The Cauchy–Neumann and Cauchy–Robin problems for a class of hyperbolic operators with double characteristics in presence of transition

Abstract

The mixed Cauchy–Neumann and Cauchy–Robin problems for a class of hyperbolic operators with double characteristics in presence of transition is investigated. Some a priori estimates in Sobolev spaces with negative indexes are proved. Subsequently, existence and uniqueness results for the mixed problems are obtained.

1 Introduction

The aim of the paper is to establish existence and uniqueness results for the Cauchy–Neumann and Cauchy–Robin problems associated to a class of hyperbolic operators with double characteristics in presence of transition.

Let $\varOmega = ]0, + \infty [ \times \varOmega _0$, where $\varOmega _0$ is an open set of ${\mathbb {R}}^2$ with Lipschitz boundary. Let $x=(x_0,x')$ where $x'=(x_1,x_2)$, let $\xi =(\xi _0, \xi ')$, where $\xi '= (\xi _1, \xi _2)$. Let $D'=( \partial _{x_1}, \beta ^2(x) \partial _{x_2})$ and let $L'=(\partial _{x_1} - a_1(x), \beta ^2(x) \partial _{x_2} - a_2(x))$, where $\beta (x) =x_0- \alpha (x')$, being $\alpha$ a real function. Let $n=(n_0,n')$ be the external normal versor to the boundary of $\varOmega$, where $n'=(n_1,n_2)$.

The mixed problems, we will study, are introduced in the sequel. Precisely, the Cauchy–Neumann problem is

$$\begin{aligned} {\left\{ \begin{array}{ll} Pu= f, \quad \mathrm{in} \ \varOmega , \\ u|_{\varOmega _0}=0, \ \frac{du}{dn}|_{\varOmega _0}=0, \ D'u \cdot n'|_{S}=0, \end{array}\right. } \end{aligned}$$

(1)

and the Cauchy–Robin problem is

$$\begin{aligned} {\left\{ \begin{array}{ll} Pu= f, \quad \mathrm{in} \ \varOmega , \\ u|_{\varOmega _0}=0, \ \frac{du}{dn}|_{\varOmega _0}=0, \ L'u \cdot n'|_{S}=0, \end{array}\right. } \end{aligned}$$

(2)

where $S=]0, + \infty [ \times \partial \varOmega _0$ and

$$\begin{aligned} P=D^2_{x_0} - D^2_{x_1} - \beta ^2(x) D^2_{x_2} + \sum _{j=0}^2 a_j(x) D_{x_j} +b(x), \quad \mathrm{in} \ \varOmega , \end{aligned}$$

(3)

with coefficients belonging in $C^{\infty }(\widetilde{\varOmega })$, where $\widetilde{\varOmega }= [0, + \infty [ \times \widetilde{\varOmega }_0$, being $\widetilde{\varOmega }_0$ an open set containing $\varOmega _0$, $\mathrm{Im} \, a_2(x)=(x_0-\alpha (x'))\widetilde{a}_2(x)$, where $\widetilde{a}_2(x)$ is a real function and $D_{x_j}= \dfrac{1}{i} \partial _{x_j}$, $j=0,1,2$.

For every $x'=(x_1,x_2)$ and $\xi =(\xi _0, \xi ')$,

$$\begin{aligned} p(x_0, x', \xi ) = - \xi _0^2+ \xi ^2_1+ (x_0 - \alpha (x'))^2 \xi ^2_2 \end{aligned}$$

is the principal symbol of P,

$$\begin{aligned} \Sigma = \left\{ \rho = (x_0, x', \xi ) \in T^* \varOmega : \ \xi ' \ne 0, \ p(\rho ) =0, \ \nabla p(\rho ) =0 \right\} \end{aligned}$$

is the characteristic set and

$$\begin{aligned} F_p(\rho )=\frac{1}{2} \left( \begin{array}{cc} p''_{x\xi }(\rho ) &{} p''_{\xi \xi }(\rho ) \\ - p''_{x x}(\rho ) &{} - p''_{\xi x}(\rho ) \end{array} \right) , \quad \forall \rho \in \Sigma . \end{aligned}$$

is the fundamental matrix of P at $\rho$. The spectrum of $F_p(\rho )$, $\mathrm{Spec}(F_p(\rho ))$, is very important to analyze the well-posedness of the mixed problems associated to P.

Hörmander proved (see [9]):

$$\begin{aligned} z \in \mathrm{Spec}(F_p(\rho )) \ \Leftrightarrow \ - z, \overline{z} \in \mathrm{Spec}(F_p(\rho )). \end{aligned}$$

We have three possible cases, see for instance [8]. There exists a positive real number $\lambda$ such that $\{ - \lambda , \lambda \} \subset \mathrm{Spec}(F_p(\rho ))$ and $\mathrm{Spec}(F_p(\rho )) {\setminus } \{ - \lambda , \lambda \} \subset i {\mathbb {R}}$, the operator P is called effectively hyperbolic at $\rho$. We introduce

$$\begin{aligned} \Sigma _+ = \left\{ \rho \in \Sigma : \ P \ \mathrm{is \ effectively \ hyperbolic \ at} \ \rho \right\} . \end{aligned}$$

Moreover, $\mathrm{Spec}(F_p(\rho )) \subset i {\mathbb {R}}$ and in the Jordan normal form of $F(\rho )$ corresponding to the eigenvalue 0, there are only Jordan blocks of dimension 2, namely $\mathrm{Ker} F_p(\rho )^2 \cap \mathrm{Im} F_p(\rho )^2 = \{ 0\}$, the operator P is called non-effectively hyperbolic of type 1 at $\rho$. We set

$$\begin{aligned} \Sigma _- = \left\{ \rho \in \Sigma : \ P \ \mathrm{is \ non{-}effectively \ hyperbolic \ of \ type} \ 1 \ \mathrm{at} \ \rho \right\} . \end{aligned}$$

Finally, $\mathrm{Spec}(F_p(\rho )) \subset i {\mathbb {R}}$ and in the Jordan normal form of $F(\rho )$ corresponding to the eigenvalue 0, there is only a Jordan blocks of dimension 4 and no block of dimension 3, namely $\mathrm{Ker} F_p(\rho )^2 \cap \mathrm{Im} F_p(\rho )^2$ is 2-dimensional, the operator P is called non-effectively hyperbolic of type 2 at $\rho$. We denote by

$$\begin{aligned} \Sigma _0 = \left\{ \rho \in \Sigma : \ P \ \mathrm{is \ non{-}effectively \ hyperbolic \ of \ type} \ 2 \ \mathrm{at} \ \rho \right\} . \end{aligned}$$

Obviously, it follows

$$\begin{aligned} \Sigma = \Sigma _- \sqcup \Sigma _0 \sqcup \Sigma _+. \end{aligned}$$

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

In [7] the authors study the well posedness of the Cauchy problem associated to hyperbolic operators with douple characteristics in presence of transition in the cases in which $\Sigma = \Sigma _0 \sqcup \Sigma _+$ or $\Sigma = \Sigma _0 \sqcup \Sigma _-$. In [2], a global existence and uniqueness theorem for the Cauchy problem related to the class of hyperbolic operators with double characteristics

$$\begin{aligned} P=D^2_{x_0} - D^2_{x_1} - (x_0 + \lambda - \alpha (x_1))^2 D^2_{x_2}, \quad \lambda \ne 0, \end{aligned}$$

