On hyperbolic equations with double characteristics in the presence of transition

Abstract

The paper deals with the study of the Cauchy problem for a class of hyperbolic second order operators with double characteristics in the presence of a transition. In particular, we obtain some a priori local estimates and, by means of these estimates, we prove local and global existence theorems.

1 Introduction

Let $\Omega= [0, + \infty[\, \times\Omega_{0}$, $\Omega_{0}$ be an open subset of $\mathbb{R}^{n}$ ($n \geq2$). Let $x=(x_{0}, x_{1}, \ldots, x_{m}, x_{m+1}, \ldots, x_{n}) = (x_{0}, x',x'')\in \Omega$, where we set $x'=(x_{1}, \ldots, x_{m}) \in\Omega'$ and $x''=(x_{m+1}, \ldots, x_{n}) \in\Omega''$, $\Omega'$ is the projection of $\Omega_{0}$ on the hyperplane $x''=0$ and $\Omega''$ is the projection of $\Omega_{0}$ on the hyperplane $x'=0$. Let us consider the following class of hyperbolic second order operators with double characteristics in the presence of a transition:

$$ P=D^{2}_{x_{0}} - \operatorname{Div}_{x'} \bigl( A\bigl(x',x''\bigr) D_{x'} \bigr) - \bigl(x_{0}+ \lambda- \alpha \bigl(x'\bigr)\bigr)^{2} \operatorname{Div}_{x''} \bigl( B\bigl(x''\bigr) D_{x''} \bigr) + \gamma(x), \quad\mbox{in } \Omega, $$

(1)

with $C^{\infty}$ coefficients, where $D_{x_{j}}= \frac{1}{i} \partial_{x_{j}}$, $j=0,1,\ldots,n$, $D_{x'}= \frac{1}{i} \nabla_{x'} = (D_{x_{1}}, \ldots, D_{x_{m}})$, $D_{x''}= \frac{1}{i} \nabla_{x''} =(D_{x_{m+1}}, \ldots, D_{x_{n}})$, $\operatorname{Div}_{x'} = \frac{1}{i} \operatorname{div}_{x'}$, $\operatorname{Div}_{x''} = \frac{1}{i} \operatorname{div}_{x''}$ and λ is a positive parameter.

For $\xi=(\xi_{0},\xi_{1}, \ldots, \xi_{m}, \xi_{m+1}, \ldots, \xi_{n})=(\xi _{0}, \xi', \xi'')$, where we set $\xi'=(\xi_{1}, \ldots, \xi_{m})$, $\xi''= (\xi_{m+1}, \ldots, \xi_{n})$ and have fixed λ, let us denote by

$$\begin{aligned} p(x, \xi) =& - \xi_{0}^{2}+ \sum _{i,j=1}^{m} a_{ij}\bigl(x',x'' \bigr) \xi'_{i} \xi'_{j} + \sum _{h=1}^{m} \partial_{x_{h}} a_{ij}\bigl(x',x''\bigr) \xi'_{h} \\ &{} +\bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr) \bigr)^{2} \sum_{i,j=m+1}^{n} b_{j}\bigl(x''\bigr) \xi''_{i} \xi''_{j} \\ &{}+ \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr) \bigr)^{2} \sum_{h=m+1}^{n} \partial_{x_{h}} b_{j}\bigl(x'' \bigr) \xi''_{h} + \gamma(x) \end{aligned}$$

the symbol of P, by Σ the characteristic set

$$\Sigma= \bigl\{ \rho= (x, \xi) \in T^{*} \Omega: p(\rho) =0, \nabla p(\rho) =0 \bigr\} , $$

where $T^{*}\Omega= \Omega\times(\mathbb{R}^{n} \setminus\{0 \})$ is the cotangent bundle related to Ω, and by $F_{p}$ the fundamental matrix of P at ρ, namely

$$F_{p}(\rho)=\frac{1}{2} \begin{pmatrix} p''_{x\xi}(\rho) & p''_{\xi\xi}(\rho)\\ - p''_{x x}(\rho) & - p''_{\xi x}(\rho)y \end{pmatrix}, \quad\forall \rho\in\Sigma. $$

The spectrum of $F(\rho)$, which we denote by $\operatorname{Spec}(F(\rho))$, has a remarkable importance for the study of the well-posedness of the Cauchy-Dirichlet problem for P.

Let us note that (see [1])

$$z \in\operatorname{Spec}\bigl(F(\rho)\bigr) \quad \Leftrightarrow \quad- z, \overline{z} \in \operatorname{Spec}\bigl(F(\rho)\bigr). $$

It is well known that $F(\rho)$ has only pure imaginary eigenvalues with a possible exception of a pair of non-zero real eigenvalues ±λ (see [1, 2]). If $F(\rho)$ has a pair of non-zero real eigenvalues, we say that P is effectively hyperbolic at ρ. If $F(\rho)$ has only pure imaginary eigenvalues and, moreover, if in the Jordan normal form of $F(\rho)$ corresponding to the eigenvalue 0, there are only Jordan blocks of dimension 2, i.e., $\operatorname{Ker} F(\rho)^{2} \cap\operatorname{Im} F(\rho)^{2} = \{ 0 \}$, we say that P is non-effectively hyperbolic of type 1 at ρ. Instead, if $F(\rho)$ has only pure imaginary eigenvalues and, moreover, if in the Jordan normal form of $F(\rho)$ corresponding to the eigenvalue 0, there is only a Jordan block of dimension 4 and no block of dimension 3, i.e., $\operatorname{Ker} F(\rho)^{2} \cap\operatorname{Im} F(\rho )^{2}$ is 2-dimensional, we say that P is non-effectively hyperbolic of type 2 at ρ. Besides let us set

$$\begin{aligned}& \Sigma_{+}= \{ \rho\in\Sigma: P \mbox{ is effectively hyperbolic at } \rho \}, \\& \Sigma_{-} = \{ \rho\in\Sigma: P \mbox{ is non-effectively hyperbolic of type } 1 \mbox{ at } \rho \}, \\& \Sigma_{0} = \{ \rho\in\Sigma: P \mbox{ is non-effectively hyperbolic of type } 2 \mbox{ at } \rho \}. \end{aligned}$$

It is easy to verify

$$\Sigma= \Sigma_{-} \sqcup\Sigma_{0} \sqcup\Sigma_{+} $$

Finally we say that we have a transition exactly when at least two among the above sets are nonempty.

The Cauchy problem for hyperbolic operators with double characteristics has been widely studied by many authors either in the case in which $F_{p}(\rho)$ has two real nonzero eigenvalues $\forall \rho \in\Sigma$ or in the case in which all the nonzero eigenvalues of $F_{p}(\rho)$ are purely imaginary numbers, $\forall\rho\in\Sigma$ (see for instance [1, 3–9]). Recently, another class of hyperbolic second order operators with double characteristics has been considered in [2]. For this class the $C^{\infty}$ well-posedness of the Cauchy problem is studied. Moreover, Carleman estimates are obtained for non-effectively hyperbolic operators. In [10], for a different class of hyperbolic second order operators some energy estimates are established and the $C^{\infty}$ well-posedness of the Cauchy problem for non-effectively hyperbolic operators is studied. We emphasize that in [2] and [10] the authors obtain a priori estimates when $\Sigma= \Sigma_{-} \sqcup \Sigma_{0}$. Instead we get a priori estimates when $\Sigma= \Sigma_{-} \sqcup\Sigma_{0} \sqcup\Sigma_{+}$ or $\Sigma= \Sigma_{-} \sqcup \Sigma_{0}$ or $\Sigma= \Sigma_{0} \sqcup\Sigma_{+}$ or $\Sigma= \Sigma_{-}$ or $\Sigma= \Sigma_{+}$. In fact, in the class of operators (1), studied also in [11, 12] and [13], both in the case in which $F_{p}(\rho)$ has two distinct real eigenvalues and in the case in which all the eigenvalues are purely imaginary numbers can occur. Namely, on the variety characteristic a transition from a case to another one can be considered. More precisely, if $p(x, \xi)= \xi_{0}^{2} - \sum_{j=1}^{m} \xi^{2}_{j} +(x_{0}+\lambda-\alpha(x'))^{2} \sum_{j=m+1}^{n} \xi^{2}_{j}$, setting $\beta(x)= x_{0}+ \lambda-\alpha(x')$, if $|\nabla_{x'} \alpha(x')|<1$ and $\beta(x)=0$ ($\xi_{0}=\xi_{1}=\cdots=\xi_{m}=0$, $\sum_{j=m+1}^{n} \xi^{2}_{j}=1$), then $F_{p}(\rho)$ has two distinct nonzero real eigenvalues. As a consequence, P is effectively hyperbolic. Instead if $|\nabla_{x'} \alpha(x')| > 1$ and $\beta(x)=0$ ($\xi_{0}=\xi_{1}=\cdots=\xi_{m}=0$, $\sum_{j=m+1}^{n} \xi^{2}_{j}=1$), $F_{p}(\rho)$ has two nonzero imaginary eigenvalues, then P is non-effectively hyperbolic. Therefore $\Sigma_{+}$ is the set of points of Σ for which $|\nabla_{x'} \alpha(x)|<1$, $\Sigma_{-}$ is the set of points of Σ for which $|\nabla_{x'} \alpha(x)|>1$ and $\Sigma_{0}$ is the set of points of Σ in which $|\nabla_{x'} \alpha(x)|=1$. Hence, even if we consider the particular class of operators (1), we have a transition from effectively hyperbolic to non-effectively hyperbolic.

In [11], an a priori estimate for solutions of a class of hyperbolic equations depending on a parameter $(-\partial^{2}_{x_{0}}+ \partial^{2}_{x_{1}}+ (x_{0}+ \lambda- \alpha(x_{1}))^{2} \partial^{2}_{x_{2}})u=f$ related to a Cauchy-Dirichlet problem is proved. Then in [14] energy estimates and existence and uniqueness results are established. For the Cauchy problem related to the same class of hyperbolic operators, a global existence and uniqueness theorem is obtained in [12, 13] and energy estimates for solutions are established in [15]. In this paper, we study the general class of hyperbolic second order operators with double characteristics in the presence of a transition (1). Under suitable assumptions on the coefficients that allow the transition on the variety characteristic, we obtain, first of all, a priori local estimate near the boundary and, then, distant from it. Such estimates allow us to prove existence theorems for the following Cauchy problem in the set Ω:

$$ \textstyle\begin{cases} Pu=f, \quad\mbox{in } \Omega, \\ u(0, x',x'') =0,\qquad \partial_{x_{0}} u(0,x',x'') =0, \end{cases} $$

(2)

see Section 6.

Let us assume that:

• (i) all the coefficients $a_{ij}(x',x'')$, $i=1, \ldots, m$, and $b_{j}(x'')$, $j=m+1, \ldots, n$ of the operator (1) belong to $C^{\infty}(\Omega_{0}) \cap L^{\infty}(\Omega_{0})$ and $C^{\infty}_{0}(\Omega'') \cap L^{\infty}(\Omega'')$, respectively, for every $k>0$;

• (ii) setting $g(x')= \dfrac{\alpha(x')}{\operatorname{div}_{x'}\overline {\alpha}(x')}$, where $\overline{\alpha}(x')$ is a vector with m components equal to $\alpha(x')$, and $h(x')= 1- \operatorname{div}_{x'} \overline{g}(x_{1})$, $g,h \in C^{\infty}$, $h(x') \in[h_{1}, h_{2}]$, $\forall x' \in\Omega'$, with $0< h_{1}< h_{2} < 4$;

• (iii) there exists $\lambda>0$ such that $|g(x')| \leq\lambda$, $\forall x' \in\Omega'$;

• (iv) setting $C(x',x'') = \operatorname{div}_{x'} \overline{A}(x',x'') g(x') + 2 [ A(x',x'') \operatorname{div}_{x'} \overline{g}(x') - \Lambda (x',x'') ]$, for every $(x',x'') \in\Omega_{0}$, where $\Lambda (x',x'')$ is a matrix with m columns equal to $A(x',x'') \nabla_{x'} g(x')$, the matrices A and B are positive definite and C is positive semidefinite, namely

  $$\begin{aligned}& \exists L_{1} \geq m: A\bigl(x',x'' \bigr) \xi' \xi' \geq L_{1} \bigl\Vert \xi' \bigr\Vert ^{2}, \quad\forall \xi' \in\mathbb{R}^{m}, \\& \exists L_{2} \geq0: B\bigl(x''\bigr) \xi'' \xi'' \geq L_{2} \bigl\Vert \xi'' \bigr\Vert ^{2}, \quad \forall\xi'' \in \mathbb{R}^{n-m}, \\& C\bigl(x',x''\bigr) \xi' \xi' \geq0, \quad \forall\xi' \in \mathbb{R}^{m}. \end{aligned}$$

It is worth remarking that in the study of hyperbolic operators considered in this note, the major difficulties in order to establish a priori estimates regard to the case in which the function $\beta(x, \lambda)= x_{0}+ \lambda- \alpha(x')$ assumes positive and negative values in Ω̅. Let us observe that if $m=1$, setting $A(x',x'')=(a(x',x''))$, as a result $C(x',x'')= \operatorname{div}_{x'} a(x',x'') g(x')$. Moreover, if $a(x',x'')$ is a constant function, then $C(x',x'')=0$. Therefore, if $m=1$ and $A(x',x'')$ is a constant function, assumption (iv) naturally occurs.

Example 1.1

Let $\alpha(x_{1})= e^{\frac{x_{1}^{3}}{3}+x_{1}}$ be a function defined in $\mathbb{R}$ and let $P= D^{2}_{x_{0}} - D^{2}_{x_{1}} - (x_{0}+ \lambda- \alpha(x_{1}))^{2} D^{2}_{x_{2}}$. It is easy to verify that $g(x_{1}) = \dfrac{1}{x_{1}^{2}+1}$ and $h(x_{1}) = \dfrac{2x_{1}}{(x_{1}^{2}+1)^{2}}+1$ in $\mathbb{R}$. Let us remark that $1- \frac{3 \sqrt{3}}{8} \leq h(x_{1}) \leq1+ \frac{3 \sqrt{3}}{8}$, $\forall x_{1} \in\mathbb{R}$. Moreover, the assumption (iii) is satisfied if we choose $\lambda \geq1$. Therefore, we have $\Sigma= \Sigma_{-} \sqcup\Sigma_{0} \sqcup \Sigma_{+}$, namely we have a transition from effectively hyperbolic to non-effectively hyperbolic.

