -4\lambda \nu \delta \overset{m}{\underset{i,j=1 }{\sum }}\left\vert \frac{\partial a_{ij}}{\partial y_{j}}\right\vert \left\vert w _{y_{i}}w _{x_{1}}\right\vert +2\lambda \nu \delta \alpha _{1}\left\vert \nabla _{y}w \right\vert ^{2} \nonumber \\ &\geqslant d_{3}\left(w \right) -2\lambda \nu \delta \frac{m^{2}M^{2}}{\varepsilon _{0}\left(x_{1}+\eta _{0}\right) } w_{x_{1}}^{2} \nonumber \\ &+ 2\lambda \nu \delta (\alpha _{1}-m(x_{1}+\eta _{0})\varepsilon _{0})\left\vert \nabla _{y}w \right\vert ^{2},\label{30} \nonumber \end{align} \tag{ B.4 }$$

where $d_{3}\left(w \right) =\lambda \nu \delta \overset{m}{\underset{i,j=1} {\sum }}(-4\left(a_{ij}w _{y_{i}}w _{x_{1}}\right) _{y_{j}}+2\left(a_{ij}w _{y_{i}}w _{y_{j}}\right) _{x_{1}}).$

Next

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle T_{4} &=4\lambda \nu \delta ^{3}\left(x_{1}+\eta _{0}\right) ^{-2}ww_{x_{1}}\phi _{1}=d_{4}(w) +4\lambda \nu \delta ^{3}w^{2}\left(x_{1}+\eta _{0}\right) ^{-3}\phi _{1} \nonumber \\ &\quad +4\lambda \nu \delta ^{4}w^{2}(\nu +1)\left(x_{1}+\eta _{0}\right) ^{-2}\phi _{4},\label{31} \nonumber \end{align} \tag{ B.5 }$$

where $\newcommand{\e}{{\rm e}} d_{4}(w) =2\lambda \nu \delta ^{3}(\left(x_{1}+\eta _{0}\right) ^{-2}w^{2}\phi _{1})_{x_{1}}$ and we set

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \phi _{4}=\lambda ^{2}\nu ^{2}\psi ^{-2\nu -3}-\frac{1}{2}\lambda \nu (\nu +2)\psi ^{-\nu -3}. \nonumber \end{align*}$$

We have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle T_{5} &=4\lambda \nu \delta w_{x_{1}}w\phi _{2} \nonumber \\ &=d_{5}(w) +4\lambda \nu \delta ^{2}(\nu +1)w^{2}\left(\phi _{4}\left\vert \nabla _{^{\prime }x}\psi \right\vert ^{2}+\frac{1}{2}\frac{(n-1)}{(\nu +1)}\lambda \nu \psi ^{-\nu -2}\right) ,\label{32} \nonumber \end{align} \tag{ B.6 }$$

where $d_{5}(w) =2\lambda \nu \delta \left(w^{2}\phi _{2}\right) _{x_{1}};$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{33} T_{6}=-4\lambda \nu \delta w_{x_{1}}w\phi _{3}=d_{6}(w) +2\lambda \nu \delta w^{2}\left(\phi _{3}\right) _{x_{1}}, \nonumber \end{align} \tag{ B.7 }$$

where $d_{6}(w) =-2\lambda \nu \delta \left(w^{2}\phi _{3}\right) _{x_{1}};$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{34} T_{7}=4\lambda \nu \sum_{i=2}^{n}\psi _{x_{i}}w_{x_{i}}w_{x_{1}x_{1}}=d_{7}(w) +2\lambda \nu (n-1)w_{x_{1}}^{2}, \nonumber \end{align} \tag{ B.8 }$$

where $d_{7}(w) =\lambda \nu \sum\limits_{i=2}^{n}(4\left(\psi _{x_{i}}w_{x_{i}}w_{x_{1}}\right) _{x_{1}}-2\lambda \nu \left(\psi _{x_{i}}w_{x_{1}}^{2}\right) _{x_{i}});$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle T_{8} &=4\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}\sum_{i,j=2}^{n}\psi _{x_{i}}w_{x_{i}}w_{x_{j}x_{j}} \nonumber \\ &=d_{8}(w) -4\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}\sum_{i,j=2}^{n}\delta _{ij}w_{x_{i}}w_{x_{j}}+2\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}\sum_{i,j=2}^{n}w_{x_{j}}^{2} \nonumber \\ &=d_{8}(w) -2\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}\left\vert \nabla _{^{\prime }x}w\right\vert ^{2}(2-(n-1)),\label{35} \nonumber \end{align} \tag{ B.9 }$$