depending on the parameter $\lambda$ in the half-space ${\mathbb {R}}^2 \times ]0, + \infty [$ is proved. In [4], the authors consider a mixed Cauchy-Dirichlet problem associated to the previous class of operators in a particular domain of ${\mathbb {R}}^3$, instead in the class, here studied, the coeffiecient $\beta$ depends only on $x_0$. In [3], a priori estimates for particular test functions useful for the study of a Cauchy-Dirichlet problem are established. In this paper, unlike [7], $\Sigma = \Sigma _- \sqcup \Sigma _0 \sqcup \Sigma _+$ where all the sets $\Sigma _-$, $\Sigma _0$ and $\Sigma _+$ can be nonempty on the same connected components of $\Sigma$. Moreover, unlike [5], here some priori estimates for general test functions (not particular test functions as in [5]) useful for proving the existence of solutions to the mixed Cauchy–Neumann and Cauchy–Robin problems are obtained. The class of operators (3) has both the case in which $F_p(\rho )$ has two distinct real eigenvalues and the case in which all the eigenvalues are purely imaginary numbers can occur (see [1, 2]). Precisely, if $| \partial _{x_1} \alpha (x')|<1$, $\beta \equiv 0$ and $\xi _0= \xi _1=0$, then $F_p(\rho )$ has two distint non-zero real eigenvalues. If $| \partial _{x_1} \alpha (x')| > 1$, $\beta \equiv 0$ and $\xi _0= \xi _1=0$, $F_p(\rho )$ has two non-zero imaginary eigenvalues. Summarizing, let $\overline{\Sigma }$ be the set of points $\rho =(x_0,x',\xi )$ of $\Sigma$ such that $\beta \equiv 0$ and $\xi _0=\xi _1=0$. It results that $\rho \in \Sigma _+$ if $\rho \in \overline{\Sigma }$ and $| \partial _{x_1} \alpha (x')|<1$, $\rho \in \Sigma _-$ if $\rho \in \overline{\Sigma }$ and $| \partial _{x_1} \alpha (x')|>1$, and $\rho \in \Sigma _0$ if $\rho \in \overline{\Sigma }$ and $| \partial _{x_1} \alpha (x')|=1$. In this paper, the special class of operators (3), we analyze, has transition from effectively hyperbolic to non-effectively hyperbolic. In [5] and [6], two classes more general of hyperbolic operators with double charateristics are investigated.

Let us introduce

$$\begin{aligned} \Gamma= & {} \left\{ x \in \overline{\varOmega }: \ \beta (x)=0 \right\} , \quad \widetilde{\Gamma } = \left\{ x \in \overline{\widetilde{\varOmega }}: \ \beta (x)=0 \right\} , \\ \Gamma '= & {} \left\{ x \in \Gamma : \ | \partial _{x_1} \alpha (x')| \ge 1 \right\} , \quad \widetilde{\Gamma }'= \left\{ x \in \widetilde{\Gamma }: \ | \partial _{x_1} \alpha (x')| \ge 1 \right\} , \\ \varOmega _0'= & {} \left\{ x' \in \varOmega _0: \ \alpha (x') \ge 0, \ | \partial _{x_1} \alpha (x')| \ge 1 \right\} , \\ \widetilde{\varOmega }_0'= & {} \left\{ x' \in \widetilde{\varOmega }_0: \ \alpha (x') \ge 0, \ | \partial _{x_1} \alpha (x')| \ge 1 \right\} , \\ S= & {} \partial \varOmega _0 \times [0,+ \infty [, \end{aligned}$$

furthermore $g(x')= \dfrac{\alpha (x')}{\partial _{x_1} \alpha (x')}$, $h(x')=1-\partial _{x_1} g(x')$, for every $x' \in \varOmega _0'$.

We consider a quadratic matrix-function $B= (b_{hk})_{h,k=0,1}$ whose elements are:

$$\begin{aligned} b_{00}(x)= & {} h(x') -2 \alpha (x') |\widetilde{a}_0(x)|, \quad \forall x \in \Gamma ', \\ b_{01}(x)= & {} b_{10}(x) = - g(x') \widetilde{a}_0(x) - \alpha (x') \widetilde{a}_1(x), \quad \forall x \in \Gamma ', \\ b_{11}(x)= & {} h(x') -2 |g(x') \widetilde{a}_1(x)|, \quad \forall x \in \Gamma ', \end{aligned}$$

where $\widetilde{a}_0$ and $\widetilde{a}_1$ are the imaginary parts of $a_0$ and $a_1$, respectively.

We suppose

• (i) $h(x') \in [h_1, h_2]$, $\forall x' \in \widetilde{\varOmega }_0'$, with $0<h_1<h_2 < 4$;

• (ii) the matrix-function B is positive definite in $\widetilde{\Gamma }'$, namely there exists $M>0$ such that $B(x') \eta \cdot \eta \ge M \Vert \eta \Vert ^2$, $\forall \eta = (\eta _1, \eta _2) \ne (0,0)$, $\forall x \in \widetilde{\Gamma }'$;

• (iii) the connected components of the curve $S\cap \Gamma '$ lie in parallel planes to $\varOmega _0$.

We observe that if $\widetilde{a}_0 = \widetilde{a}_1 =0$, on $\Gamma '$, assumption (ii) is satisfied.

The main result of the paper is the following existence and uniqueness theorem.

Theorem 1

Let $f \in H_{loc}^{r}(\overline{\varOmega })$, with $r \ge 2$. Let us suppose that assumptions (i), (ii) and (iii) hold. The Cauchy–Neumann problem (1) and Cauchy–Robin problem (2) admit a solution $u \in H^{r}_{loc}(\overline{\varOmega })$.

Example 1

Let $P= D_{x_0}^{2} - D_{x_1}^{2} - \beta ^2(x) D_{x_2}^{2} + a_0(x) D_{x_0} + \beta (x) (a_1(x) D_{x_1} + a_2(x) D_{x_2}) + b(x)$ be a hyperbolic operator in $\varOmega = ]0, + \infty [ \times \varOmega _0$ where $\beta (x)= x_0 - \dfrac{x_1^2+1}{x_2^2+4}$. It results that $\partial _{x_1} \alpha (x')= \dfrac{2x_1}{x_2^2+4}$, $g(x') = \dfrac{x_1^2+1}{2 x_1}$, $h(x') = \dfrac{x_1^2+1}{2x_1^2}$, in $\varOmega _0$. Assumption (i) is verified in $\varOmega _0$. Assumption (ii) holds if $\mathrm{Im} \, |a_0(x)| \le \dfrac{1}{2x_1^2}$ in $\Gamma '$. Assumption (iii) is fulfilled if $\Gamma '$ is constituited by arcs of hyperboles of type $x_1^2+1=a(x_2^2+4)$ ($4a>1$). For example if $\varOmega _0 = \left\{ x' \in {\mathbb {R}}^2: \ x_1^2+1 \le a(x_2^2 +4), \ x_2^2 \le \gamma (x_1) \right\}$, with $a>2$ and $\gamma \in C^1({\mathbb {R}})$ such that $\gamma (x_1) \ge 4(a-1)$, assumptions (i) and (iii) are satisfied and assumption (ii) is verified if $|\mathrm{Im} \, a_0| < \dfrac{1}{8a^2}$. Furthermore we have a transition on $\Sigma$.

Example 2

Let $P= D_{x_0}^{2} - D_{x_1}^{2} - \beta ^2(x) D_{x_2}^{2} +b(x)$ be a hyperbolic operator in $\varOmega = ]0, + \infty [ \times \varOmega _0$ with $\beta (x) = x_0 - (x_1+x_2)^2$. We have $\partial _{x_1} \alpha (x') =2x_1$, $g(x')= \dfrac{x^2_1+x^2_2}{2 x_1}$, $h(x')= \dfrac{x_1^2+x_2^2}{2x_1}$, in $\varOmega _0$. Assumption (i) is verified if $|x_2| \le \dfrac{7}{4}$ and $|x_1| \ge \dfrac{1}{2}$. Assumption (ii) is always satisfied. Assumption (iii) holds if $\Gamma '$ is constituted by arcs of circumferences with center (0, 0). For example if $\varOmega _0$ is the circle in ${\mathbb {R}}^2$ with center (0, 0) and radius r with $\dfrac{1}{2}< r< 2$, assumptions (i), (ii) and (iii) are fulfilled and we have a transition on $\Sigma$.

The paper is organized as follows. In Sect. 2 some preliminary notations and definitions are presented. Section 3 is devoted to a priori estimates near the boundary $\varOmega _0$. In Sect. 4 a priori estimates away from $\varOmega _0$ are established. Section 5 concerns estimates in Sobolev spaces making use of the pseudodifferential operator theory. Section 6 deals with some global estimates in $\varOmega$. In Sect. 7 existence and regularity results for solutions to the mixed Cauchy–Neumann and Cauchy–Robin problems are proved. At last, in Sect. 8, a uniqueness result for the mixed problems is obtained.