Example 1.2

Let us consider the function $\alpha(x_{1},x_{2})= e^{ax_{1}+bx_{2}}$ in $\mathbb{R}^{2}$, where $a+b \neq0$, and the operator

$$P=D^{2}_{x_{0}} - \bigl( 3 D_{x_{1}}^{2}+ D^{2}_{x_{1}x_{2}} + 4 D^{2}_{x_{2}} \bigr) - \bigl(x_{0}+ \lambda- \alpha(x_{1},x_{2}) \bigr)^{2} D^{2}_{x_{3}} - \bigl(x_{0}+ \lambda- \alpha(x_{1},x_{2})\bigr)^{2} D^{2}_{x_{4}} $$

in $[0,+ \infty[\, \times\mathbb{R}^{4}$. We observe that $g(x_{1},x_{2}) = \frac{1}{a+b}$ and $h(x_{1},x_{2}) = 1$ in $\mathbb{R}^{2}$. Let us remark that

$$A= \begin{pmatrix} 3 & \frac{1}{2} \\ \frac{1}{2} & 4 \end{pmatrix},\qquad B= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix},\qquad C= \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}. $$

As a consequence, $A+C=A$. The matrices A and B are defined positive with constants $L_{1}= \frac{5}{2}$ and $L_{2}= 1$, respectively. Moreover, assumption (iii) holds if $\lambda\geq \frac{1}{|a+b|}$. Therefore, $\Sigma= \Sigma_{-} \sqcup\Sigma_{0} \sqcup\Sigma_{+}$, then we have transition.

Example 1.3

Let us consider the function $\alpha(x_{1},x_{2},x_{3})= (x_{1}+x_{2}+x_{3})^{2}$ in $]{-}k,k[^{3}$, with $k>0$ and the operator

$$\begin{aligned} P={}&D^{2}_{x_{0}} - \biggl(4 D_{x_{1}}^{2} + 4 D^{2}_{x_{2}}+ 4 D^{2}_{x_{3}} + \frac{1}{2} D^{2}_{x_{1}x_{2}} + \frac{1}{2} D^{2}_{x_{1}x_{3}} + \frac{1}{2} D^{2}_{x_{2}x_{3}} \biggr) \\ &{}- \bigl(x_{0}+ \lambda- \alpha(x_{1},x_{2},x_{3}) \bigr)^{2} D^{2}_{x_{4}} \end{aligned}$$

in $[0, + \infty[\, \times\,]{-}k,k[^{3} \times\mathbb{R}$. It is easy to verify that $g(x_{1},x_{2},x_{3}) = \frac{1}{6} (x_{1}+x_{2}+x_{3})$ and $h(x_{1},x_{2}, x_{3}) = \frac{1}{2}$ in $]{-}k,k[^{3}$. Moreover, as a result

$$A= \begin{pmatrix} 4 & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{4} & 4 & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & 4 \end{pmatrix},\qquad B=(1),\qquad C= \begin{pmatrix} \frac{5}{2} & -\frac{5}{4} & -\frac{5}{4} \\ -\frac{5}{4} & \frac{5}{2} & -\frac{5}{4} \\ -\frac{5}{4} & -\frac{5}{4} & \frac{5}{2} \end{pmatrix}. $$

The matrices A and B are positive definite with constants $L_{1}= \frac{7}{2}$ and $L_{2}= 1$, respectively, and C is positive semidefinite. Finally, assumption (iii) is ensured when $\lambda\geq \frac{1}{2} k$. For k large enough, $\Sigma= \Sigma_{-} \sqcup\Sigma_{0} \sqcup\Sigma_{+}$ results, namely we have transition from effectively hyperbolic to non-effectively hyperbolic.

The paper is organized as follows. In Section 2 some preliminary notations are given. In Section 3 a priori estimates are proved. In Section 4, estimates in Sobolev spaces with $s<0$ by means of the pseudodifferential operator theory are obtained. Section 5 deals with a local existence theorem near the boundary. Then a regularity result for the solution u to the Cauchy problem (2) is shown. At last a global existence result is proved in Section 6.

2 Notations and preliminaries

Let $\alpha=(\alpha_{0}, \alpha', \alpha'') \in\mathbb{N}^{n+1}_{0}$. We denote by $\partial^{\alpha}$ the derivative of order $|\alpha|$, while $\partial^{h}_{x_{j}}$ means, as usually, the derivative of order h with respect to $x_{j}$ and $\partial^{h}_{x_{j}, x_{p}}$ denotes the derivative of order h with respect to $x_{j}$ and $x_{p}$.

Let us denote by $(\cdot, \cdot)$, $\Vert \cdot \Vert$, $\Vert \cdot \Vert _{H^{r}}$ ($r \in\mathbb{N}_{0}$) the $L^{2}$-scalar product, the $L^{2}$-norm and the $H^{r}$-norm, respectively.

$C_{0}^{\infty}(\overline{\Omega})$ is the space of the restrictions to Ω̅ of functions φ belonging to $C^{\infty}_{0}(\mathbb{R}^{n+1})$ such that φ vanishes with all the derivatives in $[0, + \infty[\, \times\partial\Omega_{0}$.

Let $s \in\mathbb{R}$, let us denote by $\Vert \cdot \Vert _{H^{0,0,s}}$ the norm given by

$$\begin{aligned} &\Vert u \Vert ^{2}_{H^{0,0,s}(\overline{\Omega})} = \frac{1}{(2 \pi)^{n-m}} \int_{0}^{+ \infty}\,dx_{0} \int_{\mathbb{R}^{m}}\,dx' \int_{\mathbb{R}^{n-m}} \bigl(1+\bigl|\xi''\bigr|^{2} \bigr)^{s} \bigl| \widehat{u}\bigl(x_{0},x', \xi''\bigr)\bigr|^{2}\,d\xi'',\\ &\quad \forall u \in C^{\infty}_{0}(\overline{\Omega}), \end{aligned}$$

where the Fourier transform is done only with respect to the variable $x_{2}$. Moreover, let us denote by $A_{s}$ the pseudodifferential operator, given by

$$ A_{s} u= \frac{1}{(2 \pi)^{n-m}} \int_{\mathbb{R}^{n-m}} e^{i x'' \cdot\xi''} \bigl(1+\bigl|\xi''\bigr|^{2} \bigr)^{\frac{s}{2}} \widehat{u}\bigl(x_{0}, x_{1}, \xi''\bigr)\,d\xi'', \quad \forall u \in C^{\infty}_{0}(\overline{\Omega}). $$

(3)

Let us recall that $A_{s}: C^{\infty}_{0}(\overline{\Omega}) \to C^{\infty}(\overline{\Omega})$. For every $\varphi(x'') \in C^{\infty}_{0}(\mathbb{R}^{n-m})$, the operator $\varphi A_{s} u$ extends as a linear continuous operator from $H^{0,0,r}_{\mathrm{comp}.}(\overline{\Omega})$ to $H^{0,0,r-s}_{\mathrm{loc}}(\overline{\Omega})$, where $r,s \in\mathbb{R}$ (see [16]). Moreover, denoted by $\mathcal{U}_{x''}$ the projection of suppu on the hyperplane $x''=0$, if $\operatorname{supp} u \subseteq \mathbb{R}^{m-n} \backslash\mathcal{U}_{x''}$, then $\varphi A_{s} u$ is regularizing with respect to the variable $x''$, namely as a result

$$\begin{aligned} &\Vert \varphi A_{s} u \Vert _{H^{0,0,r}} \leq c \Vert u \Vert _{H^{0,0,r'}},\\ &\quad \forall r,r' \in\mathbb{R}, u \in C^{\infty}(\overline{\Omega}): \operatorname{supp} u \subseteq[0, + \infty[ \times\Omega' \times \Omega'' \setminus \operatorname{supp} \varphi. \end{aligned}$$

Let us remark that the norms $\Vert u \Vert _{H^{0,0,s}(\Omega)}$ and $\Vert A_{s} u \Vert _{L^{2}(\Omega)}$ are equivalent.

Finally, let $s,p \in\mathbb{R}$, let us denote by $\| \cdot \|_{H^{s,p}}$ the norm given by

$$\begin{aligned} &\Vert u \Vert ^{2}_{H^{p,s}(\overline{\Omega})} = \sum _{|h| \leq p} \int_{0}^{+ \infty}\,dx_{0} \int_{\mathbb{R}^{m}}\,dx' \int_{\mathbb{R}^{n-m}} \frac{1}{(2 \pi)^{n-m}} \bigl(1+\bigl|\xi''\bigr|^{2} \bigr)^{p} \bigl| \partial^{h}_{x_{0},x'} \widehat{u} \bigl(x_{0}, x', \xi'' \bigr)\bigr|^{2}\,d\xi'',\\ &\quad \forall u \in C^{\infty}_{0}(\overline{\Omega}). \end{aligned}$$

3 A priori estimates

Lemma 3.1

Let $\Omega_{k} = [0,k[\, \times\Omega_{0}$, for every $k>0$, let $u \in C^{\infty}_{0} (\overline{\Omega}_{k})$, as a result

$$\begin{aligned}& \Vert u \Vert _{L^{2}(\Omega_{k})} \leq2 k \Vert \partial_{x_{0}} u \Vert , \end{aligned}$$

(4)

$$\begin{aligned}& \bigl\Vert u\bigl(0,x',x''\bigr) \bigr\Vert _{L^{2}(\Omega_{0})}^{2} \leq4 k \Vert \partial_{x_{0}} u \Vert ^{2}, \end{aligned}$$

(5)

$$\begin{aligned}& \Vert u \Vert ^{2}_{L^{2}(\Omega_{k})} + \bigl\Vert u \bigl(0,x',x''\bigr) \bigr\Vert ^{2}_{L^{2}(\Omega_{0})} \leq4 \bigl(k^{2}+k\bigr) \Vert \partial_{x_{0}} u \Vert ^{2}_{L^{2}(\Omega_{k})}. \end{aligned}$$

(6)

Proof

Let $u \in C^{\infty}_{0} (\overline{\Omega})$. We have

$$\begin{aligned} 0 =& \int_{\Omega} \partial_{x_{0}} \bigl(x_{0} u^{2}(x)\bigr)\,dx \\ =& \int_{\Omega} u^{2}(x)\,dx + 2 \int_{\Omega} x_{0} u(x) \partial_{x_{0}} u(x)\,dx, \end{aligned}$$

and therefore

$$\int_{\Omega} u^{2}(x)\,dx = - 2 \int_{\Omega} x_{0} u(x) \partial_{x_{0}} u(x)\,dx. $$

In particular, in $\Omega_{k}$ we obtain

$$\Vert u \Vert ^{2} \leq2 k \Vert u \Vert \Vert \partial_{x_{0}} u \Vert , $$

which implies (4). Analogously, by using the following equality:

$$\int_{\Omega} \partial_{x_{0}} u^{2}(x)\,dx =- \int_{\Omega_{0}} u^{2}\bigl(0,x',x'' \bigr)\,dx'\,dx'', $$

we obtain (5). Finally, collecting (4) and (5), we have (6). □

Now, we are able to prove the following a priori estimate.

Theorem 3.1

Let $\Omega_{k} = [0,k[\, \times\Omega_{0}$ be a subset of Ω, where $k>0$. Let us suppose that g, h satisfy (i), (ii), and (iii). Then there exists a constant $c>0$ such that

$$\begin{aligned} &\Vert \partial_{x_{0}} u \Vert + \sum _{j=1}^{m} \Vert \partial_{x_{j}} u \Vert + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda- \alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert \leq c \bigl( \Vert Pu\Vert + \Vert u \Vert \bigr), \\ &\quad \forall u \in C^{\infty}_{0}(\overline{ \Omega}_{k}). \end{aligned}$$

(7)

Proof

Let us integrate by parts in the inner product

$$\begin{aligned} 2\bigl((x_{0}+\lambda) \partial_{x_{0}} u(x), Pu \bigr) =& \Vert \partial_{x_{0}} u\Vert ^{2} + \lambda \int_{\Omega_{0}} \bigl(\partial_{x_{0}} u\bigl(0,x',x'' \bigr)\bigr)^{2}\,dx'\,dx'' \\ &{} -2 \int_{\Omega} A\bigl(x', x'' \bigr) \nabla_{x'} u(x) \cdot(x_{0}+\lambda) \nabla_{x'} \partial_{x_{0}} u(x)\,dx \\ &{} - 2 \int_{\Omega} \bigl(x_{0}+ \lambda- \alpha \bigl(x'\bigr)\bigr)^{2} B\bigl(x'' \bigr) \nabla_{x''} u(x) \cdot(x_{0}+ \lambda) \nabla_{x''} \partial_{x_{0}} u(x)\,dx \\ &{} - \int_{\Omega} \gamma(x) u^{2}(x)\,dx \\ &{} - \int_{\Omega} \partial_{x_{0}} \gamma(x) (x_{0}+ \lambda) u^{2}(x)\,dx \\ &{} - \lambda \int_{\Omega_{0}} \gamma\bigl(0,x',x'' \bigr) u^{2}\bigl(0,x',x'' \bigr)\,dx'\,dx'' \\ =& \Vert \partial_{x_{0}} u\Vert ^{2} + \lambda \int_{\Omega_{0}} \bigl(\partial_{x_{0}} u\bigl(0,x',x'' \bigr)\bigr)^{2}\,dx'\,dx'' \\ &{} + \int_{\Omega} A\bigl(x', x'' \bigr) \nabla_{x'} u(x) \cdot\nabla_{x'} u(x)\,dx \\ &{} + \lambda \int_{\Omega_{0}} A\bigl(x', x'' \bigr) \nabla_{x'} u\bigl(0,x',x'' \bigr) \cdot\nabla_{x'} u\bigl(0,x',x'' \bigr)\,dx'\,dx'' \\ &{} + 2 \int_{\Omega} \bigl(x_{0}+\lambda- \alpha \bigl(x'\bigr)\bigr) B\bigl(x''\bigr) \nabla_{x''}u(x) \cdot(x_{0}+ \lambda) \nabla_{x''}u(x)\,dx \\ &{} + \int_{\Omega} \bigl(x_{0}+ \lambda- \alpha \bigl(x'\bigr)\bigr)^{2} B\bigl(x'' \bigr) \nabla_{x''} u(x) \cdot\nabla_{x''}u(x)\,dx \\ &{} + \lambda \int_{\Omega_{0}} \bigl(\lambda- \alpha\bigl(x'\bigr) \bigr)^{2} B\bigl(x''\bigr) \nabla_{x''} u\bigl(0,x',x'' \bigr) \\ &{}\cdot\nabla_{x''} u\bigl(0,x',x'' \bigr)\,dx'\,dx'' \\ &{} - \int_{\Omega} \gamma(x)u^{2}(x)\,dx - \int_{\Omega} (x_{0}+\lambda) u^{2}(x) \partial_{x_{0}} \gamma(x)\,dx \\ &{} - \lambda \int_{\Omega_{0}} \gamma\bigl(0,x',x'' \bigr) u^{2}\bigl(0,x',x'' \bigr)\,dx'\,dx''. \end{aligned}$$