where $\newcommand{\e}{{\rm e}} d_{8}(w)=\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}\sum\limits_{i,j=2}^{n}\left(4(\psi _{x_{i}}w_{x_{i}}w_{x_{j}})_{x_{j}}-2\left(\psi _{x_{i}}w_{x_{j}}^{2}\right) _{x_{i}}\right) ;$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle T_{9} &=-4\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}\sum_{i=2}^{n}\psi _{x_{i}}w_{x_{i}}\sum_{k,s=1}^{m}a_{ks}w_{y_{k}y_{s}} \nonumber \\ &=d_{9}(w) +2\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}\sum_{i=2}^{n}\sum_{k,s=1}^{m}(2\left(\psi _{x_{i}}a_{ks}\right) _{y_{s}}w_{y_{k}}w_{x_{i}}-\left(\psi _{x_{i}}a_{ks}\right) _{x_{i}}w_{y_{k}}w_{y_{s}}) \nonumber \\ &\geqslant d_{9}(w) -2\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}\sum_{i=2}^{n}\sum_{k,s=1}^{m}\left(2\left\vert \left(\psi _{x_{i}}a_{ks}\right) _{y_{s}}w_{y_{k}}w_{x_{i}}\right\vert +\left\vert \left(\psi _{x_{i}}a_{ks}\right) _{x_{i}}w_{y_{k}}w_{y_{s}}\right\vert \right) \nonumber \\ &\geqslant d_{9}(w) -2\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}M(1+\sqrt{2\gamma })\left(m^{2}\left\vert \nabla _{^{\prime }x}w\right\vert ^{2}+2mn\left\vert \nabla _{y}w\right\vert ^{2}\right) ,\label{36} \nonumber \end{align} \tag{ B.10 }$$

where $\newcommand{\e}{{\rm e}} d_{9}(w) =\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}\sum\limits_{i=2}^{n}\sum\limits_{k,s=1}^{m}(-4\left(\psi _{x_{i}}a_{ks}w_{x_{i}}w_{y_{k}}\right) _{y_{s}}+2\left(\psi _{x_{i}}a_{ks}w_{y_{k}}w_{y_{s}}\right) _{x_{i}});$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle T_{10} &=4\lambda \nu \sum_{i=2}^{n}\psi _{x_{i}}w_{x_{i}}w\left(x_{1}+\eta _{0}\right) (\left(x_{1}+\eta _{0}\right) ^{-1}\delta ^{2}\phi _{1}+\left(x_{1}+\eta _{0}\right) \left(\phi _{2}-\phi _{3}\right)) \nonumber \\ &=d_{10}(w) -2\lambda \nu w^{2}\sum_{i=2}^{n}(\psi _{x_{i}}\delta ^{2}\phi _{1}+\left(x_{1}+\eta _{0}\right) ^{2}\psi _{x_{i}}(\phi _{2}-\phi _{3})){{\rm ,~}}\label{37} \nonumber \end{align} \tag{ B.11 }$$