2 Notations and preliminaries

Let $\alpha =(\alpha _0, \alpha _1, \alpha _2) \in {\mathbb {N}}^3_0$. Let $\partial ^{\alpha }$ be the derivative of order $|\alpha |$, let $\partial ^{h}_{x_j}$ be the derivative of order h with respect to $x_j$ and let $\partial ^{h}_{x_j, x_p}$ be the derivative of order h with respect to $x_j$ and $x_p$.

We indicate the $L^2$-scalar product, the $L^2$-norm and the $H^{r}$-norm by $(\cdot , \cdot )$, $\Vert \cdot \Vert$ and $\Vert \cdot \Vert _{H^{r}}$ ($r \in {{\mathbb {N}}}_0$), respectively.

Let $C_0^{\infty }(\overline{\varOmega })$ be the space of the restrictions to $\overline{\varOmega }$ of functions belonging to $C^{\infty }_0({\mathbb {R}}^3)$. For each $K \subseteq \overline{\varOmega }$ compact set, let $C^{\infty }_0 (K)$ be the set of functions $\varphi \in C^{\infty }_0 (\overline{\varOmega })$ having support contained in K. Set $\varOmega _k=[0,k[ \times \varOmega _0$, with $k>0$, let

$$\begin{aligned} C^{\infty }_0(\overline{\varOmega }_k) = \left\{ u \in C^{\infty }_0(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq [0,k[ \times \overline{\varOmega }_0 \right\} . \end{aligned}$$

Let $S({\mathbb {R}}^3)$ be the space of rapidly decreasing functions. In particular, let $S(\overline{\varOmega })$ be the space of the restrictions to $\overline{\varOmega }$ of functions belonging to $S({\mathbb {R}}^3)$.

Fixed $s \in {\mathbb {R}}$, we consider the following norm

$$\begin{aligned}&\Vert u \Vert ^2_{H^{0,s}} = \frac{1}{(2 \pi )^2} \int _0^{+ \infty } dx_0 \int _{{\mathbb {R}}^2} (1+|\xi '|^2)^{s} | \widehat{u}(x_0, \xi ')|^2 d\xi ', \\&\qquad \qquad \quad \ \forall u \in C^{\infty }_{0}(\overline{\varOmega }): \ \mathrm{supp} \ u \subseteq [0, + \infty [ \times \varOmega _0, \end{aligned}$$

where the Fourier transform is computed only with respect to the variable $x'$. Let us introduce the pseudodifferential operator $A_s: C^{\infty }_0(\varOmega ) \rightarrow C^{\infty }(\varOmega )$ given by

$$\begin{aligned}&A_s u= \frac{1}{(2 \pi )^2} \int _{{\mathbb {R}}^2} e^{i x' \cdot \xi '} (1+|\xi '|^2)^{\frac{s}{2}} \widehat{u}(x_0, \xi ') d\xi ',\\&\quad \forall u \in C^{\infty }_0(\overline{\varOmega }): \ \mathrm{supp} \ u \subseteq [0, + \infty [ \times \varOmega _0. \end{aligned}$$

For every $\varphi (x') \in C^{\infty }_0(\varOmega _0)$, the operator $\varphi A_s u$ extends as a linear continuous operator from $H^{0,r}_{comp.}(\varOmega )$ into $H^{0,r-s}_{loc}(\varOmega )$, where $r,s \in {\mathbb {R}}$. If $\mathrm{supp} \ \varphi \subseteq \varOmega _0 {\setminus } \mathrm{supp} \, u$, then $\varphi A_s u$ is a regularizing operator with respect to the variable $x'$. It results

$$\begin{aligned} \Vert \varphi A_s u \Vert _{H^{0,r}} \le c \Vert u \Vert _{H^{0,r'}}, \quad \forall r,r' \in {\mathbb {R}}, \ u \in C^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq [0, + \infty [ \times \varOmega _0. \end{aligned}$$

We note that the norms $\Vert u \Vert _{H^{0,s}(\varOmega )}$ and $\Vert A_s u \Vert _{L^2(\varOmega )}$ are equivalent for any $s \in {\mathbb {R}}$. Moreover, let $H^{0,s}(\varOmega _k)$ be the space of $u \in H^{0,s}(\varOmega _k)$ such that $\mathrm{supp} \, u \subseteq \varOmega _k$.

Let $s \in {\mathbb {R}}$ and $p \ge 0$. Let $H^{p,s}({\mathbb {R}}^3)$ be the space of all the distributions on ${\mathbb {R}}^3$ such that

$$\begin{aligned} \Vert u \Vert ^2_{H^{p,s}({\mathbb {R}}^3)} = \frac{1}{(2 \pi )^2} \sum _{|h| \le p} \int _{{\mathbb {R}}^3} (1+|\xi '|^2)^{s} | \partial ^h_{x_0} \widehat{u}(x_0, \xi ')|^2 dx_0 d\xi ' < + \infty . \end{aligned}$$

Let $H^{p,s}(\varOmega )$ be the space of all the restrictions to $\varOmega$ of elements of $H^{p,s}({\mathbb {R}}^3)$ endowed with the norm

$$\begin{aligned} \Vert u \Vert _{H^{p,s}(\varOmega )} = \inf _{{\begin{array}{c} U \in H^{p,s}({\mathbb {R}}^3) \\ U|_{\varOmega } =u \end{array}}} \Vert U \Vert _{H^{p,s}({\mathbb {R}}^3)}. \end{aligned}$$

Analogously we can define the space $H^{p,s}(\varOmega _k)$.

Finally, let us consider the transposed operator $\, ^t P$ given by:

$$\begin{aligned} {}^{t}P= & {} - \partial _{x_0}^2 + \partial _{x_1}^2 + (x_0-\alpha (x'))^2 \partial ^2_{x_2} - 4 (x_0-\alpha (x')) (\partial _{x_2} \alpha ) \partial _{x_2} \\&- \frac{1}{i} \sum _{j=0}^2 a_j(x) \partial _{x_j} - \frac{1}{i} \sum _{j=0}^2 \partial _{x_j} a_j(x) - 2 (\partial _{x_2} \alpha )^2 +b(x). \end{aligned}$$

3 A priori estimates near the boundary $\varOmega _0$

We enunciate the following preliminary result which synthesizes Lemmas 3.1 and 3.2 proved in [5].

Lemma 1

Let $u \in S(\overline{\varOmega })$ and let $p, \alpha _0, \alpha _1, \alpha _2 \in {\mathbb {N}}_0$. Then

$$\begin{aligned} \Vert x_0^{\frac{p}{2}} \partial ^{\alpha _0, \alpha _1, \alpha _2} u \Vert \le \frac{2}{p+1} \Vert x_0^{\frac{p+2}{2}} \partial ^{\alpha _0+1, \alpha _1, \alpha _2} u \Vert \end{aligned}$$

(4)

and

$$\begin{aligned} \int _{\varOmega _0} |u(0,x')|^2 dx' \le 2 \Vert u \Vert \Vert \partial _{x_0} u \Vert . \end{aligned}$$

The proof of the following preliminary result is analogous to the one of Lemma 3.3 in [5] with some modification, therefore for reader’s convenience we write it. As in Lemma 3.3 in [5], let us consider the set

$$\begin{aligned} I_{k, \delta } = \left\{ x \in \overline{\varOmega }: \ x_0 < k, \ |x_0 - \alpha (x')| > \delta \right\} , \end{aligned}$$

with $k, \delta$ positive and small enough.

Lemma 2

For every $\varepsilon , \delta >0$, there exists $k>0$ such that

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert + \Vert u \Vert \le \varepsilon \Vert \ ^{t} Pu \Vert , \\&\qquad \ \, \forall u \in C^{\infty }_0(\overline{\varOmega }): \ \mathrm{supp} \ u \subseteq I_{k, \delta }, \ D'u \cdot n'|_S=0. \end{aligned}$$

Proof

Let us denote the principal part of $^{t}P$ by $^{t}P_2$, the part of the first order of $^{t}P$ by $^{t}P_1$ and the part of the zero order of $^{t}P$ by $^{t}P_0$.