(8)

On the other hand, by integrating by parts in the inner product, we obtain

$$\begin{aligned} 2\bigl(g\bigl(x'\bigr) \operatorname{div}_{x'} \overline{u}(x), Pu\bigr) =& - 2 \bigl(g\bigl(x'\bigr) \operatorname{div}_{x'} \overline{u}(x), \partial_{x_{0}}^{2} u(x)\bigr) \\ &{} + 2 \bigl(g\bigl(x'\bigr) \operatorname{div}_{x'} \overline{u}(x), \operatorname{div}_{x'} A\bigl(x',x'' \bigr) \nabla _{x'}u(x)\bigr) \\ &{} + 2 \bigl(g\bigl(x'\bigr) \operatorname{div}_{x'} \overline{u}(x), \bigl(x_{0}+\lambda-\alpha \bigl(x' \bigr)\bigr)^{2} \operatorname{div}_{x''}B\bigl(x'' \bigr) \nabla_{x''} u(x)\bigr) \\ &{} + 2 \bigl(g\bigl(x'\bigr) \operatorname{div}_{x'} \overline{u}(x), \gamma(x) u(x)\bigr). \end{aligned}$$

(9)

Let us compute separately every inner product:

$$\begin{aligned} -2 \bigl(g\bigl(x'\bigr) \operatorname{div}_{x'} \overline{u}(x), \partial^{2}_{x_{0}} u(x)\bigr) =& 2 \bigl(g \bigl(x'\bigr) \operatorname{div}_{x'} \partial_{x_{0}} \overline{u}(x), \partial_{x_{0}}u(x)\bigr) \\ &{} + 2 \int_{\Omega_{0}} \partial_{x_{0}} u\bigl(0,x',x'' \bigr) g\bigl(x'\bigr) \operatorname{div}_{x'} \overline{u}\bigl(0,x',x''\bigr)\,dx'\,dx'' \\ =& - \int_{\Omega} \operatorname{div}_{x'} \overline{g} \bigl(x'\bigr) \bigl(\partial_{x_{0}} u(x) \bigr)^{2}\,dx \\ &{} + 2 \int_{\Omega_{0}} \partial_{x_{0}} u\bigl(0, x', x''\bigr) g\bigl(x'\bigr) \operatorname{div}_{x'} \overline{u}\bigl(0,x',x'' \bigr)\,dx'\,dx''. \end{aligned}$$

(10)

For the second one, we have

$$\begin{aligned} & 2\bigl(g\bigl(x'\bigr) \operatorname{div}_{x'} \overline{u}(x), \operatorname{div}_{x'} A\bigl(x',x'' \bigr) \nabla_{x'} u(x)\bigr) \\ &\quad = - 2 \sum_{h=1}^{m} \int_{\Omega}A\bigl(x',x'' \bigr) \nabla_{x'} u(x) \cdot g\bigl(x'\bigr) \nabla_{x'} \partial_{x_{h}} u(x)\,dx \\ &\qquad{} -2 \sum_{h=1}^{m} \int_{\Omega} A\bigl(x',x'' \bigr) \nabla_{x'} u(x) \cdot\partial_{x_{h}} u(x) \nabla_{x'} g\bigl(x'\bigr)\,dx. \end{aligned}$$

(11)

Moreover, as a result

$$\begin{aligned} &{-} 2 \sum_{h=1}^{m} \int_{\Omega}A\bigl(x',x'' \bigr) \nabla_{x'} u(x) \cdot g\bigl(x'\bigr) \nabla_{x'} \partial_{x_{h}} u(x)\,dx \\ &\quad = 2 \sum_{h=1}^{m} \int_{\Omega}A\bigl(x',x'' \bigr) \nabla_{x'} \partial_{x_{h}} u(x) \cdot g \bigl(x'\bigr) \nabla_{x'} u(x)\,dx \\ &\qquad{} + 2 \sum_{h=1}^{m} \int_{\Omega} \partial_{x_{h}} A\bigl(x',x'' \bigr) \nabla_{x'} u(x) \cdot g\bigl(x'\bigr) \nabla_{x'} u(x)\,dx \\ &\qquad{} + 2 \sum_{h=1}^{m} \int_{\Omega} A\bigl(x',x'' \bigr) \nabla_{x'} u(x) \cdot\partial_{x_{h}} g \bigl(x'\bigr) \nabla_{x'} u(x)\,dx, \end{aligned}$$

and that implies

$$\begin{aligned} &{-} 2 \sum_{h=1}^{m} \int_{\Omega} A\bigl(x',x'' \bigr) \nabla_{x'} u \cdot g\bigl(x'\bigr) \nabla_{x'} \partial_{x_{h}} u(x)\,dx \\ &\quad = \sum_{h=1}^{m} \int_{\Omega} \partial_{x_{h}} A\bigl(x',x'' \bigr) \nabla_{x'} u(x) \cdot g\bigl(x'\bigr) \nabla_{x'} u(x)\,dx \\ &\qquad{} + \sum_{h=1}^{m} \int_{\Omega} A\bigl(x',x'' \bigr) \nabla_{x'} u(x) \cdot\partial_{x_{h}} g \bigl(x'\bigr) \nabla_{x'} u(x)\,dx. \end{aligned}$$

(12)

Substituting (12) in (11), we obtain

$$\begin{aligned} & 2\bigl(g\bigl(x'\bigr) \operatorname{div}_{x'} \overline{u}(x), \operatorname{div}_{x'} \bigl(A\bigl(x',x'' \bigr) \nabla_{x'}u(x)\bigr)\bigr) \\ &\quad = \sum_{h=1}^{m} \int_{\Omega} \partial_{x_{h}} A\bigl(x',x'' \bigr) \nabla_{x'} u(x) \cdot g\bigl(x'\bigr) \nabla_{x'} u(x)\,dx \\ &\qquad{} + \sum_{h=1}^{m} \int_{\Omega} A\bigl(x',x'' \bigr) \nabla_{x'} u(x) \cdot\partial_{x_{h}} g \bigl(x'\bigr) \nabla_{x'} u(x)\,dx \\ &\qquad{} -2 \sum_{h=1}^{m} \int_{\Omega} A\bigl(x',x'' \bigr) \nabla_{x'} u(x) \cdot\partial_{x_{h}} u(x) \nabla_{x'} g\bigl(x'\bigr)\,dx. \end{aligned}$$

(13)

We compute

$$\begin{aligned} & \sum_{h=1}^{m} \partial_{x_{h}} A\bigl(x',x'' \bigr) \nabla_{x'} u(x) \cdot g\bigl(x'\bigr) \nabla_{x'} u(x) + \sum_{h=1}^{m} A \bigl(x',x''\bigr) \nabla_{x'} u(x) \cdot \partial_{x_{h}} g\bigl(x'\bigr) \nabla_{x'} u(x) \\ &\quad = \operatorname{div}_{x'} \overline{A}\bigl(x',x'' \bigr) \nabla_{x'} u(x) \cdot g\bigl(x'\bigr) \nabla_{x'} u(x) + A\bigl(x',x'' \bigr) \nabla_{x'} u(x) \cdot\operatorname{div}_{x'} \overline{g}\bigl(x'\bigr) \nabla_{x'} u(x) \\ &\quad = \bigl( \operatorname{div}_{x'} \overline{A} \bigl(x',x''\bigr) g \bigl(x'\bigr) + A\bigl(x',x'' \bigr) \operatorname{div}_{x'} \overline{g}\bigl(x'\bigr) \bigr) \nabla_{x'} u(x) \cdot\nabla_{x'} u(x). \end{aligned}$$

(14)

Moreover, as a result

$$ \sum_{h=1}^{m} A \bigl(x',x''\bigr) \nabla_{x'} u \cdot \partial_{x_{h}} u \nabla_{x'} g\bigl(x' \bigr)= \Lambda\bigl(x',x''\bigr) \nabla_{x'} u(x) \cdot\nabla_{x'} u(x), $$

(15)

where $\Lambda(x',x'')$ is a matrix with m columns equal to $A(x',x'') \nabla_{x'} g(x')$. Taking into account (13), (14), and (15), we obtain

$$\begin{aligned} &2\bigl(g\bigl(x'\bigr) \operatorname{div}_{x'} \overline{u}(x), \operatorname{div}_{x'} A\bigl(x',x'' \bigr) \nabla_{x'}u(x)\bigr) \\ &\quad = \int_{\Omega} \bigl[ \operatorname{div}_{x'} \overline{A}\bigl(x',x''\bigr) g \bigl(x'\bigr) + 2 A\bigl(x',x'' \bigr) \operatorname{div}_{x'} \overline{g}\bigl(x'\bigr) - 2 \Lambda\bigl(x',x''\bigr) \bigr] \nabla_{x'}u(x) \cdot \nabla_{x'} u(x)\,dx \\ &\qquad{} - \int_{\Omega}A\bigl(x',x'' \bigr) \operatorname{div}_{x'} \overline{g}\bigl(x'\bigr) \nabla_{x'} u \cdot\nabla_{x'} u \,dx \\ &\quad = \int_{\Omega} C\bigl(x',x'' \bigr) \nabla_{x'}u(x) \cdot\nabla_{x'} u(x)\,dx - \int_{\Omega}A\bigl(x',x'' \bigr) \nabla_{x'} u \cdot\nabla_{x'} u \,dx \\ &\qquad{} + \int_{\Omega}h\bigl(x'\bigr) A\bigl(x',x'' \bigr) \nabla_{x'} u \cdot\nabla _{x'} u \,dx, \end{aligned}$$

(16)

where we have set $C(x',x'') = \operatorname{div}_{x'} \overline{A}(x',x'') g(x') + 2 [ A(x',x'') \operatorname{div}_{x'} \overline{g}(x') - \Lambda(x',x'') ]$, for every $(x',x'') \in\Omega_{0}$.

Let us consider

$$\begin{aligned} & 2 \bigl( g\bigl(x'\bigr) \operatorname{div}_{x'} \overline{u}(x), \bigl(x_{0}+ \lambda-\alpha \bigl(x' \bigr)\bigr)^{2} \operatorname{div}_{x''} \bigl( \overline{B} \bigl(x''\bigr) \nabla_{x''}u(x) \bigr) \bigr) \\ &\quad = - 2 \sum_{h=1}^{m} \int_{\Omega} \bigl(x_{0}+ \lambda-\alpha \bigl(x'\bigr)\bigr)^{2} B\bigl(x'' \bigr) \nabla_{x''} u(x) \cdot g\bigl(x'\bigr) \nabla_{x''} \partial_{x_{h}} u(x)\,dx \\ &\quad = 2 \sum_{h=1}^{m} \int_{\Omega} \bigl(x_{0}+ \lambda- \alpha \bigl(x'\bigr)\bigr)^{2} B\bigl(x'' \bigr) \nabla_{x''} \partial_{x_{h}} u(x) \cdot g \bigl(x'\bigr) \nabla_{x''} u(x)\,dx \\ &\qquad{} - 4 \sum_{h=1}^{m} \int_{\Omega} \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr) \partial_{x_{h}} \alpha \bigl(x'\bigr) B\bigl(x''\bigr) \nabla_{x''} u(x) \cdot g\bigl(x'\bigr) \nabla_{x''}u(x)\,dx \\ &\qquad{} + 2 \sum_{h=1}^{m} \int_{\Omega} \bigl(x_{0}+\lambda -\alpha \bigl(x'\bigr)\bigr)^{2} B\bigl(x'' \bigr) \nabla_{x''} u(x) \cdot\partial_{x_{h}} g \bigl(x'\bigr) \nabla_{x''} u(x)\,dx, \end{aligned}$$

(17)

from which it follows that

$$\begin{aligned} & {-} 2 \sum_{h=1}^{m} \int_{\Omega} \bigl(x_{0}+ \lambda-\alpha \bigl(x'\bigr)\bigr)^{2} B\bigl(x'' \bigr) \nabla_{x''} u \cdot g\bigl(x'\bigr) \nabla_{x''} \partial_{x_{h}} u \,dx \\ &\quad = - 2 \sum_{h=1}^{m} \int_{\Omega} \bigl(x_{0}+ \lambda-\alpha \bigl(x'\bigr)\bigr) \partial_{x_{h}} \alpha \bigl(x'\bigr) B\bigl(x''\bigr) \nabla_{x''} u \cdot g\bigl(x'\bigr) \nabla_{x''} \partial_{x_{h}} u \,dx \\ &\qquad{} + \sum_{h=1}^{m} \int_{\Omega} \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr)^{2} B\bigl(x'' \bigr) \nabla_{x''}u(x) \cdot\partial_{x_{h}} g \bigl(x'\bigr) \nabla_{x''} u(x)\,dx. \end{aligned}$$

(18)