where $\newcommand{\e}{{\rm e}} d_{10}(w) =2\lambda \nu \sum\limits_{i=2}^{n}(w^{2}\psi _{x_{i}}\left(x_{1}+\eta _{0}\right) ^{2}(\left(x_{1}+\eta _{0}\right) ^{-2}\delta ^{2}\phi _{1}+\phi _{2}-\phi _{3}))_{x_{i}};$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle T_{11} &=-4\lambda \nu \sum_{i,j=1}^{m}a_{ij}\psi _{y_{i}}w_{y_{j}}w_{x_{1}x_{1}} \nonumber \\ &=d_{11}(w) +4\lambda \nu \sum_{i,j=1}^{m}\frac{\partial a_{ij} }{\partial x_{1}}\psi _{y_{i}}w_{y_{j}}w_{x_{1}}-2\lambda \nu \sum_{i,j=1}^{m}\left(\frac{\partial a_{ij}}{\partial y_{j}}\psi _{y_{i}}+a_{ij}\delta _{ij}\right) w_{x_{1}}^{2} \nonumber \\ &\geqslant d_{11}(w) -4\lambda \nu \sum_{i,j=1}^{m}\left\vert \frac{\partial a_{ij}}{\partial x_{1}}\psi _{y_{i}}w_{y_{j}}w_{x_{1}}\right\vert -2\lambda \nu \sum_{i,j=1}^{m}\left\vert \left(\frac{\partial a_{ij}}{\partial y_{j}}\psi _{y_{i}}+a_{ij}\delta _{ij}\right) \right\vert w_{x_{1}}^{2} \nonumber \\ &\geqslant d_{11}(w) -2\lambda \nu (M^{2}m^{2}\left(x_{1}+\eta _{0}\right) ^{-1}\sqrt{2\gamma }+Mm^{2}\sqrt{2\gamma }+Mm)w_{x_{1}}^{2} \nonumber \\ &-2\lambda \nu \left(x_{1}+\eta _{0}\right) \sqrt{2\gamma }m\left\vert \nabla _{y}w\right\vert ^{2},\label{38} \nonumber \end{align} \tag{ B.12 }$$

where $d_{11}(w) =\lambda \nu \sum\limits_{i,j=1}^{m}(-4\left(a_{ij}\psi _{y_{i}}w_{y_{j}}w_{x_{1}}\right) _{x_{1}}+2\left(a_{ij}\psi _{y_{i}}w_{x_{1}}^{2}\right) _{y_{j}});$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle T_{12} &=-4\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}\sum\limits_{i,j=1}^{m}a_{ij}\psi _{y_{i}}w_{y_{j}}\sum\limits_{s=2}^{n}w_{x_{s}x_{s}} \nonumber \\ &=d_{12}(w) +4\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}\sum\limits_{i,j=1}^{m}\sum\limits_{s=2}^{n}\frac{\partial a_{ij}}{\partial x_{s}}\psi _{y_{i}}w_{y_{j}}w_{x_{s}} \nonumber \\ &-2\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}\sum\limits_{i,j=1}^{m}\sum\limits_{s=2}^{n}\left(\frac{\partial a_{ij} }{\partial y_{j}}\psi _{y_{i}}+a_{ij}\delta _{ij}\right) w_{x_{s}}^{2} \nonumber \\ &\geqslant d_{12}(w) -2\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}M\sqrt{2\gamma }\left(m^{2}\left\vert \nabla _{^{\prime }x}w\right\vert ^{2}+(n-1)m\left\vert \nabla _{y}w\right\vert ^{2}\right) \nonumber \\ &-2\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}Mm(\sqrt{2\gamma } m+1)\left\vert \nabla _{^{\prime }x}w\right\vert ^{2},\label{39} \nonumber \end{align} \tag{ B.13 }$$

where $\newcommand{\e}{{\rm e}} d_{12}(w) =\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}\sum\nolimits_{i,j=1}^{m}\sum\nolimits_{s=2}^{n}(-4\left(a_{ij}\psi _{y_{i}}w_{y_{j}}w_{x_{s}}\right) _{x_{s}}+2\left(a_{ij}\psi _{y_{i}}w_{x_{s}}^{2}\right) _{y_{j}}).$

Similarly to (B.10) we obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle T_{13} &=4\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}\sum\limits_{i,j,k,s=1}^{m}a_{ij}\psi _{y_{i}}w _{y_{j}}a_{ks}w _{y_{k}y_{s}} \nonumber \\ &\geqslant d_{13}\left(w \right) -6\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}M^{2}\left(2\sqrt{2\gamma }+1\right) m^{3}\left\vert \nabla _{y}w \right\vert ^{2},\label{40} \nonumber \end{align} \tag{ B.14 }$$

where $\newcommand{\e}{{\rm e}} d_{13}\left(w \right) =2\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}{{\sum }^{m}_{i,j,k,s=1}}(a_{ij}\psi _{y_{i}}w _{y_{j}}w _{y_{k}}a_{ks})_{y_{s}}.$