Integrating by parts and taking into account the boundary conditions, we obtain

$$\begin{aligned}&(e^{\tau x_0} \partial _{x_0} u, \ ^{t} P_2u) + ( \ ^{t}P_2 u, e^{\tau x_0} \partial _{x_0} u) \nonumber \\&\quad = 2(e^{\tau x_0} \partial _{x_0} u(x), \ ^{t}P_2u) \nonumber \\&\quad = \tau \Vert e^{\frac{1}{2} \tau x_0} \partial _{x_0} u\Vert ^2 + \tau \Vert e^{\frac{1}{2} \tau x_0} \partial _{x_1} u\Vert ^2 + \tau \Vert e^{\frac{1}{2} \tau x_0} (x_0 -\alpha (x')) \partial _{x_2} u\Vert ^2 \nonumber \\&\quad \quad +2 \left( e^{\tau x_0} ( x_0 - \alpha (x')) \partial _{x_2} u, \partial _{x_2} u \right) \nonumber \\&\quad \quad + 4 \left( e^{\tau x_0} \partial _{x_0} u, ( x_0 - \alpha (x')) \partial _{x_2} \alpha \partial _{x_2} u \right) \nonumber \\&\quad \quad + 2 \int _{S} e^{\tau x_0} \partial _{x_1} u \cdot n_1 \partial _{x_0} u d\sigma + 2 \int _{S} e^{\tau x_0} \beta ^2(x) \partial _{x_2} u \cdot n_2 \partial _{x_0} u d\sigma \nonumber \\&\quad \quad + \int _{\varOmega _0} (\partial _{x_0} u)^2 dx' + \int _{\varOmega _0} (\partial _{x_1} u)^2 dx' + \int _{\varOmega _0} ((x_0 - \alpha (x')) \partial _{x_2}u)^2 dx' \nonumber \\&\quad \ge \tau \Vert e^{\frac{1}{2} \tau x_0} \partial _{x_0} u\Vert ^2 + \tau \Vert e^{\frac{1}{2} \tau x_0} \partial _{x_1} u\Vert ^2 + \tau \Vert e^{\frac{1}{2} \tau x_0} (x_0-\alpha (x')) \partial _{x_2} u\Vert ^2 \nonumber \\&\quad \quad - \frac{2}{\delta } \Vert e^{\frac{1}{2} \tau x_0} (x_0-\alpha (x')) \partial _{x_2} u \Vert ^2 -2 \Vert e^{\frac{1}{2} \tau x_0} \partial _{x_0} u \Vert ^2 \nonumber \\&\quad \quad - 2 \Vert e^{\frac{1}{2} \tau x_0} |\partial _{x_2} \alpha (x')|^{\frac{1}{2}} (x_0-\alpha (x')) \partial _{x_2} u \Vert ^2,\nonumber \\&\quad \quad \forall u \in C^{\infty }_0(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq I_{k, \delta }, \ D'u \cdot n'|_S=0. \end{aligned}$$

(5)

Moreover, we have

$$\begin{aligned}&(e^{\tau x_0} \partial _{x_0} u, \, ^{t}(P-P_2)u) + (\, ^{t}(P-P_2)u, e^{\tau x_0} \partial _{x_0} u) \nonumber \\&\quad \ge -c \big ( \Vert e^{\frac{1}{2} \tau x_0} \partial _{x_0} u\Vert ^2 - \Vert e^{\frac{1}{2} \tau x_0} \partial _{x_1} u\Vert ^2 - \Vert e^{\frac{1}{2} \tau x_0} (x_0-\alpha (x')) \partial _{x_2} u\Vert ^2 \nonumber \\&\qquad + \Vert e^{\frac{1}{2} \tau x_0} u \Vert \big ). \end{aligned}$$

(6)

Adding (5) and (6) and applying Lemma 1, it results, for $\frac{1}{2} \tau x_0 < 1$,

$$\begin{aligned}&(e^{\tau x_0} \partial _{x_0} u, \ ^{t} Pu) + ( \ ^{t}P u, e^{\tau x_0} \partial _{x_0} u) \nonumber \\&\quad \ge \tau \left( \Vert \partial _{x_0} u\Vert ^2 + \Vert \partial _{x_1} u\Vert ^2 + \Vert (x_0-\alpha (x')) \partial _{x_2} u\Vert ^2 \right) - c \Big ( \dfrac{1}{\delta } \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert ^2 \nonumber \\&\quad \quad + \Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert ^2 + \Vert x_0^{\frac{1}{2}} \partial _{x_0} u \Vert ^2 \Big ). \end{aligned}$$

(7)

Making use of (7) and choosing $x_0< \dfrac{1}{\tau }$, we have

$$\begin{aligned}&\Vert \partial _{x_0} u\Vert ^2 + \Vert \partial _{x_1} u\Vert ^2 + \Vert (x_0-\alpha (x')) \partial _{x_2} u\Vert ^2 \nonumber \\&\quad \le \frac{1}{\tau } \left\| \ ^{t} Pu \right\| \Vert \partial _{x_0} u \Vert + \frac{c}{\tau \delta } \Vert (x_0-\alpha (x')) \partial _{x_2} u \Vert ^2 + \frac{c}{\tau } \Vert \partial _{x_0} u \Vert ^2 \nonumber \\&\quad \quad + \frac{c}{\tau } \Vert (x_0-\alpha (x')) \partial _{x_2} u \Vert ^2 + \frac{c}{\tau } \Vert \partial _{x_1} u \Vert ^2 + \frac{c}{\tau } \Vert u \Vert ^2. \end{aligned}$$

(8)

Taking into account Lemma 1 and considering $\tau$ large enough, the claim is archieved. $\square$

Now, we establish the following result.

Theorem 2

Let us suppose that assumptions (i), (ii) and (iii) hold. Then, for every $\varepsilon >0$, there exists $k>0$ such that

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert + \Vert u \Vert \le \varepsilon \Vert \ ^{t} Pu\Vert , \\&\, \forall u \in C^{\infty }_0(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _k= [0,k[ \times \overline{\varOmega }_0, \ D'u \cdot n'|_S=0. \end{aligned}$$

Proof

If $\Gamma \cap \varOmega _0 = \emptyset$, we are able to use Lemma 2. Hence, the claim is proved. If $\Gamma \cap \varOmega _0 \ne \emptyset$, we distinguish two regions. More precisely, for every $\dfrac{4}{5}< \eta < 1$, let

$$\begin{aligned} \varOmega _{k,\eta } = \left\{ x \in \overline{\varOmega }: \ x_0 \in [0,k[, \ (1-\eta ) \alpha (x') \le x_0 \le \left( \frac{1}{5} + \eta \right) \alpha (x') \right\} \end{aligned}$$

and let $\varOmega _k {\setminus } \varOmega _{k,\eta }$. Then, let us set

$$\begin{aligned} \varOmega _{k,\eta }' = \left\{ x \in \varOmega _{k,\eta }: \ | \partial _{x_1} \alpha (x')| \ge 1 \right\} , \quad \varOmega _{k,\eta , \eta '}' = \left\{ x \in \varOmega _{k,\eta }: \ | \partial _{x_1} \alpha (x')| \ge 1 - \eta ' \right\} . \end{aligned}$$

Evidently $\varOmega _{k,\eta , \eta '}' \supseteq \varOmega _{k,\eta }'$. Moreover, we choose $k, \eta$ and $\eta '$ such that assumptions (i) and (ii) are satisfied. Let us consider a function $u \in C_0^{\infty }(\overline{\varOmega })$ with $\mathrm{supp} \, u \subseteq \varOmega _{k,\eta , \eta '}'$ and $D'u \cdot n'|_S = 0$. Let us remark that $\varOmega _{k,\eta } \cap \varOmega _0$ has measure zero in ${\mathbb {R}}^2$. Moreover, $\varOmega _{k,\eta } \cap S$ is empty or has measure zero in ${\mathbb {R}}^2$, for k small enough. Let us consider the inner products:

$$\begin{aligned} ( \, ^{t}Pu, Au) + (Au, \, ^{t} Pu), \quad \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{k,\eta , \eta '}', \ D'u \cdot n'|_S=0, \end{aligned}$$

where $Au = x_0 \partial _{x_0}u + g(x') \partial _{x_1}u$. It results

$$\begin{aligned} ( \, ^{t}Pu, Au) + (Au, \, ^{t} Pu)= & {} (\, ^{t}P_2u, Au) + (Au, \, ^{t}P_2u) +(\, ^{t}P_1u, Au) \nonumber \\&+ (Au,\, ^{t} P_1u) + (\, ^{t}P_0u, Au) + (Au,\, ^{t} P_0u), \nonumber \\&\forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{k,\eta , \eta '}', \ D'u \cdot n'|_S=0.\nonumber \\ \end{aligned}$$