Substituting (18) in (17) and using assumption (ii), we have

$$\begin{aligned} &{-}2\bigl(g\bigl(x'\bigr) \operatorname{div}_{x'} \overline{u}(x), \bigl(x_{0}+ \lambda-\alpha\bigl(x' \bigr)\bigr)^{2} \operatorname{div}_{x''} \overline{B} \bigl(x''\bigr) \nabla_{x''}u(x)\bigr) \\ &\quad = - 2 \sum_{h=1}^{m} \int_{\Omega} \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr) \partial_{x_{h}} \alpha \bigl(x'\bigr) B\bigl(x''\bigr) \nabla_{x''} u(x) \cdot g(x) \nabla _{x''} u(x)\,dx \\ &\qquad{} + \sum_{h=1}^{m} \int_{\Omega } \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr)^{2} B\bigl(x'' \bigr) \nabla_{x''} u(x) \cdot\partial _{x_{h}} g \bigl(x'\bigr) \nabla_{x''} u(x)\,dx \\ &\quad = - 2 \int_{\Omega} \bigl(x_{0}+\lambda- \alpha \bigl(x'\bigr)\bigr) B\bigl(x''\bigr) \nabla_{x''} u(x) \cdot\alpha(x) \nabla_{x''} u(x)\,dx \\ &\qquad{} + \int_{\Omega} \bigl(x_{0}+\lambda- \alpha \bigl(x'\bigr)\bigr)^{2} B\bigl(x'' \bigr) \nabla_{x''}u(x) \cdot\operatorname{div}_{x'} \overline{g}\bigl(x'\bigr) \nabla_{x''} u(x)\,dx \\ &\quad = - 2 \int_{\Omega} \bigl(x_{0}+\lambda- \alpha \bigl(x'\bigr)\bigr) B\bigl(x''\bigr) \nabla _{x''} u(x) \cdot\alpha(x) \nabla_{x''} u(x)\,dx \\ &\qquad{} + \int_{\Omega} \bigl(x_{0}+\lambda- \alpha \bigl(x'\bigr)\bigr)^{2} B\bigl(x'' \bigr) \nabla_{x''}u(x) \cdot\nabla_{x''} u(x)\,dx \\ &\qquad{} - \int_{\Omega} \bigl(x_{0}+\lambda- \alpha \bigl(x'\bigr)\bigr)^{2} B\bigl(x'' \bigr) \nabla_{x''}u(x) \cdot h\bigl(x'\bigr) \nabla_{x''} u(x)\,dx. \end{aligned}$$

(19)

Finally, we have

$$\begin{aligned} 2\bigl(g\bigl(x'\bigr) \operatorname{div}_{x'} \overline{u}(x), \gamma(x) u(x)\bigr) =& - \int _{\Omega} \operatorname{div}_{x'} \overline{ \gamma}(x) g\bigl(x'\bigr) u^{2}(x)\,dx \\ &{} - \int_{\Omega} \gamma(x) \operatorname{div}_{x'} \overline{g}\bigl(x'\bigr) u^{2}(x)\,dx \\ =& - \int_{\Omega} \operatorname{div}_{x'} \overline{ \gamma}(x) g\bigl(x'\bigr) u^{2}(x)\,dx \\ &{} - \int_{\Omega} \gamma(x) u^{2}(x)\,dx \\ &{} + \int_{\Omega} \gamma(x) h\bigl(x'\bigr) u^{2}(x)\,dx. \end{aligned}$$

(20)

By adding (10), (16), (19), and (20), we have

$$\begin{aligned} & 2\bigl((x_{0}+ \lambda) \partial_{x_{0}} u + g \bigl(x'\bigr) \operatorname{div}_{x''} \overline {u}, Pu \bigr) \\ &\quad = \Vert \partial_{x_{0}} u \Vert ^{2} + \int_{\Omega} \bigl(4 - h\bigl(x'\bigr)\bigr) \bigl(x_{0}+\lambda- \alpha\bigl(x'\bigr) \bigr)^{2} B\bigl(x''\bigr) \nabla_{x''} u(x) \cdot \nabla_{x''} u(x)\,dx \\ &\qquad{} + \int_{\Omega_{0}} \bigl\{ \lambda \bigl[ \bigl(\partial_{x_{0}} u\bigl(0,x',x''\bigr) \bigr)^{2} + A\bigl(x',x'' \bigr) \nabla_{x'} u\bigl(0,x',x'' \bigr) \cdot\nabla_{x'} u\bigl(0,x',x'' \bigr) \bigr] \\ &\qquad{} + 2 \partial_{x_{0}} u\bigl(0,x',x'' \bigr) g\bigl(x'\bigr) \operatorname{div}_{x'} \overline{u}\bigl(0,x',x''\bigr) \bigr\} \,dx'\,dx'' \\ &\qquad{} + \int_{\Omega} \bigl(h\bigl(x'\bigr)A \bigl(x',x''\bigr) +C \bigl(x',x''\bigr)\bigr) \nabla_{x'} u \cdot\nabla_{x'}u \,dx \\ &\qquad{} + \int_{\Omega} \bigl(2h\bigl(x'\bigr)-3\bigr) \gamma(x) u^{2}(x)\,dx \\ &\qquad{} - \int_{\Omega} (x_{0}+\lambda) \partial_{x_{0}} \gamma(x) u^{2}(x)\,dx \\ &\qquad{} - \lambda \int_{\partial\Omega} \gamma\bigl(0,x',x'' \bigr) u^{2}\bigl(0,x',x'' \bigr)\,dx'\,dx'' \\ &\qquad{} - \int_{\Omega} \operatorname{div}_{x'} \overline{g} \bigl(x'\bigr) u^{2}(x)\,dx. \end{aligned}$$

Making use of assumptions (i), (ii), (iii), and (iv), we obtain

$$\begin{aligned} &\bigl((x_{0}+ \lambda) \partial_{x_{0}} u + g \bigl(x'\bigr) \operatorname{div}_{x'} \overline {u}(x), Pu\bigr) \\ &\quad\geq h_{1} \Vert \partial_{x_{0}} u \Vert ^{2} + L_{1} h_{1} \sum _{j=1}^{m} \Vert \partial_{x_{j}} u \Vert ^{2} \\ &\qquad{} + L_{2} (4-h_{2}) \sum _{j=1}^{n-m} \bigl\Vert \bigl(x_{0}+ \lambda- \alpha\bigl(x'\bigr)\bigr) \partial_{x_{j+m}} u \bigr\Vert ^{2} + 4\bigl(k^{2}+k\bigr) c \Vert u \Vert ^{2},\quad \forall u \in C^{\infty}_{0} (\overline{ \Omega}_{k}), \end{aligned}$$

where $\Omega_{k} = [0,k[\, \times\Omega_{0}$, with $k>0$, from which (7) follows. □

As a consequence, we have the following corollary.

Corollary 3.1

Under the same assumptions of Theorem  3.1 and for k small enough, there exists a constant $c>0$ such that

$$\begin{aligned} &\Vert \partial_{x_{0}} u \Vert + \sum _{j=1}^{m} \Vert \partial_{x_{j}} u \Vert + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda- \alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert + \Vert u \Vert \leq c \Vert Pu \Vert , \\ &\quad\forall u \in C^{\infty}_{0}(\overline{ \Omega}_{k}). \end{aligned}$$

(21)

Proof

Taking into account (4) and (7) and choosing a positive number k small enough, we obtain (21). □

4 Estimates in Sobolev spaces with $s<0$ by means of pseudodifferential operator theory

Let us, first, prove some preliminary results.

Lemma 4.1

Let $u \in C^{\infty}_{0}(\overline{\Omega})$, with $\overline{\Omega}= [0, + \infty[\, \times\Omega' \times \mathbb{R}^{n-m}$, and let $\varphi\in C^{\infty}_{0}(\mathbb{R}^{n-m})$, with $\operatorname{supp} \varphi \subseteq\mathbb{R}^{n-m} \backslash\mathcal{U}_{x''}$. As a result

$$\Vert \varphi A_{s} u \Vert _{L^{2}(\overline{\Omega})} \leq \frac{c_{q,r,s}}{L^{q}} \Vert u \Vert _{H^{0,0,r}(\overline{\Omega})}, \quad \forall s \in \mathbb{R}, r \in\mathbb{Z}^{-}, q \geq s+r, $$

where L is the distance between suppφ and $\mathcal{U}_{x''}$, supposed to be greater than 1.

Proof

Let us consider

$$\begin{aligned} \varphi A_{s} u =& \frac{1}{(2 \pi)^{n-m}} \int_{\mathbb{R}^{n-m}} e^{i x'' \cdot\xi''} \varphi\bigl(x'' \bigr) \bigl(1 +\bigl|\xi''\bigr|^{2} \bigr)^{\frac{1}{2}} \widehat {u}\bigl(x_{0}, x', \xi''\bigr)\,d\xi'' \\ =& \frac{1}{(2 \pi)^{n-m}} \int_{\mathbb {R}^{2(n-m)}} e^{i (x''-y'') \cdot\xi''} \varphi\bigl(x'' \bigr) \bigl(1 +\bigl|\xi ''\bigr|^{2} \bigr)^{\frac{1}{2}} u\bigl(x_{0}, x', y''\bigr)\,dy''\,d\xi'' \\ =& \frac{i^{2p}}{(2 \pi)^{m-n}} \int_{\mathbb{R}^{2(n-m)}} e^{i (x''-y'') \cdot\xi''} \frac{\varphi(x'') u(x_{0}, x', y'')}{|x''-y''|^{2p}} \bigl( \triangle_{\xi''} \bigl(1 +\bigl|\xi''\bigr|^{2} \bigr)^{\frac {s}{2}}\bigr)^{[p]}\,dy''\,d\xi'' \\ =& \frac{i^{2p}}{(2 \pi)^{m-n}} \int_{\mathbb{R}^{n-m}} \bigl(\triangle_{\xi''} \bigl(1 +\bigl| \xi''\bigr|^{2}\bigr)^{\frac{s}{2}} \bigr)^{[p]}\,d\xi''\\ &{}\times \int_{\mathbb{R}^{n-m}} e^{i (x''-y'') \cdot\xi''} u\bigl(x_{0}, x', y''\bigr) \frac{\psi ( \frac{|x''-y''|}{L} ) \varphi(x'')}{|x''-y''|^{2p}}\,dy'', \end{aligned}$$

where $m \in\mathbb{N}$ and $\psi\in C^{\infty}(\mathbb{R})$ such that $\psi(\tau)=1$ if $|\tau|\geq1$, $\psi(\tau)=0$ if $|\tau| \leq\frac{1}{2}$.

This implies

$$\varphi A_{s} u= \frac{i^{2p} \varphi(x'')}{(2 \pi)^{n-m}} \int_{\mathbb{R}^{n-m}} \bigl(\triangle_{\xi''} \bigl(1 +\bigl| \xi''\bigr|^{2}\bigr)^{\frac{s}{2}} \bigr)^{[p]} u\bigl(x_{0}, x', x''\bigr) * \psi \biggl( \frac{|x''|}{L} \biggr) \frac{e^{i x'' \cdot\xi''}}{|x''|^{2p}}\,d\xi'', $$

where the convolution is done with respect to $x''$, and also

$$\begin{aligned} \mathcal{F}_{x''}(\varphi A_{s} u)={}& \frac{i^{2p} \widehat{\varphi}(\eta'')}{2 \pi} \\ &{} * \int_{\mathbb{R}^{n-m}} \bigl(\triangle_{\xi''} \bigl(1 +\bigl| \xi''\bigr|^{2}\bigr)^{\frac{s}{2}} \bigr)^{[p]} \widehat{u}\bigl(x_{0}, x', \eta''\bigr) \mathcal{F}_{x''} \biggl(\psi \biggl( \frac{|x''|}{L} \biggr) \frac{e^{i x'' \cdot\xi''}}{|x''|^{2p}} \biggr)\,d\xi'', \end{aligned}$$

(22)

where

$$\mathcal{F}_{x''} \biggl(\psi \biggl( \frac{|x''|}{L} \biggr) \frac{e^{i x'' \cdot\xi''}}{|x''|^{2p}} \biggr)= \int_{\mathbb{R}^{n-m}} e^{i x'' \cdot(\xi''- \eta'')} \psi \biggl( \frac{|x''|}{L} \biggr) \frac{1}{|x''|^{2p}}\,dx''. $$

It results

$$\begin{aligned} &\bigl(1+\bigl|\xi''-\eta''\bigr|^{2r'} \bigr) \mathcal{F}_{x''} \biggl(\psi \biggl( \frac {|x''|}{L} \biggr) \frac{e^{i x'' \cdot\xi''}}{|x''|^{2p}} \biggr) \\ &\quad = \int_{\mathbb{R}^{n-m}} e^{i x'' \cdot(\xi''- \eta'')} \psi \biggl( \frac{|x''|}{L} \biggr) \frac{1}{|x''|^{2p}}\,dx'' \\ &\qquad{} + (-1)^{r} \int_{\mathbb{R}^{n-m}} e^{i x'' \cdot(\xi''- \eta'')} \biggl( \triangle_{x''} \biggl( \psi \biggl( \frac{|x''|}{L} \biggr) \frac{1}{|x''|^{2p}} \biggr) \biggr)^{[r]}\,dx'' \end{aligned}$$

and then

$$ \biggl\vert \mathcal{F}_{x''} \biggl(\psi \biggl( \frac{|x''|}{L} \biggr) \frac{e^{i x'' \cdot\xi''}}{|x''|^{2p}} \biggr) \biggr\vert \leq \frac{c_{r,p}}{(1+|\xi''-\eta''|^{2r'})} \biggl( \frac{1}{L} \biggr)^{2p-n+m+1}. $$

(23)