Finally we obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle T_{14} &=-4\lambda \nu \left(x_{1}+\eta _{0}\right) \sum_{i,j=1}^{m}a_{ij}\psi _{y_{i}}w _{y_{j}}w (\left(x_{1}+\eta _{0}\right) ^{-1}\delta ^{2}\phi _{1}+\left(x_{1}+\eta _{0}\right) (\phi _{2}-\phi _{3})) \nonumber \\ &=d_{14}\left(w \right) +2\lambda \nu w ^{2}\sum_{i,j=1}^{m}(a_{ij}\psi _{y_{i}}(\delta ^{2}\phi _{1}+\left(x_{1}+\eta _{0}\right) ^{2}(\phi _{2}-\phi _{3})))_{y_{j}},\label{41} \nonumber \end{align} \tag{ B.15 }$$

where $\newcommand{\e}{{\rm e}} d_{14}\left(w \right) =-2\lambda \nu \sum\nolimits_{i,j=1}^{m}(a_{ij}\psi _{y_{i}}w ^{2}\left(x_{1}+\eta _{0}\right) ^{2}(\left(x_{1}+\eta _{0}\right) ^{-2}\delta ^{2}\phi _{1}+\phi _{2}-\phi _{3}))_{y_{j}}.$

Then by relations (B.2)–(B.15), we can write

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \psi ^{\nu +1}\left(L_{0}u \right) ^{2}\chi ^{2} &\geqslant 2\lambda \nu \delta \left(x_{1}+\eta _{0}\right) ^{-3}\beta _{1}w _{x_{1}}^{2}-2\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}(\beta _{0}-1)\left\vert \nabla _{^{\prime }x}w \right\vert ^{2} \nonumber \\ &+2\lambda \nu \left(x_{1}+\eta _{0}\right) \beta _{2}\left\vert \nabla _{y}w \right\vert ^{2}+\left(\lambda ^{3}\nu ^{3}\beta _{3}+\lambda ^{2}\nu ^{2}\beta _{4}\right) w ^{2}+D_{1}\left(w \right) , \nonumber \\ &\label{42} \nonumber \end{align} \tag{ B.16 }$$

where

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \beta _{1} &=2-Mm\left(x_{1}+\eta _{0}\right) ^{2}(Mm(\varepsilon _{0}^{-1}+\sqrt{2\gamma }\delta ^{-1})+\delta ^{-1}(\sqrt{2\gamma } m+1)(x_{1}+\eta _{0})), \nonumber \\ \beta _{2} &=\delta (\alpha _{1}-m\varepsilon _{0})-\beta _{21}, \nonumber \\ \beta _{21} &=\sqrt{2\gamma }m+(2n(1+\sqrt{2\gamma })+(n-1)\sqrt{2\gamma } +3m^{2}M(2\sqrt{2\gamma }+1))\left(x_{1}+\eta _{0}\right) mM, \nonumber \end{align*}$$

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \beta _{3} =&~4\delta ^{3}\left(x_{1}+\eta _{0}\right) ^{-3}\psi ^{-2\nu -2}+4\delta ^{4}(\nu +1)\left(x_{1}+\eta _{0}\right) ^{-2}\psi ^{-2\nu -3} \nonumber \\ &+4\delta ^{2}(\nu +1)\psi ^{-2\nu -3}\left\vert \nabla _{^{\prime }x}\psi \right\vert ^{2}+\beta _{31}, \nonumber \\ \beta _{31} =&~2\delta (\phi _{31})_{x_{1}}+2\left\vert \nabla _{^{\prime }x}\psi \right\vert ^{2}(\left(x_{1}+\eta _{0}\right) ^{2}(\phi _{31}-\phi _{21})-\delta ^{2}\psi ^{-2\nu -2}\phi _{1}) \nonumber \\ &+2\sum_{i,j=1}^{m}(a_{ij}\psi _{y_{i}}(\left(x_{1}+\eta _{0}\right) ^{2}(\phi _{21}-\phi _{31})+\delta ^{2}\psi ^{-2\nu -2}))_{y_{j}}, \nonumber \end{align*}$$