(9)

Integrating by parts, for every $u \in C_0^{\infty }(\overline{\varOmega })$ such that $\mathrm{supp} \, u \subseteq \varOmega _{k,\eta , \eta '}'$, and $D'u \cdot n'|_S=0$, we have

$$\begin{aligned}&(\, ^{t}P_2u, Au) + (Au, \, ^{t}P_2u) \\&\quad = 2 (\, ^{t}P_2u, Au) \\&\quad = 2 ( \, ^{t}P_2u, x_0 \partial _{x_0} u) + 2 (\, ^{t} P_2u, g(x') \partial _{x_1} u) \\&\quad = (\partial _{x_0} u, \partial _{x_0} u) + (\partial _{x_1} u, \partial _{x_1} u) + \left( (x_0 - \alpha (x'))^2 \partial _{x_2} u, \partial _{x_2} u \right) \\&\quad \quad + 2 \left( (x_0- \alpha (x')) x_0 \partial _{x_2}u, \partial _{x_2}u \right) + 4 \left( (x_0-\alpha (x')) \partial _{x_2} \alpha (x') \partial _{x_2} u, x_0 \partial _{x_0} u \right) \\&\quad \quad - (\partial _{x_0} u, \partial _{x_1} g(x') \partial _{x_0}u) - (\partial _{x_1} g(x') \partial _{x_1} u, \partial _{x_1} u) \\&\quad \quad + 4 ((x_0-\alpha (x')) \partial _{x_2} \alpha (x') g(x') \partial _{x_2} u, \partial _{x_1}u) \\&\quad \quad - 2 \left( (x_0-\alpha (x')^2 \partial _{x_2} g(x') \partial _{x_2}u, \partial _{x_1} u \right) \\&\quad \quad + \left( (x_0 -\alpha (x'))^2 \partial _{x_1} g(x') \partial _{x_2}u, \partial _{x_2}u \right) \\&\quad \quad -2 \left( (x_0 - \alpha (x'))^2 \partial _{x_1} \alpha (x') g(x') \partial _{x_2}u, \partial _{x_2} u \right) . \end{aligned}$$

From which, it follows

$$\begin{aligned}&2 (\, ^{t}P_2u,Au) \nonumber \\&\quad = \Vert h^{\frac{1}{2}}(x') \partial _{x_0} u \Vert ^2 + \Vert h^{\frac{1}{2}}(x') \partial _{x_1} u \Vert ^2 + \Vert (4- h(x'))^{\frac{1}{2}} \left( x_0-\alpha (x')\right) \partial _{x_2} u \Vert ^2 \nonumber \\&\quad \quad + 4 \left( (x_0-\alpha (x')) \partial _{x_2} \alpha (x') \partial _{x_2} u, x_0 \partial _{x_0}u\right) \nonumber \\&\quad \quad + 4 \left( (x_0-\alpha (x')) \partial _{x_2} \alpha (x') g(x') \partial _{x_1} u, \partial _{x_2} u \right) \nonumber \\&\quad \quad - 24 \left( (x_0-\alpha (x'))^2 \partial _{x_2} g(x') \partial _{x_2} u, \partial _{x_1} u \right) , \end{aligned}$$

(10)

since we used that the boundary integrals are zero because the set $\varOmega _{k,\eta ,\eta '}' \cap \partial \varOmega$ has zero measure.

Let us consider

$$\begin{aligned}&(\, ^{t}P_1u,Au) + (Au, \, ^t P_1u) \nonumber \\&\quad = -8 \left( (x_0-\alpha (x')) \partial _{x_2} \alpha (x') \partial _{x_2} u, x_0 \partial _{x_0}u +g(x') \partial _{x_1} u \right) \nonumber \\&\quad \quad - 2 \left( \widetilde{a}_0(x) \partial _{x_0} u + \widetilde{a}_1(x) \partial _{x_1} u + (x_0 - \alpha (x')) \widetilde{a}_2(x) \partial _{x_2} u, x_0 \partial _{x_0} u + g(x') \partial _{x_1} u \right) \nonumber \\&\quad = -8 \left( (x_0-\alpha (x'))^2 \partial _{x_2} \alpha (x') \partial _{x_2} u, \partial _{x_0}u \right) \nonumber \\&\quad \quad -8 \left( (x_0-\alpha (x')) \alpha (x') \partial _{x_2} \alpha (x') \partial _{x_2} u, \partial _{x_0}u \right) \nonumber \\&\quad \quad -8 \left( (x_0-\alpha (x')) \partial _{x_2} \alpha (x') g(x') \partial _{x_2} u, \partial _{x_1}u \right) \nonumber \\&\quad \quad -2 \left( (x_0-\alpha (x')) \widetilde{a}_0(x) \partial _{x_0} u, \partial _{x_0}u \right) \nonumber \\&\quad \quad - 2 \left( \widetilde{a}_0(x) \alpha (x') \partial _{x_0} u, \partial _{x_0}u \right) -2 \left( \widetilde{a}_0(x) g(x') \partial _{x_0} u, \partial _{x_1}u \right) \nonumber \\&\quad \quad - 2 \left( \widetilde{a}_1(x) (x_0 - \alpha (x')) \partial _{x_1} u, \partial _{x_0}u \right) - 2 \left( \widetilde{a}_1(x) \alpha (x') \partial _{x_1} u, \partial _{x_0}u \right) \nonumber \\&\quad \quad - 2 \left( \widetilde{a}_1(x) g(x') \partial _{x_1} u, \partial _{x_1}u \right) -2 \left( \widetilde{a}_2(x) (x_0-\alpha (x'))^2 \partial _{x_2} u, \partial _{x_0}u \right) \nonumber \\&\quad \quad - 2 \left( \widetilde{a}_2(x) (x_0- \alpha (x')) \alpha (x') \partial _{x_2} u, \partial _{x_0}u \right) \nonumber \\&\quad \quad -2 \left( \widetilde{a}_2(x) (x_0-\alpha (x')) g(x') \partial _{x_2} u, \partial _{x_1}u \right) . \end{aligned}$$

(11)

Making use of assumptions (i) and (ii), it follows

$$\begin{aligned}&\Vert h^{\frac{1}{2}}(x') \partial _{x_0} u \Vert ^2 + \Vert h^{\frac{1}{2}}(x') \partial _{x_1} u \Vert ^2 + \Vert (4-h(x'))^{\frac{1}{2}} (x_0- \alpha (x')) \partial _{x_2} u \Vert ^2 \nonumber \\&\quad \quad - 2 \left( \widetilde{a}_0(x) \alpha (x') \partial _{x_0} u, \partial _{x_0}u \right) -2 \left( \widetilde{a}_0(x) g(x') \partial _{x_1} u, \partial _{x_0}u \right) \nonumber \\&\quad \quad - 2 \left( \widetilde{a}_1(x) \alpha (x') \partial _{x_1} u, \partial _{x_0}u \right) - 2 \left( \widetilde{a}_1(x) g(x') \partial _{x_1} u, \partial _{x_1}u \right) \nonumber \\&\quad \ge L \left( \Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 \right) + (4-h_2) \Vert (x_0- \alpha (x')) \partial _{x_2} u \Vert ^2, \nonumber \\&\quad \quad \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{k,\eta ,\eta '}', \ D'u \cdot n'|_S=0. \end{aligned}$$

(12)