Making use of (22) and (23), we obtain

$$\begin{aligned} & \bigl\Vert \mathcal{F}_{x''}(\varphi A_{s} u) \bigr\Vert \\ &\quad = \Vert \varphi A_{s} u \Vert \\ &\quad \leq\frac{1}{(2\pi)^{n-m}} \bigl\Vert \widehat{\varphi}\bigl( \eta''\bigr)\bigr\Vert _{L^{1}(\mathbb{R}^{n-m})} \\ &\qquad{}\cdot \biggl\Vert \int_{\mathbb{R}^{n-m}} \bigl(\triangle_{\xi''} \bigl(1 +\bigl| \xi''\bigr|^{2}\bigr)^{\frac{s}{2}} \bigr)^{[p]} \widehat{u}\bigl(x_{0}, x', \eta''\bigr) \mathcal{F}_{x''} \biggl(\psi \biggl( \frac{|x''|}{L} \biggr) \frac{e^{i x'' \cdot\xi''}}{|x''|^{2p}} \biggr)\,d\xi'' \biggr\Vert _{L^{2}(\Omega)} \\ &\quad \leq c \int_{\mathbb{R}^{n-m}} \biggl\Vert \bigl(\triangle_{\xi''} \bigl(1 +\bigl|\xi''\bigr|^{2}\bigr)^{\frac{s}{2}} \bigr)^{[p]} \widehat{u}\bigl(x_{0}, x', \eta''\bigr) \mathcal {F}_{x''} \biggl(\psi \biggl( \frac{|x''|}{L} \biggr) \frac{e^{i x'' \cdot\xi''}}{|x''|^{2p}} \biggr) \biggr\Vert _{L^{2}(\Omega)}\,d\xi'' \\ &\quad \leq\frac{c_{r,p}}{L^{2p-n+m+1}} \int_{\mathbb{R}^{n-m}} \biggl\Vert \frac{(\triangle_{\xi''} (1 +|\xi''|^{2})^{\frac{s}{2}})^{[p]} \widehat{u}(x_{0}, x', \eta'')}{(1+|\xi'' - \eta''|^{2r'})} \biggr\Vert _{L^{2}(\Omega)}\,d\xi''. \end{aligned}$$

Taking into account the previous inequality and the Peetre inequality (see [16], p. 17), it follows that

$$ \Vert \varphi A_{s} u \Vert _{L^{2}(\overline{\Omega})} \leq \frac{c_{r,p,s}}{L^{2p-n+m}} \int_{\mathbb{R}^{n-m}} \frac{\Vert (1 +|\xi''|^{2})^{\frac{s}{2} - \frac{2p+n-m}{2}} \widehat{u}(x_{0}, x', \eta'') \Vert _{L^{2}(\Omega)}}{(1+|\xi'' - \eta''|^{2r})}\,d\xi''. $$

(24)

If $p \geq\frac{s+r+n-m+1}{2}$, setting $q=2p-n+m+1$ in (24), $r=2r'$, results in

$$\Vert \varphi A_{s} u \Vert _{L^{2}(\Omega)} \leq \frac{c_{q,r,s}}{L^{q}} \Vert u \Vert _{H^{0,0,r}(\Omega)}, $$

where the constant $c_{q,r,s}$ is independent on L. □

Taking into account Lemma 4.1, it is easy to show the following.

Lemma 4.2

Let $\varphi\in C^{\infty}_{0}(\mathbb{R}^{n-m})$ such that $\varphi(|\tau''|)=0$ if $|\tau''| \leq1$. For every $\varepsilon >0$, $r \in\mathbb{Z}^{-}$, $s\in\mathbb{R}$ and for every $u \in C^{\infty}_{0} (\overline{\Omega})$, with $\overline{\Omega}= [0, + \infty[\, \times\Omega' \times\mathbb{R}^{n-m}$, there exists $L>0$ such that

$$\biggl\Vert \varphi \biggl( \frac{|x''|}{L} \biggr) A_{s} u \biggr\Vert _{L^{2}(\Omega)} \leq\varepsilon\|u \|_{H^{0,0,r}(\Omega)}. $$

Furthermore, we are able to prove the following.

Lemma 4.3

Let $\varphi\in C^{\infty}_{0}(\mathbb{R}^{n-m})$ such that $\varphi(|\tau''|)=1$ if $|\tau''| \leq 1$. For every $\varepsilon >0$, $r \in \mathbb{Z}^{-}$, $s\in\mathbb{R}$ and for every $u \in C^{\infty}_{0} (\overline{\Omega})$, with $\overline{\Omega}= [0, + \infty[\, \times \Omega' \times\mathbb{R}^{n-m}$, there exists $L>0$ such that

$$\biggl\Vert \biggl( 1- \varphi \biggl( \frac{|x''|}{L} \biggr) \biggr) A_{s} u \biggr\Vert _{L^{2}(\Omega)} \leq\varepsilon\|u \|_{H^{0,0,r}(\Omega)}. $$

Proof

In order to establish this result we can proceed as Lemma 4.1, but in (22) we need to consider the Fourier transform of the function $\psi(|x''|)= 1- \varphi ( \frac{|x''|}{L} )$ instead of $\widehat{\varphi}(\eta'')$ and keep in mind that

$$\widehat{\psi g} = \widehat{\psi} * \widehat{g} = ( 2\pi\delta- \widehat{ \varphi}) * \widehat{g} = 2 \pi\widehat{g} - \widehat{\varphi} * \widehat{g}, \quad \forall g \in S'(\mathbb{R}), $$

where $S'(\mathbb{R})$ is the space of tempered distributions defined in $\mathbb{R}$. □

Next, we prove a result concerning estimates near the boundary.

Lemma 4.4

Let $\Omega= \,]0, + \infty[ \times\Omega_{0}$, where $\Omega_{0}$ is an open subset of $\mathbb{R}^{n}$. For every ε and δ positive, there exists $k>0$ such that if

$$I_{k,\delta} = \bigl\{ x \in\overline{\Omega}: x_{0} < k, \bigl|x_{0}+ \lambda- \alpha\bigl(x'\bigr) \bigr| > \delta \bigr\} , $$

as a result

$$\begin{aligned} &\Vert \partial_{x_{0}} u \Vert + \sum _{j=1}^{m} \Vert \partial_{x_{j}} u \Vert + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda- \alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert + \Vert u \Vert \leq\varepsilon \Vert Pu \Vert , \\ &\quad \forall u \in C^{\infty}_{0}(\overline{\Omega}): \operatorname{supp} u \subseteq I_{k, \delta}. \end{aligned}$$

(25)

Proof

Integrating by parts and proceeding as in the first part of Theorem 3.1, we have

$$\begin{aligned} 2\bigl(e^{\tau x_{0}} \partial_{x_{0}} u(x), Pu\bigr) =& \tau\bigl\Vert e^{\frac{1}{2} \tau x_{0}} \partial_{x_{0}} u\bigr\Vert ^{2} + \int_{\Omega_{0}} \bigl(\partial_{x_{0}} u\bigl(0,x',x'' \bigr)\bigr)^{2}\,dx'\,dx'' \\ &{} + \int_{\Omega} e^{\tau x_{0}} A\bigl(x', x''\bigr) \nabla_{x'} u(x) \cdot\nabla _{x'} u(x)\,dx \\ &{} + \int_{\Omega_{0}} A\bigl(x', x'' \bigr) \nabla_{x'} u\bigl(0,x',x'' \bigr) \cdot \nabla_{x'} u\bigl(0,x',x'' \bigr)\,dx'\,dx'' \\ &{} + 2 \int_{\Omega} e^{\tau x_{0}} \bigl(x_{0}+\lambda- \alpha\bigl(x'\bigr)\bigr) B\bigl(x'' \bigr) \nabla_{x''} u(x) \cdot\nabla_{x''}u(x)\,dx \\ &{}+ \tau \int_{\Omega} e^{\tau x_{0}} \bigl(x_{0}+ \lambda- \alpha\bigl(x'\bigr)\bigr)^{2} B\bigl(x'' \bigr) \nabla_{x''} u(x) \cdot\nabla_{x''}u(x)\,dx \\ &{} + \lambda \int_{\Omega_{0}} \bigl(\lambda- \alpha\bigl(x'\bigr) \bigr)^{2} B\bigl(x''\bigr) \nabla_{x''} u\bigl(0,x',x'' \bigr) \cdot\nabla_{x''} u\bigl(0,x',x'' \bigr)\,dx'\,dx'' \\ &{} - \int_{\Omega} \tau e^{\tau x_{0}} \gamma(x) u^{2}(x)\,dx - \int_{\Omega} e^{\tau x_{0}} u^{2}(x) \partial_{x_{0}} \gamma(x)\,dx \\ &{} - \int_{\Omega_{0}} \gamma\bigl(0,x',x'' \bigr) u^{2}\bigl(0,x',x'' \bigr)\,dx'\,dx''. \end{aligned}$$

(26)

This implies

$$\begin{aligned} & \tau \Biggl( \bigl\Vert e^{\frac{1}{2} \tau x_{0}} \partial_{x_{0}} u \bigr\Vert ^{2} + L_{1} \sum_{j=1}^{m} \bigl\Vert e^{\frac{1}{2} \tau x_{0}} \partial_{x_{j}} u \bigr\Vert ^{2} + L_{2} \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr) \bigr) e^{\frac{1}{2} \tau x_{0}} \partial_{x_{j}} u \bigr\Vert ^{2} \Biggr) \\ &\quad \leq c \sum_{j=m+1}^{n} \biggl\Vert \frac{(x_{0}+\lambda-\alpha (x'))}{(x_{0}+\lambda-\alpha(x'))^{\frac{1}{2}}} e^{\frac{1}{2} \tau x_{0}} \partial_{x_{j}} u \biggr\Vert ^{2} + \tau\bigl\Vert e^{\frac{1}{2} \tau x_{0}} \bigl| \gamma (x)\bigr|^{\frac{1}{2}} u(x) \bigr\Vert ^{2} \\ &\qquad{} + \bigl\Vert e^{\frac{1}{2} \tau x_{0}} \bigl|\partial_{x_{0}} \gamma(x)\bigr|^{\frac{1}{2}} u(x) \bigr\Vert ^{2} + \int_{\Omega_{0}} \bigl|\gamma\bigl(0,x',x'' \bigr)\bigr| u^{2}\bigl(0,x',x'' \bigr)\,dx'\,dx'' \\ &\qquad{} + \bigl(ce^{\tau x_{0}} Pu, \partial_{x_{0}} u\bigr) \\ &\quad \leq\frac{c}{\delta} \sum_{j=m+1}^{n} \bigl\Vert e^{\frac{1}{2}\tau x_{0}} \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert ^{2} + c \biggl( \frac{1}{\tau^{2}} + \frac{1}{\tau} \biggr) \Vert \partial_{x_{0}} u \Vert ^{2} + \frac{1}{2} \bigl\Vert e^{\frac{1}{2} \tau x_{0}} Pu \bigr\Vert ^{2} \\ &\qquad{} + \frac{1}{2} \bigl\Vert e^{\frac{1}{2} \tau x_{0}} \partial_{x_{0}} u \bigr\Vert ^{2}. \end{aligned}$$

Choosing $x_{0}< \frac{1}{\tau}$, it follows that

$$\begin{aligned} & \Vert \partial_{x_{0}} u \Vert ^{2} + \sum _{j=1}^{m} \Vert \partial_{x_{j}} u \Vert ^{2} + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} u \bigr\Vert ^{2} \\ &\quad \leq\frac{c}{\tau\delta} \sum_{j=m+1}^{n} \bigl\Vert e^{\frac{1}{2}\tau x_{0}} \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert ^{2} + c \frac{1}{\tau} \| Pu \|^{2} + \frac{c}{\tau} \| \partial_{x_{0}} u \|^{2}. \end{aligned}$$

(27)

For τ large enough, making use of (27) and (4) we obtain the claim. □

As a consequence, we establish the following result.

Lemma 4.5

For every ε and δ positive, there exists $k>0$ such that if

$$I_{k,\delta} = \bigl\{ x \in\overline{\Omega}: x_{0} < k, \bigl|x_{0}+ \lambda- \alpha\bigl(x'\bigr) \bigr| > \delta \bigr\} , $$

for every $s<0$, as a result

$$\begin{aligned} &\Vert \partial_{x_{0}} u \Vert _{H^{0,0,s}(\Omega)} + \sum _{j=1}^{m} \Vert \partial_{x_{j}} u \Vert _{H^{0,0,s}(\Omega)} + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda- \alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega)} + \Vert u \Vert _{H^{0,0,s}(\Omega)} \\ &\quad \leq\varepsilon \Vert Pu \Vert _{H^{0,0,s}(\Omega)},\quad \forall u \in C^{\infty}_{0}(\overline{\Omega}): \operatorname{supp} u \subseteq I_{k, \delta}. \end{aligned}$$

(28)

Proof

Let $\varphi\in C^{\infty}_{0}(\mathbb{R})$, set $v_{s}= \varphi(|x''|) A_{s} u$, as a result $\operatorname{supp} v_{s} \subseteq I_{k, \delta}$ and $\varphi(|x''|)=1$ in $\mathcal{U}_{x''}$. Therefore we can rewrite (25) with $v_{s}$, namely

$$ \Vert \partial_{x_{0}} v_{s} \Vert + \sum _{j=1}^{m} \Vert \partial_{x_{j}} v_{s} \Vert + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} v_{s} \bigr\Vert + \Vert v_{s} \Vert \leq\varepsilon \Vert Pv_{s} \Vert . $$

(29)

Let us compute

$$\begin{aligned} \Vert \partial_{x_{0}} v_{s} \Vert \geq& \Vert \partial_{x_{0}} A_{s} u \Vert - \bigl\Vert ( \varphi- 1) \partial_{x_{0}} A_{s} u \bigr\Vert \\ =& \Vert \partial_{x_{0}} A_{s} u \Vert - \Vert Ru \Vert , \end{aligned}$$

(30)

where $(\varphi- 1) \partial_{x_{0}} A_{s} =R$ is a regularizing operator. In the same way,

$$ \Vert \partial_{x_{j}} v_{s} \Vert \geq c \Vert A_{s} \partial_{x_{j}} u \Vert - \Vert Ru \Vert , \quad \forall j=1, \ldots, m. $$

(31)

Similarly, we obtain

$$\begin{aligned} \bigl\Vert \bigl(x_{0} +\lambda-\alpha \bigl(x'\bigr)\bigr) \partial_{x_{j}} v_{s} \bigr\Vert =& \bigl\Vert \bigl(x_{0}+\lambda -\alpha \bigl(x'\bigr)\bigr) \partial_{x_{j}} \varphi A_{s} u \bigr\Vert \\ =& \bigl\| A_{s} \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr) \partial_{x_{j}} u + (\varphi-1) A_{s} \bigl(x_{0}+\lambda-\alpha\bigl(x' \bigr)\bigr) \partial_{x_{j}} u \\ &{} + \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr)\bigr) [ \partial_{x_{j}}, \varphi] (A_{s}u) \bigr\| \\ \geq& \bigl\Vert A_{s} \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr) \bigr) \partial_{x_{j}} u \bigr\Vert - c \Vert Ru\Vert , \end{aligned}$$

(32)

where we take into account that $(\varphi-1) A_{s} \partial_{x_{j}}$ and $[\partial_{x_{j}}, \varphi] A_{s}$ are regularizing operators. Finally, we have

$$\begin{aligned} \Vert v_{s} \Vert =& \bigl\Vert A_{s} u + (\varphi-1) A_{s} u \bigr\Vert \\ \geq& \Vert A_{s} u \Vert - \Vert Ru \Vert , \end{aligned}$$

(33)

$(\varphi-1) A_{s}$ being a regularizing operator.