$$\begin{align*} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle \beta _{4} =&~-4\delta ^{3}\left(x_{1}+\eta _{0}\right) ^{-3}(\nu +1)\psi ^{-\nu -2}+4\delta ^{2}(\nu +1) \biggl\{\frac{1}{2}\frac{(n-1)}{(\nu +1)} \psi ^{-\nu -2} \nonumber \\ &-\frac{1}{2}(\nu +2)\psi ^{-\nu -3}\left\vert \nabla _{^{\prime }x}\psi \right\vert^{2}-\delta ^{2}\left(x_{1}+\eta _{0}\right) ^{-2}\frac{1}{2} (\nu +2)\psi ^{-\nu -3}\biggr\} - 2\delta (\phi _{32})_{x_{1}} \nonumber \\ &+2\left\vert \nabla _{^{\prime }x}\psi \right\vert ^{2}(\left(x_{1}+\eta _{0}\right) ^{2}(\phi _{22}-\phi _{32})+\delta ^{2}(\nu +1)\psi ^{-\nu -2}\phi _{1}) \nonumber \\ &+2\sum_{i,j=1}^{m}(a_{ij}\psi _{y_{i}}(\left(x_{1}+\eta _{0}\right) ^{2}(\phi _{32}-\phi _{22})-\delta ^{2}(\nu +1)\psi ^{-\nu -2}))_{y_{j}} \nonumber \end{align*}$$

and $D_{1}\left(w \right) =\sum\limits_{k=1}^{14}d_{k}\left(w \right) .$

Now we shall evaluate the expressions $\beta _{1},$ $\beta _{2},$ $\beta _{3},$ $\beta _{4}$ in (B.16), respectively.

Since $\delta \geqslant 4$, $\newcommand{\e}{{\rm e}} x_{1}+\eta _{0}<\frac{3}{4}\gamma$ and $\sqrt{2\gamma }<1$ by (7), we obtain

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \beta _{1}>2-Mm\left(\frac{3}{4}\gamma \right) ^{2}\left(Mm(\varepsilon _{0}^{-1}+1)+(m+1)\frac{3}{4}\gamma \right) >1, \nonumber \end{align*}$$

which implies

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle 2\lambda \nu \delta \left(x_{1}+\eta _{0}\right) ^{-3}\beta _{1}w_{x_{1}}^{2}\geqslant 2\lambda \nu \delta \left(x_{1}+\eta _{0}\right) ^{-3}w_{x_{1}}^{2}{{\rm .}}\label{43} \nonumber \end{align} \tag{ B.17 }$$

Next, by choosing $0<\varepsilon _{0}<\frac{\alpha _{1}}{4m},$ and setting $\delta _{3}=\frac{4m}{\alpha _{1}}\sqrt{2\gamma }l_{1},$ we see that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{44} \beta _{2}>\delta \left(\alpha _{1}-m\varepsilon _{0}\right) -m\sqrt{2\gamma }l_{1}\geqslant \frac{1}{2}\delta \alpha _{1} \nonumber \end{align} \tag{ B.18 }$$

for $\delta \geqslant \delta _{3},$ where $\alpha_1$ is the constant in (5) and

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle l_{1}=1+(n-1)M\frac{3}{4}\gamma +(2nM+3M^{2}m^{2})(1+2\sqrt{2\gamma })\frac{3 }{4}\left(\frac{\gamma }{2}\right) ^{1/2}. \nonumber \end{align*}$$

It is clear that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \beta _{3} &>4\delta ^{4}(\nu +1)\left(x_{1}+\eta _{0}\right) ^{-2}\psi ^{-2\nu -3}-\left\vert \beta _{31}\right\vert \nonumber \\ &=4\delta ^{4}(\nu +1)\left(x_{1}+\eta _{0}\right) ^{-2}\psi ^{-2\nu -3}\left(1-\frac{1}{\delta ^{2}}\left\vert \widetilde{\beta } _{31}\right\vert \right), \label{45} \nonumber \end{align} \tag{ B.19 }$$

where $\newcommand{\e}{{\rm e}} \widetilde{\beta }_{31}=\frac{1}{4\delta ^{2}}\left(x_{1}+\eta _{0}\right) ^{2}\psi ^{2\nu +3}\frac{1}{\nu +1}\beta _{31}.$ Since the functions a_ij and $\psi$ are bounded in the space $C^{2}\left(\overline{\Omega }_{\gamma }\right) ,$ the function $\widetilde{\beta }_{31}$ is bounded uniformly with respect to $(x,y)\in \overline{\Omega }_{\gamma }$ and $\delta ,$ that is there exists a number M₄  >  0 such that $\left\vert \widetilde{\beta }_{31}\right\vert \leqslant M_{4}.$ Then