Moreover, we remark that the functions $\alpha , g, \beta$ are zero on $\varOmega \cap \Gamma$. As a consequence, we can choose k small enough and an appropriate $\eta$ such that (12) holds and it results

$$\begin{aligned}&4 \left( (x_0 - \alpha (x'))^2 \partial _{x_2} \alpha (x') \partial _{x_2} u, \partial _{x_0}u \right) + 4 \left( (x_0 - \alpha (x')) \partial _{x_2} \alpha (x') g(x') \partial _{x_1} u, \partial _{x_2}u \right) \nonumber \\&\quad \quad + 4 \left( (x_0 - \alpha (x')) \alpha (x') \partial _{x_2} \alpha (x') \partial _{x_2} u, \partial _{x_0}u \right) \nonumber \\&\quad \quad -2 \left( (x_0 - \alpha (x')) \partial _{x_2} \alpha (x') g(x') \partial _{x_2} u, \partial _{x_1}u \right) \nonumber \\&\quad \quad - 2 \left( \widetilde{a}_0(x) (x_0 - \alpha (x')) \partial _{x_0} u, \partial _{x_0}u \right) -2 \left( \widetilde{a}_1(x) (x_0 - \alpha (x')) \partial _{x_1} u, \partial _{x_0}u \right) \nonumber \\&\quad \quad - 2 \left( \widetilde{a}_2(x) (x_0 - \alpha (x'))^2 \partial _{x_2} u, \partial _{x_0}u \right) -2 \left( \widetilde{a}_2(x) (x_0 - \alpha (x')) \alpha (x') \partial _{x_2} u, \partial _{x_0}u \right) \nonumber \\&\quad \quad - 2 \left( \widetilde{a}_2(x) (x_0 - \alpha (x')) g(x') \partial _{x_2} u, \partial _{x_1}u \right) \nonumber \\&\quad \ge - \delta \left( \Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \Vert (x_0- \alpha (x')) \partial _{x_2} u \Vert ^2 \right) , \end{aligned}$$

(13)

with $\delta < \min (L,4-h_2)$. Adding (9), (10), (11) and taking into account (13), we have

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \Vert (x_0- \alpha (x')) \partial _{x_2} u \Vert ^2\\&\quad \le c |(\, ^tPu, Au) + (Au, \, ^tPu) | \\&\quad \le c \Big ( \Vert x_0^{\frac{1}{2}} \, ^t Pu \Vert \Vert x_0^{\frac{1}{2}} \partial _{x_0} u \Vert \\&\quad + \Vert g^{\frac{1}{2}}(x') \, ^tPu \Vert + \Vert g^{\frac{1}{2}}(x') \partial _{x_1} u \Vert \Big ). \end{aligned}$$

For k small enough and an appropriate $\eta$, taking into account the previous inequality and Lemma 1, we obtain

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0- \alpha (x')) \partial _{x_2} u \Vert + \Vert u \Vert \le \varepsilon \Vert \, ^t Pu \Vert , \nonumber \\&\, \qquad \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{k,\eta ,\eta '}', \ D'u \cdot n'|_S=0, \end{aligned}$$

(14)

with $ck < \varepsilon$ and $c|g(x')| < \varepsilon$ in $\varOmega _{k,\eta ,\eta '}'$.

Now, we consider $\mathrm{supp} \, u \subseteq \varOmega _{k,\eta } {\setminus } \overline{\varOmega }_{k,\eta ,\eta '}'$ and we remind that it results $|\partial _{x_1} \alpha (x')| <1$ in $\varOmega _{k,\eta } {\setminus } \overline{\varOmega }_{k,\eta ,\eta '}'$. Integrating by parts, we obtain

$$\begin{aligned}&( \, ^t Pu, x_0 \partial _{x_0} u) + (x_0 \partial _{x_0} u, \, ^t Pu) \\&\quad = \Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \Vert (x_0- \alpha (x')) \partial _{x_2} u \Vert ^2 + 2 (x_0 (x_0 - \alpha (x')) \partial _{x_2} u, \partial _{x_2} u) \\&\quad \quad + 4 ((x_0 - \alpha (x')) \partial _{x_2} \alpha (x'), x_0 \partial _{x_0} u) + ( \, {{}^{t}}P_1 u, x_0 \partial _{x_0} u) + ( x_0 \partial _{x_0} u, \, ^t P_1u) \\&\quad \quad + ( \, {{}^{t}}P_0 u, x_0 \partial _{x_0} u) + ( x_0 \partial _{x_0} u, \, ^t P_0u), \\&\quad \quad \forall u \in C_0^{\infty }(\varOmega _{k,\eta } {\setminus } \overline{\varOmega }_{k,\eta ,\eta '}'): \ D'u \cdot n'|_S=0. \end{aligned}$$

Making use of the previous inequality for k small enough and Lemma 1, it follows

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \Vert (x_0- \alpha (x')) \partial _{x_2} u \Vert ^2 + \Vert u \Vert ^2 \nonumber \\&\qquad \le \varepsilon \Vert \, ^tPu \Vert ^2 + \varepsilon \Vert (x_0 - \alpha (x'))^{\frac{1}{2}} \partial _{x_2} u \Vert ^2. \end{aligned}$$

(15)

In order to estimate $\Vert (x_0 - \alpha (x'))^{\frac{1}{2}} \partial _{x_2} u \Vert ^2$, we consider the inner products $(\partial _{x_0} u, \, ^t Pu) + ( \, ^t Pu, \partial _{x_0} u)$ and integrate by parts in the principal part for $x_0 \le \alpha (x')$ and, then, for $x_0 \ge \alpha (x')$. In particular, for $x_0 \le \alpha (x')$, it results

$$\begin{aligned}&- ((x_0 - \alpha (x')) \partial _{x_2} u, \partial _{x_2} u) + \int _{\Gamma } ((\partial _{x_0} u)^2 + 2 \partial _{x_1} \alpha (x') \partial _{x_0} u \partial _{x_1} u + (\partial _{x_1} u)^2 ) d \sigma \nonumber \\&\quad \quad - 4 ((x_0 - \alpha (x')) \partial _{x_2} \alpha (x') \partial _{x_2} u, \partial _{x_0} u) \nonumber \\&\quad = - ( \, ^t (P-P_2) u, \partial _{x_0} u) - (\partial _{x_0} u, \, ^t (P-P_2) u) \nonumber \\&\quad \quad + (\, ^t Pu, \partial _{x_0} u) + ( \partial _{x_0} u, \, ^t Pu), \quad \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{k,\eta } {\setminus } \varOmega _{k,\eta ,\eta '}'.\nonumber \\ \end{aligned}$$

(16)

Moreover, for $x_0 \ge \alpha (x')$, we have

$$\begin{aligned}&- ((x_0 - \alpha (x')) \partial _{x_2} u, \partial _{x_2} u) + \int _{\Gamma } ((\partial _{x_0} u)^2 + 2 \partial _{x_1} \alpha (x') \partial _{x_0} u \partial _{x_1} u + (\partial _{x_1} u)^2 ) d \sigma \nonumber \\&\quad \quad - 4 ((x_0 - \alpha (x')) \partial _{x_2} \alpha (x') \partial _{x_2} u, \partial _{x_0} u) \nonumber \\&\quad = - ( \, ^t (P-P_2) u, \partial _{x_0} u) - (\partial _{x_0} u, \, ^t (P-P_2) u) + (\, ^t Pu, \partial _{x_0} u) \nonumber \\&\qquad + ( \partial _{x_0} u, \, ^t Pu), \quad \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{k,\eta } {\setminus } \varOmega _{k,\eta ,\eta '}'. \end{aligned}$$

(17)

Adding (16) and (17), and taking into account that $|\partial _{x_1} \alpha (x')| < 1$ in the considered part, we get

$$\begin{aligned}&\Vert (x_0- \alpha (x'))^{\frac{1}{2}} \partial _{x_2} u \Vert ^2 \nonumber \\&\qquad \le c ( \Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \Vert (x_0- \alpha (x')) \partial _{x_2} u \Vert ^2 + \Vert u \Vert ^2 + \Vert \, ^t Pu \Vert ^2 ). \end{aligned}$$

(18)

Making use of (18), (15) and Lemma 1, for k small enough and, hence, $\varepsilon$ small enough, it follows

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \Vert (x_0- \alpha (x')) \partial _{x_2} u \Vert ^2 \le \varepsilon \Vert \, ^t Pu \Vert ^2,\nonumber \\&\forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{k. \eta } {\setminus } \varOmega _{k,\eta , \eta '}', \ D'u \cdot n'|_S=0. \end{aligned}$$

(19)