Making use of (30), (31), (32), and (33), it follows that

$$\begin{aligned} &\Vert \partial_{x_{0}} v_{s} \Vert + \sum _{j=1}^{m} \Vert \partial_{x_{j}} v_{s} \Vert + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} v_{s} \bigr\Vert + \Vert v_{s} \Vert \\ &\quad \geq \Vert \partial_{x_{0}} u \Vert _{H^{0,0,s}(\Omega_{k})} + c \Biggl( \sum_{j=1}^{m} \Vert \partial_{x_{j}} u \Vert _{H^{0,0,s}(\Omega_{k})} + \sum _{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda-\alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega_{k})} \Biggr) \\ &\qquad{} + \Vert u \Vert _{H^{0,0,s}(\Omega_{k})} - c \Vert Ru \Vert . \end{aligned}$$

Since $\Vert Ru \Vert \leq c \Vert u \Vert _{H^{0,0,s}(\Omega_{k})} \leq c k \Vert \partial_{x_{0}} u \Vert _{H^{0,0,s}(\Omega_{k})}$, choosing k small enough, as a result

$$\begin{aligned} &\Vert \partial_{x_{0}} v_{s} \Vert + \sum _{j=1}^{m} \Vert \partial_{x_{j}} v_{s} \Vert + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} v_{s} \bigr\Vert + \Vert v_{s} \Vert \\ &\quad \geq c \Biggl( \Vert \partial_{x_{0}} u \Vert _{H^{0,0,s}(\Omega_{k})} + \sum_{j=1}^{m} \Vert \partial_{x_{j}} u \Vert _{H^{0,0,s}(\Omega_{k})} + \sum _{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda-\alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega_{k})} \\ &\qquad{} + \Vert u \Vert _{H^{0,0,s}(\Omega_{k})} \Biggr). \end{aligned}$$

(34)

On the other hand, we have

$$Pv_{s} = \varphi\bigl(\bigl|x''\bigr|\bigr) A_{s} Pu+ \varphi\bigl(\bigl|x''\bigr|\bigr) [P,A_{s}](u) + \bigl[\varphi\bigl(\bigl|x''\bigr| \bigr),P\bigr](A_{s} u). $$

As a consequence, we obtain

$$ \Vert P v_{s} \Vert = \bigl\Vert \varphi \bigl(\bigl|x''\bigr|\bigr) A_{s} Pu+ \varphi \bigl(\bigl|x''\bigr|\bigr) B_{s+1}u + R u \bigr\Vert , $$

(35)

where we set $[P,A_{s}] = B_{s+1}$, this being a pseudodifferential operator endowed with the symbol with respect to the variable $x''$ of order $s+1$. Such a symbol has the following principal part:

$$c\bigl(x, \xi''\bigr) = - \frac{1}{i} \sum _{p=m+1}^{n} \bigl(x_{0}+ \lambda- \alpha\bigl(x'\bigr)\bigr)^{2} \sum _{i,j=m+1}^{n} \partial_{x_{p}} b_{j} \bigl(x''\bigr) \xi_{i}'' \xi_{j}'' \partial_{\xi_{p}} \bigl(1+\bigl| \xi''\bigr|^{2}\bigr)^{\frac{s}{2}}. $$

Therefore the symbol $b(x,\xi'')$ can be written as

$$b\bigl(x,\xi''\bigr) = c\bigl(x, \xi''\bigr) +d\bigl(x,\xi'' \bigr), $$

where $d(x,\xi'')$ is a symbol of order s and we set

$$B_{s+1}= C_{s+1}+D_{s}. $$

Moreover, we set $R=[\varphi(|x''|),P]A_{s}$, which is a regularizing operator.

At last, we remark that

$$C_{s+1} u = C_{s}' \bigl(x_{0}+ \lambda-\alpha\bigl(x'\bigr)\bigr) \sum _{i=m+1}^{n} \partial_{x_{j}} u $$

and the symbol $c_{s}'$ of $C_{s}'$ is given by

$$c_{s}'\bigl(x, \xi''\bigr) = - \frac{1}{i} \sum_{p=m+1}^{n} \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr)\bigr) \sum _{j=m+1}^{n} \partial_{x_{p}} b_{j}\bigl(x''\bigr) \xi_{j}'' \partial_{\xi_{p}} \bigl(1+ \bigl|\xi''\bigr|^{2} \bigr)^{\frac{s}{2}}. $$

By such insights and by (35), as a result

$$ \begin{aligned}[b] \Vert Pv_{s} \Vert &\leq c \Biggl( \Vert A_{s} Pu \Vert + \sum_{j=m+1}^{n} \bigl\Vert C_{s}'\bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert + \Vert D_{s} u \Vert + \Vert Ru \Vert \Biggr) \\ &\leq c \Biggl( \Vert Pu \Vert _{H^{0,0,s}(\Omega)} + \sum _{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda-\alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega)} + \Vert u \Vert _{H^{0,0,s}(\Omega)} \Biggr). \end{aligned} $$

(36)

By using (29), (34), (36), and for ε small enough, the claim follows. □

Now, we are able to prove the following theorem.

Theorem 4.1

Let $\Omega_{k} = [0,k[\, \times\Omega_{0}$, with k such that (21) holds, let $\Omega_{0}= \Omega' \times \mathbb{R}^{n-m}$, let $\Omega'$ be an open set of $\mathbb{R}^{n-m}$ and let $B(x'')$ be a constant. Under assumptions (i) and (ii), for every $s \in\mathbb{Z}^{-}$ there exists $c>0$ such that

$$\begin{aligned} &\Vert \partial_{x_{0}} u \Vert _{H^{0,0,s}(\Omega)} + \sum _{j=1}^{m} \Vert \partial_{x_{j}} u \Vert _{H^{0,0,s}(\Omega)} + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda- \alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega)} + \Vert u \Vert _{H^{0,0,s}(\Omega)} \\ &\quad \leq c \Vert Pu \Vert _{H^{0,0,s}(\Omega_{k})},\quad \forall u \in C^{\infty}_{0}(\overline{\Omega}): \operatorname{supp} u \subseteq \Omega_{k}. \end{aligned}$$

(37)

Proof

Let $\varphi\in C^{\infty}_{0}(\mathbb{R})$ and $\varphi(\tau')=1$ if $|\tau'| \leq1$ and $\mathcal{U}_{x''} \subseteq[-L, L]^{n-m}$. Setting $v_{s} = \varphi ( \frac{|x''|}{L} ) A_{s} u$ in (21), as a result

$$ \Vert \partial_{x_{0}} v_{s} \Vert + \sum _{j=1}^{m} \Vert \partial_{x_{j}} v_{s} \Vert + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda- \alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} v_{s} \bigr\Vert + \Vert v_{s} \Vert \leq c \Vert P v_{s} \Vert . $$

(38)

Furthermore, we have

$$\begin{aligned} \Vert \partial_{x_{0}} v_{s} \Vert =& \bigl\Vert \partial_{x_{0}} A_{s} u + (\varphi-1) \partial_{x_{0}} A_{s} u \bigr\Vert \\ \geq& \Vert A_{s} \partial_{x_{0}} u \Vert - \bigl\Vert (\varphi-1) A_{s} \partial_{x_{0}} u \bigr\Vert . \end{aligned}$$

Taking into account Lemma 4.3, for L large enough, it follows that

$$ \Vert \partial_{x_{0}} v_{s} \Vert \geq c \Vert A_{s} \partial_{x_{0}} u \Vert . $$

(39)

Making use of the same technique, we have

$$ \Vert \partial_{x_{j}} v_{s} \Vert \geq c \Vert A_{s} \partial_{x_{j}} u \Vert ,\quad \forall j=1, \ldots, m, $$

(40)

and similarly

$$\begin{aligned} \bigl\Vert \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr) \partial_{x_{j}} v_{s} \bigr\Vert =& \biggl\| A_{s}\bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr) \partial_{x_{j}} u + (\varphi-1) A_{s}\bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} u \\ &{}+ \biggl[ \partial_{x_{j}}, \varphi \biggl( \frac{|x''|}{L} \biggr) \biggr] A_{s}\bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr) u \biggr\| \\ \geq& c \bigl\Vert A_{s}\bigl(x_{0}+ \lambda-\alpha \bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert - \frac{c}{L} \bigl\Vert A_{s}\bigl(x_{0}+\lambda- \alpha\bigl(x'\bigr)\bigr) u \bigr\Vert , \end{aligned}$$

(41)

where L is large enough. Finally, we have

$$\begin{aligned} \Vert v_{s} \Vert =& \Vert A_{s} u \Vert + \biggl\Vert \biggl( 1- \varphi \biggl( \frac {|x''|}{L} \biggr) \biggr) A_{s} u \biggr\Vert \\ \geq& (1- \varepsilon) \Vert u \Vert _{H^{0,0,s}(\Omega)}, \end{aligned}$$

(42)

having used Lemma 4.2.

On the other hand we have the result

$$ Pv_{s} = \varphi \biggl( \frac{|x''|}{L} \biggr) A_{s} Pu + \biggl[ \varphi \biggl( \frac{|x''|}{L} \biggr), P \biggr] (A_{s}u). $$

(43)

Let us observe that

$$ \Vert A_{s}Pu\Vert =\Vert Pu\Vert _{H^{0,0,s}(\Omega)}, $$

(44)

from the continuity property of the pseudodifferential operators (see [16], Theorem 2.1) and making use of Lemma 4.2

$$ \biggl\Vert \biggl[ \varphi \biggl( \frac{|x''|}{L} \biggr) , P \biggr] (A_{s} u ) \biggr\Vert \leq\varepsilon \Vert u\Vert _{H^{0,0,s}(\Omega)}. $$

(45)

By using (43), (44), and (45), we have

$$\begin{aligned} \Vert Pv_{s} \Vert \leq& \biggl\Vert \varphi \biggl( \frac{|x''|}{L} \biggr) A_{s} Pu \biggr\Vert + \biggl\Vert \varphi \biggl( \frac{|x''|}{L} \biggr) B_{s} u \biggr\Vert + \biggl\Vert \biggl[ \varphi \biggl( \frac{|x''|}{L} \biggr), P \biggr] (A_{s} u) \biggr\Vert \\ \leq& c \|Pu \|_{H^{0,0,s}(\Omega)} + \varepsilon\| u \|_{H^{0,0,s}(\Omega)}. \end{aligned}$$

(46)

Making use of (38), (39), (40), (41), and (42), choosing ε small enough and taking into account Lemma 3.1, as a result

$$\begin{aligned} & \Vert \partial_{x_{0}} u \Vert _{H^{0,0,s}(\Omega)} + \sum _{j=1}^{m} \Vert \partial _{x_{j}} u \Vert _{H^{0,0,s}(\Omega)} + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega)} + \Vert u \Vert _{H^{0,0,s}(\Omega)} \\ &\quad \leq c \Vert Pu\Vert _{H^{0,0,s}(\Omega)} + \varepsilon \Vert u \Vert _{H^{0,0,s}(\Omega)} \\ &\quad \leq c \Vert Pu\Vert _{H^{0,0,s}(\Omega)} + \varepsilon \Vert \partial_{x_{0}} u \Vert _{H^{0,0,s}(\Omega_{k})}, \end{aligned}$$

(47)

and for ε small enough the claim is established. □

Now, we prove an estimate in Sobolev spaces with $s<0$.

Theorem 4.2

Under assumptions (i), (ii), (iii), and (iv), for every $s \in \mathbb{R}_{0}^{-}$ there exists $c>0$ such that

$$\begin{aligned} & \Vert \partial_{x_{0}} u \Vert _{H^{0,0,s}(\Omega)} + \sum_{j=1}^{m} \Vert \partial_{x_{j}} u \Vert _{H^{0,0,s}(\Omega)} + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda- \alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega)} + \Vert u \Vert _{H^{0,0,s}(\Omega)} \\ &\quad \leq c \Vert Pu \Vert _{H^{0,0,s}(\Omega_{k})},\quad \forall u \in C^{\infty}_{0}(\overline{\Omega}): \operatorname{supp} u \subseteq \Omega_{k}= [0,k[ \times\Omega_{0}. \end{aligned}$$

(48)

Proof

Let $\varphi\in C^{\infty}_{0}(\mathbb{R})$ such that $\varphi(|x''|)=1$ on $\mathcal{U}_{x''}$ and let k such that (21) holds.

Then, for every $u \in C^{\infty}_{0}(\overline{\Omega})$ such that $\operatorname{supp} u \subseteq\Omega_{k}$, as a result

$$\Vert \partial_{x_{0}} v_{s} \Vert + \sum _{j=1}^{m} \Vert \partial_{x_{j}} v_{s} \Vert + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda- \alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} v_{s} \bigr\Vert + \Vert v_{s} \Vert \leq c \Vert P v_{s} \Vert , $$

where $v_{s}= \varphi(|x''|) A_{s} u$.