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{46} \beta _{3}>2\delta ^{4}(\nu +1)\left(x_{1}+\eta _{0}\right) ^{-2}\psi ^{-2\nu -3} \nonumber \end{align} \tag{ B.20 }$$

for $\delta \geqslant \delta _{4}=\sqrt{2M_{4}}.$ On the other hand, $\beta _{4}$ is also bounded on $\overline{\Omega }_{\gamma }$ for fixed $\delta$ and $\nu$, say, $\left\vert \beta _{4}\right\vert \leqslant M_{5}$, M₅  >  0, then by (B.20) we obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle &\lambda ^{3}\nu ^{3}\beta _{3}+\lambda ^{2}\nu ^{2}\beta _{4}\geqslant 2\lambda ^{3}\nu ^{3}\delta ^{4}(\nu +1)\left(x_{1}+\eta _{0}\right) ^{-2}\psi ^{-2\nu -3}-\lambda ^{2}\nu ^{2}M_{5} \nonumber \\ &=2\lambda ^{3}\nu ^{4}\delta ^{4}\left(x_{1}+\eta _{0}\right) ^{-2}\psi ^{-2\nu -3}+2\lambda ^{3}\nu ^{3}\delta ^{4}\left(x_{1}+\eta _{0}\right) ^{-2}\psi ^{-2\nu -3}-\lambda ^{2}\nu ^{2}M_{5} \nonumber \\ &\geqslant 2\lambda ^{3}\nu ^{4}\delta ^{4}\left(x_{1}+\eta _{0}\right) ^{-2}\psi ^{-2\nu -3}\label{47} \nonumber \end{align} \tag{ B.21 }$$

for $\delta \geqslant \delta _{4},$ $\lambda \geqslant \lambda _{0}=M_{5}\left(\delta ,\nu ,\gamma \right) ,$ $\nu >1.$

Thus, inequalities (B.17)–(B.21) lead to

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle &\psi ^{\nu +1}\left(L_{0}u \right) ^{2}\chi ^{2}\geqslant 2\lambda \nu \delta \left(x_{1}+\eta _{0}\right) ^{-3}w _{x_{1}}^{2}-2\lambda \nu \left(x_{1}+\eta _{0}\right) ^{2}(\beta _{0}-1)\left\vert \nabla _{^{\prime }x}w \right\vert ^{2} \nonumber \\ &+\lambda \nu \delta \alpha _{1}\left(x_{1}+\eta _{0}\right) \left\vert \nabla _{y}w \right\vert ^{2}+2\lambda ^{3}\nu ^{4}\delta ^{4}\left(x_{1}+\eta _{0}\right) ^{-2}\psi ^{-2\nu -3}w ^{2}+D_{1}\left(w \right)\label{48} \nonumber \end{align} \tag{ B.22 }$$

for $\delta \geqslant \delta _{0}=\max \left\{4,\delta _{3},\delta _{4}\right\}$, $\lambda \geqslant \lambda _{0}$, $\nu >1.$

Finally, returning to the original function $u=w\chi ^{-1}$ in (B.22), we obtain (21), where

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle D_{2}(\chi u) =&-2\lambda ^{2}\nu ^{2}\delta ^{2}\left(\left(x_{1}+\eta _{0}\right) ^{-3}u^{2}\psi ^{-\nu -1}\chi ^{2}\right) _{x_{1}} \nonumber \\ &+2\lambda ^{2}\nu ^{2}\left(x_{1}+\eta _{0}\right) ^{2}(\beta _{0}-1)\sum_{i=2}^{n}\left(u^{2}\psi _{x_{i}}\psi ^{-\nu -1}\chi ^{2}\right) _{x_{i}} \nonumber \\ &-\lambda ^{2}\nu ^{2}\delta \alpha _{1}\sum_{i=1}^{m}\left(\left(x_{1}+\eta _{0}\right) u^{2}\psi _{y_{i}}\psi ^{-\nu -1}\chi ^{2}\right) _{y_{i}},\label{49} \nonumber \end{align} \tag{ B.23 }$$

and $\nabla _{x,y}\cdot U_{1}=D_{1}(\chi u)+D_{2}(\chi u).$ Thus the proof of lemma 2 is completed.