Let us consider $u \in C_0^{\infty }(\overline{\varOmega })$ such that $\mathrm{supp} \, u \subseteq \varOmega _k {\setminus } \overline{\varOmega }_{k, \eta }$ and compute the following inner products

$$\begin{aligned}&( \, ^t Pu, x_0 \partial _{x_0} u) + (x_0 \partial _{x_0} u, \, ^t Pu) \nonumber \\&\quad = \Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \frac{1}{2} \Vert (x_0- \alpha (x')) \partial _{x_2} u \Vert ^2 \nonumber \\&\quad \quad + \left( \left( \frac{1}{2} (x_0- \alpha (x'))^2 + 2 x_0 (x_0 - \alpha (x')) \right) \partial _{x_2} u, \partial _{x_2} u \right) \nonumber \\&\quad \quad + 2 \int _S ( \partial _{x_1} u \cdot n_1+ \beta ^2(x) \partial _{x_2} u \cdot n_2) x_0 \partial _{x_0} u d \sigma \nonumber \\&\quad \quad + 4 (x_0 (x_0 - \alpha (x')) \partial _{x_2} \alpha (x') \partial _{x_2} u, \partial _{x_0} u) \nonumber \\&\quad \quad + (\, ^t (P- P_2) u, x_0 \partial _{x_0} u ) + (x_0 \partial _{x_0} u, \, ^t (P-P_2) u) \nonumber \\&\quad \ge \Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \Vert (x_0- \alpha (x')) \partial _{x_2} u \Vert ^2 \nonumber \\&\quad \quad - c \left( \Vert x_0^{\frac{1}{2}} \partial _{x_0} u \Vert ^2 + \Vert x_0^{\frac{1}{2}} \partial _{x_1} u \Vert ^2 + \Vert x_0^{\frac{1}{2}} (x_0- \alpha (x')) \partial _{x_2} u \Vert ^2 + \Vert x_0^{\frac{1}{2}} u \Vert ^2 \right) \nonumber \\&\quad \quad + \left( \left( \frac{1}{2} (x_0- \alpha (x'))^2 + 2 x_0 (x_0 - \alpha (x')) \right) \partial _{x_2} u, \partial _{x_2} u \right) . \end{aligned}$$

(20)

Making use of Lemma 1 and taking $\frac{4}{5}< \eta < 1$, we have $\frac{1}{2} (x_0 - \alpha (x'))^2 +2x_0 (x_0 \alpha (x')) > 0$, in $\varOmega _k {\setminus } \overline{\varOmega }_{k, \eta }$. For k small enough, it results

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0- \alpha (x')) \partial _{x_2} u \Vert + \Vert u \Vert \le \varepsilon \Vert \, ^t Pu \Vert , \nonumber \\&\quad \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _k {\setminus } \varOmega _{k,\eta }, \ D'u \cdot n'|_S=0. \end{aligned}$$

(21)

Since $\varOmega _0 \cap \Gamma$ has zero measure, without lost generality, we consider $u \in C_0^{\infty }(\overline{\varOmega })$ with $\mathrm{supp} \, u \subseteq \varOmega _k {\setminus } (\varOmega _0 \cap \Gamma )$. Let $\varphi \in C_0^{\infty }(\overline{\varOmega })$, with $\varphi \equiv 1$ on $\varOmega _{k, \frac{4}{5}} \cap \mathrm{supp} \, u$, $\mathrm{supp} \, \varphi \subseteq \varOmega _{k, \eta _1}$, with $\eta _1 > \frac{4}{5}$ and $0 \le \varphi \le 1$ in $\overline{\varOmega }$. Furthermore, let $\varphi ' \in C_0^{\infty }(\overline{\varOmega })$, with $\varphi ' \equiv 1$ on $\varOmega _{k, \eta }'$ and $\mathrm{supp} \, \varphi \varphi ' \subseteq \varOmega _{k, \eta _1, \eta '}$. We rewrite (14) for $\varphi \varphi ' u$, with $u \in C_0^{\infty }(\overline{\varOmega })$, and for k small enough:

$$\begin{aligned} \Vert \partial _{x_0} \varphi \varphi ' u \Vert + \Vert \partial _{x_1} \varphi \varphi ' u \Vert + \Vert (x_0- \alpha (x') \partial _{x_2} \varphi \varphi ' u \Vert \le \varepsilon \Vert \, ^t P \varphi \varphi ' u \Vert . \end{aligned}$$

Taking into account (19), we have

$$\begin{aligned}&\Vert \partial _{x_0} \varphi (1- \varphi ') u \Vert + \Vert \partial _{x_1} \varphi (1- \varphi ') u \Vert \\&+ \Vert (x_0- \alpha (x') \partial _{x_2} \varphi (1- \varphi ') u \Vert \le \varepsilon \Vert \, ^t P \varphi (1- \varphi ') u \Vert . \end{aligned}$$

Adding the previous inequalities and taking $\varepsilon$ small enough, we obtain

$$\begin{aligned} \Vert \partial _{x_0} \varphi u \Vert + \Vert \partial _{x_1} \varphi u \Vert + \Vert (x_0- \alpha (x')) \partial _{x_2} \varphi u \Vert + \Vert \varphi u \Vert \le \varepsilon \Vert \, ^t P \varphi u \Vert . \end{aligned}$$

With analogous techniques, it follows

$$\begin{aligned}&\Vert \partial _{x_0} \varphi u \Vert + \Vert \partial _{x_1} \varphi u \Vert + \Vert (x_0- \alpha (x') \partial _{x_2} \varphi u \Vert + \Vert \varphi u \Vert \nonumber \\&\quad \le \varepsilon \left( \Vert \varphi \, ^t P u \Vert + \Vert [\, ^t P, \varphi ] u \Vert \right) \nonumber \\&\quad \le \varepsilon \Vert \, ^t Pu \Vert + c \varepsilon \left( \Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert + \Vert u \Vert \right) . \end{aligned}$$

(22)

Then, set $\psi = 1- \varphi$, we rewrite (21) for $\psi u$

$$\begin{aligned}&\Vert \partial _{x_0} \psi u \Vert + \Vert \partial _{x_1} \psi u \Vert + \Vert (x_0- \alpha (x') \partial _{x_2} \psi u \Vert + \Vert \psi u \Vert \nonumber \\&\quad \le \varepsilon \Vert \, ^t P \psi u \Vert \nonumber \\&\quad \le \varepsilon \left( \Vert \psi \, ^t P u \Vert + \Vert [\, ^t P, \psi ] u \Vert \right) \nonumber \\&\quad \le \varepsilon \Vert \, ^t Pu \Vert + c \varepsilon \left( \Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert + \Vert u \Vert \right) . \end{aligned}$$

(23)

Adding (22) and (23), it follows

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0- \alpha (x') \partial _{x_2} u \Vert + \Vert u \Vert \\&\quad \le \varepsilon \Vert \, ^t Pu \Vert + c \varepsilon \left( \Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert + \Vert u \Vert \right) . \end{aligned}$$

Then, taking $\varepsilon$ small enough, the claim is achieved. $\square$

Now, we prove the counterpart results for the Cauchy–Robin problem.

Lemma 3

For every $\varepsilon , \delta >0$ there exists $k>0$ such that

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0 -\alpha (x')) \partial _{x_2} u \Vert + \Vert u \Vert \le \varepsilon \Vert \, {{}^{t}}P u \Vert , \\&\qquad \ \, \, \forall u \in C^{\infty }_0 (\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq I_{k, \delta }, \ L'u \cdot n'|_S=0. \end{aligned}$$

Proof

Integrating by parts, for every $u \in C^{\infty }_0(\overline{\varOmega })$ such that $\mathrm{supp} \, u \subseteq I_{k, \delta }$ and $L'u \cdot n'|_S=0$, we have

$$\begin{aligned}&(\, {{}^{t}}P_2u, x_0 \partial _{x_0} u) + (x_0 \partial _{x_0} u, \, {{}^{t}}P_2u) \nonumber \\&\quad = \Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \Vert (x_0- \alpha (x')) \partial _{x_2} u \Vert ^2 +2 \left( x_0 (x_0- \alpha (x')) \partial _{x_2} u, \partial _{x_2} u \right) \nonumber \\&\quad \quad + 4 \left( x_0 (x_0- \alpha (x')) \partial _{x_2} \alpha (x') \partial _{x_2} u, \partial _{x_0} u \right) \nonumber \\&\quad \quad + 2 \int _S \left( \partial _{x_1} u \cdot n_1 + \beta ^2(x) \partial _{x_2} u \cdot n_2 \right) x_0 \partial _{x_0} u \, d\sigma \nonumber \\&\quad = \Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \Vert (x_0- \alpha (x')) \partial _{x_2} u \Vert ^2 +2 \left( x_0 (x_0- \alpha (x')) \partial _{x_2} u, \partial _{x_2} u \right) \nonumber \\&\quad \quad + 4 \left( x_0 (x_0- \alpha (x')) \partial _{x_2} \alpha (x') \partial _{x_2} u, \partial _{x_0} u \right) \nonumber \\&\quad \quad + 2 \int _S \left( \widetilde{a}_1(x) n_1 + \beta ^2(x) \widetilde{a}_2(x) n_2 \right) u x_0 \partial _{x_0} u \, d\sigma , \end{aligned}$$