Proceeding as in the proof of (34) (see from (29) to (34)), we obtain

$$\begin{aligned} &\Vert \partial_{x_{0}} v_{s} \Vert + \sum _{j=1}^{m} \Vert \partial_{x_{j}} v_{s} \Vert + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} v_{s} \bigr\Vert + \Vert v_{s} \Vert \\ &\quad \geq c \Biggl( \Vert \partial_{x_{0}} u \Vert _{H^{0,0,s}(\Omega_{k})} + \sum_{j=1}^{m} \Vert \partial_{x_{j}} u \Vert _{H^{0,0,s}(\Omega_{k})} + \sum _{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda-\alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega_{k})} \\ &\qquad{} + \Vert u \Vert _{H^{0,0,s}(\Omega_{k})} \Biggr). \end{aligned}$$

On the other hand, we have

$$\begin{aligned} P v_{s} =& \varphi\bigl(\bigl|x''\bigr|\bigr) A_{s} Pu + \bigl[\varphi\bigl(\bigl|x'\bigr|\bigr), P \bigr](A_{s} u) + \varphi \bigl(\bigl|x'\bigr|\bigr) [P, A_{s}]u = \varphi\bigl(\bigl|x'\bigr|\bigr) A_{s} Pu + Ru + B_{s+1} u, \end{aligned}$$

where $R = [\varphi(|x'|), P] A_{s}$ is a regularizing operator and $B_{s+1} = [P, A_{s}]$ is a pseudodifferential operator with respect to the variables $x''$ of order $s+1$ endowed with symbol $b(x, \xi'')$ with principal part equal to

$$c\bigl(x, \xi''\bigr) = - \frac{1}{i} \sum _{p=m+1}^{n} \bigl(x_{0}+ \lambda- \alpha\bigl(x'\bigr)\bigr)^{2} \sum _{i,j=m+1}^{n} \partial_{x_{p}} b_{j} \bigl(x''\bigr) \xi_{i}'' \xi_{j}'' \partial_{\xi_{p}} \bigl(1+\bigl| \xi''\bigr|^{2}\bigr)^{\frac{s}{2}}. $$

Hence,

$$b\bigl(x, \xi''\bigr) = c\bigl(x, \xi''\bigr) +d\bigl(x, \xi'' \bigr), $$

where $c(x, \xi'')$ is the symbol of order s. Therefore, we have

$$B_{s+1} =C_{s+1} +D_{s}. $$

Then as a result

$$ [P, A_{s}](u) = C_{s+1}u +D_{s} u. $$

(49)

Taking into account (41), (42), (43), and (49), we obtain

$$ \begin{aligned}[b] \Vert Pv_{s} \Vert &\leq \biggl\Vert \varphi \biggl( \frac{|x''|}{L} \biggr) A_{s} Pu \biggr\Vert + \biggl\Vert \varphi \biggl( \frac{|x''|}{L} \biggr) B_{s+1}u \biggr\Vert + \biggl\Vert \biggl[ \varphi \biggl( \frac{|x''|}{L} \biggr), P \biggr] (A_{s} u) \biggr\Vert \\ &\leq c \| Pu \|_{H^{0,0,s}(\Omega)} + \biggl\Vert \varphi \biggl( \frac{|x''|}{L} \biggr) C_{s+1}u \biggr\Vert + \biggl\Vert \varphi \biggl( \frac{|x''|}{L} \biggr) D_{s}u \biggr\Vert + \varepsilon \| u \|_{H^{0,0,s}(\Omega)} \\ &\leq c \bigl( \| Pu \|_{H^{0,0,s}(\Omega)} + \| u \|_{H^{0,0,s}(\Omega)} \bigr) + \varepsilon\| u \|_{H^{0,0,s}(\Omega)} + \biggl\Vert \varphi \biggl( \frac{|x''|}{L} \biggr) C_{s+1}u \biggr\Vert . \end{aligned} $$

(50)

We remember that $C_{s+1}$ is a pseudodifferential operator endowed with the symbol

$$c\bigl(x,\xi'\bigr) = - \frac{1}{i} \sum _{p=m+1}^{n} \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr)^{2} \sum _{i,j=m+1}^{n} \partial_{x_{p}} b_{j} \bigl(x''\bigr) \xi_{i}'' \xi_{j}'' \partial_{\xi_{p}} \bigl(1+\bigl|\xi''\bigr|^{2}\bigr)^{\frac{1}{2}}. $$

Therefore, we have

$$\begin{aligned} C_{s+1}u =& \frac{1}{(2\pi)^{n-m}} \int_{\mathbb{R}^{n-m}} e^{i x'' \cdot\xi''} c\bigl(x,\xi'' \bigr) \widehat{u}\bigl(x_{0},x',\xi'' \bigr)\,d\xi'' \\ =& \frac{1}{(2\pi)^{n-m}} \int_{\mathbb{R}^{n-m}} e^{i x'' \cdot\xi ''} \chi\bigl(\bigl|\xi''\bigr| \bigr) c\bigl(x,\xi''\bigr) \widehat{u} \bigl(x_{0},x',\xi''\bigr)\,d\xi'' \\ &{} + \frac{1}{(2\pi)^{n-m}} \int_{\mathbb{R}^{n-m}} e^{i x'' \cdot \xi''} \bigl( 1- \chi\bigl(\bigl| \xi''\bigr|\bigr) \bigr) c\bigl(x,\xi'' \bigr) \widehat{u}\bigl(x_{0},x',\xi'' \bigr)\,d\xi'', \end{aligned}$$

where $\chi\in C^{\infty}_{0}(\mathbb{R})$ such that $\chi(t)=1$ for $|t|<1$. Therefore, we have

$$\begin{aligned} C_{s+1}u =& Ru \\ &{} + \frac{1}{(2\pi)^{n-m}} \int_{\mathbb{R}^{n-m}} e^{i x'' \cdot \xi''} \bigl( 1- \chi\bigl(\bigl| \xi''\bigr|\bigr) \bigr) c\bigl(x,\xi'' \bigr) \widehat{u}\bigl(x_{0},x',\xi'' \bigr)\,d\xi'', \end{aligned}$$

where R is a regularizing operator. On the other hand, we have

$$\begin{aligned} & \int_{\mathbb{R}^{n-m}} e^{i x'' \cdot\xi''} \bigl( 1- \chi\bigl(\bigl| \xi''\bigr|\bigr) \bigr) c\bigl(x,\xi'' \bigr) \widehat{u}\bigl(x_{0},x',\xi'' \bigr)\,d\xi'' \\ &\quad = \sum_{j=m+1}^{n} \int_{\mathbb{R}^{n-m}} e^{i x'' \cdot \xi''} c'\bigl(x, \xi''\bigr) \widehat{\bigl(x_{0}+\lambda- \alpha\bigl(x'\bigr)\bigr)^{2} \partial_{x_{j}}u \bigl(x_{0},x',\xi''\bigr)}\,d\xi'', \end{aligned}$$

where $c'(x,\xi'')= \dfrac{ ( 1- \chi(|\xi''|) ) c(x,\xi'')}{(x_{0}+\lambda-\alpha(x'))^{2} |\xi''|^{2}} (\xi_{1}+ \cdots+\xi_{n})$ is a symbol of order s. As a consequence, it follows that

$$\begin{aligned} C_{s+1}u =& Ru + \sum_{j=1}^{n} C'_{s} \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr)^{2} \partial _{x_{j}} u \\ =& Ru + \sum_{j=m+1}^{n} C'_{s} \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr)^{2} \partial_{x_{j}} \chi \biggl( \frac{|x_{0}+\lambda-\alpha(x')|}{\delta} \biggr) u \\ &{} + \sum_{j=m+1}^{n} C'_{s} \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr) \bigr)^{2} \partial_{x_{j}} \biggl[ 1- \chi \biggl( \frac{|x_{0}+\lambda -\alpha(x')|}{\delta} \biggr) \biggr] u. \end{aligned}$$

Then, taking into account Lemma 4.5,

$$\begin{aligned} \Vert C_{s+1}u \Vert \leq& \Vert Ru \Vert + c \sum_{j=m+1}^{n} \biggl\Vert \frac{(x_{0}+\lambda-\alpha(x'))^{2}}{\delta} \delta\chi \biggl( \frac{|x_{0}+\lambda-\alpha(x')|}{\delta} \biggr) \partial_{x_{j}} u \biggr\Vert _{H^{0,0,s(\Omega)}} \\ &{} +c \sum_{j=m+1}^{n} \biggl\Vert \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr) \bigr)^{2} \partial_{x_{j}} \biggl[ 1- \chi \biggl( \frac{|x_{0}+\lambda-\alpha(x')|}{\delta} \biggr) \biggr] u \biggr\Vert _{H^{0,0,s}(\Omega)} \\ \leq& \Vert u \Vert _{H^{0,0,s}(\Omega)} + c \delta\sum _{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda-\alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega)} \\ &{} + c \varepsilon\biggl\Vert P \biggl( 1-\chi \biggl( \frac{|x_{0}+\lambda -\delta(x')|}{\delta} \biggr) \biggr) u \biggr\Vert \\ \leq& \|u \|_{H^{0,0,s}(\Omega)} + c \delta\sum_{j=m+1}^{n} \|\bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \|_{H^{0,0,s}(\Omega)} + c \varepsilon\| Pu \|_{H^{0,0,s}(\Omega)} \\ &{}+ c \varepsilon\biggl\Vert \biggl[ P, \biggl( 1- \chi \biggl( \frac{|x_{0}+\lambda-\alpha(x')|}{\delta} \biggr) \biggr) \biggr] (u) \biggr\Vert _{H^{0,0,s}(\Omega)} \\ \leq& \Vert u \Vert _{H^{0,0,s}(\Omega)} + c \delta\sum _{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda-\alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega)} \\ &{} + c \varepsilon \Biggl( \| Pu \|_{H^{0,0,s}(\Omega)} + \Vert \partial_{x_{0}} u \Vert _{H^{0,0,s}(\Omega)} + \sum _{j=1}^{m} \| \partial_{x_{j}} u \|_{H^{0,0,s}(\Omega)} \Biggr). \end{aligned}$$

(51)

Making use of (50), (51), and Lemma 3.1, we have

$$\begin{aligned} \| Pv_{s} \| \leq& c \bigl( \| Pu \|_{H^{0,0,s}(\Omega_{k})} + k \| \partial_{x_{0}} u \|_{H^{0,0,s}(\Omega_{k})} \bigr) \\ &{} + c \varepsilon \Biggl( \| Pu \|_{H^{0,0,s}(\Omega_{k})} + \| \partial_{x_{0}} u \|_{H^{0,0,s}(\Omega_{k})} + \sum_{j=1}^{m} \| \partial_{x_{j}} u \|_{H^{0,0,s}(\Omega_{k})} \Biggr) \\ &{} + c \delta\sum_{j=m+1}^{n} \bigl\| \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\| _{H^{0,0,s}(\Omega_{k})}. \end{aligned}$$

(52)

Finally, by (52), (38), (39), (40), and (41), for δ and ε small enough and, hence, k small enough, as a result

$$\begin{aligned} & \Vert \partial_{x_{0}} u \Vert _{H^{0,0,s(\Omega_{k})}} + \sum _{j=1}^{m} \Vert \partial_{x_{j}} u \Vert _{H^{0,0,s}(\Omega_{k})} + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+\lambda- \alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega_{k})} \\ &\quad{} + \Vert u \Vert _{H^{0,0,s}(\Omega_{k})}\leq c \Vert Pu \Vert _{H^{0,0,s}(\Omega_{k})}, \quad\forall u \in C^{\infty}_{0} (\overline{\Omega}): \operatorname{supp} u \subseteq\Omega_{k}. \end{aligned}$$

 □

5 A local existence theorem near the boundary and a regularity result

Let $\Omega_{k}=[0, k[\, \times\Omega_{0}$, with $k>0$; the following local existence theorem near the boundary holds.

Theorem 5.1

Let $f \in H^{0,0,s}(\Omega)$, with $s \geq0$. Then there exists $w \in H^{0,0,s}(\Omega_{k})$ such that

$$\bigl(w, {}^{t}Pu\bigr)= (f,u), \quad\forall u \in C^{\infty}_{0}( \overline{\Omega}): \operatorname{supp} u \subseteq \Omega_{k}. $$

Proof

Let S be the space

$$S= \bigl\{ \psi\in C^{\infty}_{0}(\overline{ \Omega}_{k}): \psi= {}^{t}Pu, \forall u \in C^{\infty}_{0}(\overline{\Omega}): \operatorname{supp} u \subseteq\Omega_{k} \bigr\} . $$

Let T be the linear functional defined as

$$T(\psi) = T\bigl({}^{t} Pu\bigr)= (f,u), \quad \forall\psi\in S. $$

Making use of Theorems 4.1 and 4.2, we have

$$\begin{aligned} \bigl|T(\psi)\bigr| =& \bigl|(f, u)\bigr| \\ \leq& \Vert f \Vert _{H^{0,0,s}(\Omega)} \Vert u \Vert _{H^{0,0,-s}(\Omega_{k})} \\ \leq& c \Vert f \Vert _{H^{0,0,s}(\Omega)} \bigl\Vert {}^{t}Pu \bigr\Vert _{H^{0,0,-s}(\Omega _{k})} \\ =& c' \Vert \psi \Vert _{H^{0,0,-s}(\Omega_{k})},\quad \forall\psi\in S, \end{aligned}$$

where $c'= c \Vert f \Vert _{H^{0,0,s}(\Omega)}$. Hence T is continuous on S and can be extended to a linear continuous functional in $H^{0,0,-s}(\overline{\Omega}_{k})$. Making use of the representation theorems, there exists $w \in H^{0,0,s}(\overline{\Omega}_{k})$ such that

$$\begin{aligned} T(\psi) =& ( w,\phi) \\ =& \bigl(w, {}^{t}Pu\bigr) \\ =& (f,u), \quad\forall u \in C^{\infty}_{0}(\Omega): \operatorname{supp} u \subseteq\Omega_{k}. \end{aligned}$$

 □

Now, let us study the regularity of the solution w. To this aim, we set

$$L=D^{2}_{x_{0}} - \operatorname{Div}_{x'} \bigl(A \bigl(x',x''\bigr) D_{x'} \bigr) $$

and, for every $x'' \in\Omega''$, we consider the Cauchy problem

$$\textstyle\begin{cases} Lv=h, \quad\mbox{in } ]0,k[ \times\Omega', \\ v(0, x')=0, \qquad v_{x_{0}}(0,x')=0. \end{cases} $$

Since L is a strictly hyperbolic operator, it is well known that if $h \in H^{s}$ then the solution $v \in H^{s+1}$. As a consequence, since $Pw=f$ in the sense of distributions, $Lw=h$, with $h=f+(x_{0}+\lambda-\alpha(x'))^{2} \operatorname{Div}_{x''}(B(x'')D_{x''}) w - \gamma(x) w$. Moreover, having $f \in H^{s, 2(r-s)}$, with $0 \leq s \leq r$ and $r \geq2$, it follows that

$$ w \in H^{1, 2(r-1)}(\Omega_{k}). $$

(53)

Let us proceed by induction. We prove

$$w \in H^{s-1,2(r-1)} (\Omega_{k}) \quad\Longrightarrow\quad w \in H^{s,2(r-s)} (\Omega_{k}),\quad 2 \leq s \leq r. $$

Hence, we compute

$$\partial^{s-1,2(r-s+1)} Lw = \partial^{s-1,2(r-s+1)} h, $$

from which we have

$$\begin{aligned} L \partial^{s-1,2(r-s+1)} w =& \partial^{s-1,2(r-s+1)} h - \bigl[ \partial^{s-1,2(r-s+1)}, L \bigr] w \\ =& \partial^{s-1,2(r-s+1)} f + \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr)^{2} \operatorname{Div}_{x''} \bigl(B\bigl(x''\bigr)D_{x''}\bigr) \partial^{s-1,2(r-s+1)} w \\ &{} + \bigl[ \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr) \bigr)^{2} \operatorname{Div}_{x''} \bigl(B \bigl(x''\bigr)D_{x''}\bigr), \partial^{s-1,2(r-s+1)} \bigr] w \\ &{}- \bigl[ \partial^{s-1,2(r-s+1)}, L \bigr] w. \end{aligned}$$

This implies

$$w \in H^{s, 2(r-s+1)}(\Omega_{k}) \subseteq H^{s, 2(r-s)}( \Omega_{k}). $$

Since $s \leq r-1$, as a result

$$w \in H^{r}(\Omega_{k}). $$

Therefore, we proved that if w is solution to the equation:

$$ \bigl(w, {}^{t}P \varphi\bigr) =(f, \varphi),\quad \forall\varphi\in C^{\infty}_{0}(\overline{\Omega}_{k}), $$

(54)

then the distribution $w \in H^{r+1}(\overline{\Omega}_{k})$ ($r\geq 2$). Integrating by part the left-hand side of (54), as a result, for every $\varphi\in C^{\infty}_{0}(\overline{\Omega}_{k})$ with $\operatorname{supp} \varphi\subseteq\Omega_{k}$,

$$(Pw, \varphi)= (f, \varphi), $$

and that implies

$$ Pw=f, \quad \mbox{a.e. in } \Omega_{k}. $$

(55)

Moreover, integrating by parts the left-hand side of (54), for every $\varphi\in C^{\infty}_{0}(\overline{\Omega}_{k})$ with $\varphi(0,x',x'')=0$, we have

$$(Pw, \varphi) - \int_{\Omega_{0}} w\bigl(0,x',x'' \bigr) \varphi_{x_{0}}\bigl(0,x',x'' \bigr)\,dx'\,dx'' = (f, \varphi), $$

and combining with (55), it follows that

$$w\bigl(0,x',x''\bigr)=0. $$

Finally, integrating by part the left-hand side of (54), for every $\varphi\in C^{\infty}_{0}(\Omega_{k})$ with $\varphi_{x_{0}}(0,x', x'')=0$, we obtain

$$(Pw, \varphi) - \int_{\Omega_{0}} w_{x_{0}}\bigl(0,x',x'' \bigr) \varphi\bigl(0,x',x''\bigr)\,dx'\,dx'' = (f, \varphi), $$

and making use of (55), as a result

$$w_{x_{0}}\bigl(0,x',x'' \bigr)=0. $$

Hence, we proved that $w \in H^{r}(\Omega_{k})$ ($r\geq2$) is a solution to the problem

$$\textstyle\begin{cases} Pw=f, \quad\mbox{in } \Omega_{k}, \\ w(0,x',x'')=0,\qquad w_{x_{0}}(0,x',x'')=0, \end{cases} $$

for k small enough.

6 A global existence result

Let $\overline{x}_{0} >0$ and let $\Omega_{\overline{x}_{0}} = [\overline{x}_{0}, + \infty[\, \times\Omega_{0}$, by means of the change of variables $x_{0}=y_{0}+ \overline{x}_{0}$, the problem

$$\textstyle\begin{cases} Pw=f, \quad\mbox{in } \Omega_{\overline{x}_{0}}, \\ w(\overline{x}_{0},x',x'')=0, \qquad w_{x_{0}}(\overline{x}_{0},x',x'')=0, \end{cases} $$

becomes

$$\textstyle\begin{cases} Pv=f, \quad\mbox{in } \Omega, \\ v(0,x',x'')=0, \qquad v_{y_{0}}(0,x',x'')=0, \end{cases} $$

where $v(y_{0},x',x'')=v(x_{0}- \overline{x}_{0},x',x'') = w(x_{0},x',x'')$.

According to the results of Section 5, for k small enough, there exists a solution $v \in H^{r}(\Omega_{k})$, $r \geq2$, verifying the problem

$$\textstyle\begin{cases} Pv=f, \quad\mbox{in } \Omega_{k}, \\ v(0,x',x'')=0, \qquad v_{y_{0}}(0,x',x'')=0. \end{cases} $$

Hence, there exists a solution $w \in H^{r}(\Omega_{\overline{x}_{0}, k})$, where $\Omega_{\overline{x}_{0}, k}= [\overline{x}_{0}, \overline{x}_{0}+k[\, \times\Omega_{0}$ verifying the problem

$$ \textstyle\begin{cases} Pw=f, \quad\mbox{in } \Omega_{\overline{x}_{0},k} ,\\ w(\overline{x}_{0},x',x'')=0,\qquad w_{x_{0}}(\overline{x}_{0},x',x'')=0. \end{cases} $$

(56)

Now, if $B(x'')$ is constant and $\Omega_{0}= \Omega' \times \mathbb{R}^{n-m}$, k does not depend on s. Then we can proceed in the following way. From the existence of a solution $w \in H^{r}(\Omega_{\overline{x}_{0},k})$ to problem (56), it follows that also the problem

$$ \textstyle\begin{cases} Pw=f, \quad\mbox{in } \Omega_{\overline{x}_{0},k} ,\\ w(\overline{x}_{0},x',x'')=g_{1}(x',x''), \qquad w_{x_{0}}(\overline{x}_{0},x',x'')=g_{2}(x',x''), \end{cases} $$

(57)

where $f \in C^{\infty}(\Omega)$, $g_{1} \in C^{\infty}(\Omega_{0})$, $g_{2} \in C^{\infty}(\Omega_{0})$, admits a solution $w \in C^{\infty}(\overline{\Omega}_{\overline{x}_{0}, k})$. In fact, let $h(x_{0},x',x'')$ be a function belonging to $C^{\infty}(\Omega_{\overline{x}_{0},k})$ such that $h(\overline{x}_{0}, x',x'')= g_{1}(x',x'')$ and $h_{x_{0}}(\overline{x}_{0}, x',x'')= g_{2}(x',x'')$, the solution to (57) is $w=h+ \overline{w}$, where w̅ is solution to

$$\textstyle\begin{cases} P\overline{w}=f+Ph, \quad\mbox{in } \Omega_{\overline{x}_{0},k}, \\ \overline{w}(\overline{x}_{0},x',x'')=0, \qquad \overline{w}_{x_{0}}(\overline{x}_{0},x',x'')=0. \end{cases} $$

Set $\Omega_{h}=[0,h[\, \times\Omega_{0}$, with $h>0$, by means of compactness theorems and the arbitrariness of $\overline{x}_{0}$, we can decompose $\overline{\Omega}_{k}$ in the union of a finite number of compacts $\overline{\Omega}_{i}=[k_{i-1}, k_{i}] \times\Omega_{0}$, for $i=1, \ldots, p$, where $k_{0}=0$, and such that there exists a solution $w_{i} \in C^{\infty}(\Omega_{i})$ to the problem

$$\textstyle\begin{cases} Pw_{i}=f, \quad\mbox{in } \Omega_{i}, \\ w_{i}(k_{i},x',x'')=w_{i-1}(k_{i},x',x''),\qquad \partial_{x_{0}} w_{i}(k_{i},x',x'')= \partial_{x_{0}} w_{i-1}(k_{i},x',x''), \end{cases} $$

where $i=1, \ldots, p-1$, $w_{0}(0,x',x'')=0$ and $\partial_{x_{0}} w_{0}(0,x',x'')=0$. By construction, it follows that the function

$$w(x_{0},x',x'')= \sum_{i=0}^{p} w_{i} \bigl(x_{0},x',x''\bigr) \chi_{i}\bigl(x_{0},x',x'' \bigr), $$

where

$$\chi_{i}\bigl(x_{0},x',x'' \bigr) = \textstyle\begin{cases} 1 & \mbox{in } [k_{i}, k_{i+1}] \times\Omega_{0}, \\ 0 & \mbox{otherwise}, \end{cases} $$

is a solution to the problem

$$\textstyle\begin{cases} Pw=f, \quad\mbox{in } \Omega_{h}, \\ w(0,x',x'')=0, \qquad w_{x_{0}}(0,x',x'')=0, \end{cases} $$

with $f \in C^{\infty}(\overline{\Omega})$ and $w \in C^{\infty}(\Omega_{h})$. For the arbitrariness of h, we have proved that under assumptions (i), (ii), (iii), and (iv), if $\Omega_{0}= \Omega' \times\mathbb{R}^{n-m}$ and $B(x'')$ is a constant, then the problem

$$\textstyle\begin{cases} Pu=f, \quad\mbox{in } \Omega, \\ u(0,x',x'')=0,\qquad u_{x_{0}}(0,x',x'')=0, \end{cases} $$

with $f \in C^{\infty}(\overline{\Omega})$, admits a solution $u \in C^{\infty}(\overline{\Omega})$.

If $B(x'')$ is not constant, since c depends on s in (37), we proceed as follows. For every $h>0$ and for every $\overline{x}_{0} \in[0,h[$, we set $\Omega_{\overline{x}_{0},k} = [\overline{x}_{0}, \overline{x}_{0} +k[\, \times\Omega_{0}$. By means of a change of variables $x_{0}=y_{0} + \overline{x}_{0}$, we show, as done before, (37) for every $u \in C^{\infty}_{0}(\overline{\Omega}_{\overline{x}_{0},k})$ and k small enough. Then it is possible to divide $\Omega_{h}$ in a finite number of subsets $\Omega_{0}=[0, k_{0}[\, \times\Omega_{0}$, $\Omega_{1}=[k_{1}, k_{2}[\, \times\Omega_{0}$, … , $\Omega_{p}=[k_{p}, h[\, \times\Omega_{0}$, with $k_{i+1} < k_{i} < k_{j}$, for every $i=0, \ldots, p$, $k_{p+1}=h$ and $j \geq i+2$, such that (48) holds in every $\Omega_{i}$, namely (37) holds for every $u \in C^{\infty}_{0}(\overline{\Omega}_{i})$, $i=0, \ldots,p$. Now, for every $u \in C^{\infty}(\overline{\Omega})$ with $\operatorname{supp} u \subseteq \Omega_{h}$, as a result

$$\begin{aligned} \Vert u \Vert _{H^{0,0,s}([k_{i},k_{i+2}[ \times\Omega_{0})} \leq& \Vert u \psi \Vert _{H^{0,0,s}([k_{i},k_{i+1}[ \times\Omega_{0})} \\ \leq& c \Vert Pu \psi \Vert _{H^{0,0,s}([k_{i},k_{i+1}[ \times\Omega_{0})} \\ \leq& c \Vert Pu \Vert _{H^{0,0,s}(\Omega_{h})} \\ &{}+ c \bigl( \Vert \partial_{x_{0}} u \Vert _{H^{0,0,s}([k_{i+2},k_{i+4}[ \times\Omega_{0})} + \Vert u \Vert _{H^{0,0,s}([k_{i+2},k_{i+4}[ \times\Omega_{0})} \bigr), \end{aligned}$$

(58)

where $\psi\in C^{\infty}([0,h[)$, $\psi=1$ on $[k_{i}, k_{i+2}]$ and $\operatorname{supp} \psi\subseteq[k_{i}, k_{i+1}]$, for every i odd with $i=-1, \ldots, p-2$, $k_{-1}=0$ and $k_{p+2} > k_{p+1}$. By (58) it follows that

$$\Vert u \Vert _{H^{0,0,s}([k_{i}, k_{i+2}[ \times\Omega_{0})} \leq c \Vert Pu \Vert _{H^{0,0,s}(\Omega_{h})}, $$

from which, adding with respect to i, with i odd and $i=-1, \ldots, p-2$, as a result

$$\Vert u \Vert _{H^{0,0,s}(\Omega_{h})} \leq c \Vert Pu \Vert _{H^{0,0,s}(\Omega_{h})}. $$

Making using of the previous inequality and proceeding as in Section 5, we see that there exists $w \in H^{0,0,s}(\Omega_{h})$ such that

$$ \bigl(w, {}^{t}Pu\bigr) = (f,u),\quad \forall u \in C^{\infty}_{0}(\overline{\Omega}_{h}), $$

(59)

and $w \in H^{r}(\Omega_{h}) \cap H^{r, 2(r-s)}(\Omega_{h})$, with $0 \leq s \leq r$, $r \geq2$. Integrating by parts in (59) (see Section 5), for the arbitrariness of h, we can prove that for every $h>0$ the problem

$$\textstyle\begin{cases} Pw=f, \quad\mbox{in } \Omega_{h}, \\ w(0,x',x'')=0,\qquad w_{x_{0}}(0,x',x'')=0, \end{cases} $$

with $f \in H^{s,2(r-s)}(\Omega)$, for every $0 \leq s \leq r$, $r \leq2$, admits a solution $w \in H^{r}(\Omega_{h}) \cap H^{r, 2(r-s)}(\Omega_{h})$, with $0 \leq s \leq r$, $r \geq2$.