(24)

where we toke into account that $L'u \cdot n'|_S=0$. It results

$$\begin{aligned}&\int _S (\widetilde{a}_1(x) n_1 + \beta ^2(x) \widetilde{a}_2(x) n_2) u x_0 \partial _{x_0} u d \sigma \nonumber \\&\quad = \int _{\varOmega _k} x_0 \partial _{x_1}(\widetilde{a}_1(x) u \partial _{x_0} u) dx + \int _{\varOmega _k} x_0 \partial _{x_2}(\widetilde{a}_2(x) (x_0- \alpha (x')) u \partial _{x_0} u) dx \nonumber \\&\quad = \frac{1}{2} (x_0 \partial _{x_1} \widetilde{a}_1(x), \partial _{x_0} u^2) - (\widetilde{a}_1(x) u, \partial _{x_1} u) - (x_0 \partial _{x_0} \widetilde{a}_0(x) u, \partial _{x_1} u) \nonumber \\&\quad \quad + \frac{1}{2} (x_0 \partial _{x_2}( \widetilde{a}_2(x) (x_0 - \alpha (x'))), \partial _{x_0} u^2) - (\widetilde{a}_2(x) (x_0- \alpha (x')) u, \partial _{x_2} u) \nonumber \\&\quad \quad - (x_0 \partial _{x_0}( \widetilde{a}_2(x) (x_0 - \alpha (x'))) u, \partial _{x_2} u). \end{aligned}$$

(25)

Making use of (24), (25) and Lemma 1, it follows

$$\begin{aligned}&(\, {{}^{t}}P_2 u, x_0 \partial _{x_0} u) + (x_0 \partial _{x_0} u, \, {{}^{t}}P_2 u) \nonumber \\&\quad \ge \Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \Vert (x_0 -\alpha (x')) \partial _{x_2} u \Vert ^2 - \frac{1}{\delta } \Vert x_0^{\frac{1}{2}} (x_0 -\alpha (x')) \partial _{x_2} u \Vert ^2 \nonumber \\&\quad \quad - c \big ( \Vert x_0^{\frac{1}{2}} \partial _{x_0} u \Vert ^2 - \Vert x_0^{\frac{1}{2}} \partial _{x_1} u \Vert ^2 - \Vert x_0^{\frac{1}{2}} (x_0 -\alpha (x')) \partial _{x_2} u \Vert ^2 - \Vert x_0^{\frac{1}{2}} u \Vert ^2 \big ). \end{aligned}$$

(26)

On the other hand, we have

$$\begin{aligned}&(\, ^t(P-P_2) u, x_0 \partial _{x_0} u) + (x_0 \partial _{x_0} u, \, ^t(P-P_2) u) \nonumber \\&\quad \ge - c \big ( \Vert x_0^{\frac{1}{2}} \partial _{x_0} u \Vert ^2 - \Vert x_0^{\frac{1}{2}} \partial _{x_1} u \Vert ^2 - \Vert x_0^{\frac{1}{2}} (x_0 -\alpha (x')) \partial _{x_2} u \Vert - \Vert x_0^{\frac{1}{2}} u \Vert ^2 \big ). \end{aligned}$$

(27)

Adding (26) and (27) and using Lemma 1, we obtain

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \Vert (x_0 -\alpha (x')) \partial _{x_2} u \Vert ^2 + \Vert u \Vert ^2 \\&\quad \le \Vert x_0^{\frac{1}{2}} \partial _{x_0} u \Vert \Vert x_0^{\frac{1}{2}} \, {{}^{t}}P u \Vert + \frac{1}{\delta } \Vert x_0^{\frac{1}{2}} (x_0 -\alpha (x')) \partial _{x_2} u \Vert ^2 + c \big ( \Vert x_0^{\frac{1}{2}} \partial _{x_0} u \Vert ^2 \\&\quad \quad + \Vert x_0^{\frac{1}{2}} \partial _{x_1} u \Vert ^2 + \Vert x_0^{\frac{1}{2}} (x_0 -\alpha (x')) \partial _{x_2} u \Vert ^2 + \Vert x_0^{\frac{1}{2}} u \Vert ^2 \big ), \end{aligned}$$

from which the claim follows taking $x_0 \le k$ and k small enough. $\square$

We prove the following result by using similar arguments as above.

Theorem 3

Let us suppose that assumptions (i), (ii) and (iii) hold. It results that for every $\varepsilon > 0$ there exists $k > 0$ such that

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0- \alpha (x')) \partial _{x_2} u \Vert + \Vert u \Vert \le \varepsilon \Vert \, ^t Pu \Vert , \\&\, \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _k =[0,k[ \times \varOmega _0, \ L'u \cdot n'|_S =0. \end{aligned}$$

Proof

We can proceed as in the proof of Theorem 2 making use of Lemma 3 instead of Lemma 2. Moreover, the integral $2 \int _S \left( \partial _{x_1} u \cdot n_1 + \beta ^2(x) \partial _{x_2} u \cdot n_2 \right) x_0 \partial _{x_0} u \, d\sigma$ in (20) has be estimated as in (25). More precisely, using the same arguments in (25), we obtain

$$\begin{aligned}&2 \int _S \left( \partial _{x_1} u \cdot n_1 + \beta ^2(x) \partial _{x_2} u \cdot n_2 \right) x_0 \partial _{x_0} u \, d\sigma \\&\quad \ge -c \left( \Vert x_0^{\frac{1}{2}} \partial _{x_0} u \Vert ^2 + \Vert x_0^{\frac{1}{2}} \partial _{x_1} u \Vert ^2 + \Vert x_0^{\frac{1}{2}} (x_0- \alpha (x')) \partial _{x_2} u \Vert ^2 - \Vert x_0^{\frac{1}{2}} u \Vert ^2 \right) ^2. \end{aligned}$$

As a consequence, the anologous estimate of (20) can be deduced. $\square$

Taking into account Theorems 2 and 3, we deduce easily the next theorem.

Theorem 4

Let us suppose that assumptions (i), (ii) and (iii) hold. It results that for every $\varepsilon > 0$ there exists $k > 0$ such that

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0- \alpha (x')) \partial _{x_2} u \Vert + \Vert u \Vert \le \varepsilon \Vert \, ^t Pu \Vert , \\&\qquad \quad \ \, \forall u \in C_0^{\infty }(\overline{\widetilde{\varOmega }}): \ \mathrm{supp} \, u \subseteq \widetilde{\varOmega }_k =[0,k[ \times \widetilde{\varOmega }_0. \end{aligned}$$

4 A priori estimates array from $\varOmega _0$

Let us set

$$\begin{aligned}&\varOmega _{\overline{x}_0, k, \eta } = \Bigg \{ x \in \overline{\varOmega }: \quad x_0 \in [\overline{x}_0, \overline{x}_0 + k[, \\&\quad \eta \overline{x}_0+ (1+ \eta ) \alpha (x') \le x_0 \le \left( \frac{1}{5} + \eta \right) \alpha (x') + \left( \frac{4}{5} - \eta \right) \overline{x}_0 \Bigg \},\\&\quad \varOmega _{\overline{x}_0, k, \eta }' = \left\{ x \in \varOmega _{\overline{x}_0, k, \eta }: \ |\partial _{x_1} \alpha (x') | \ge 1 \right\} ,\\&\quad \varOmega _{\overline{x}_0, k, \eta , \eta '}' = \left\{ x \in \varOmega _{\overline{x}_0, k, \eta }: \ |\partial _{x_1} \alpha (x') | \ge 1 - \eta ' \right\} , \end{aligned}$$

where $\overline{x}_0 >0$, $\frac{4}{5}< \eta < 1$, $k>0$. Evidently $\varOmega _{\overline{x}_0, k, \eta , \eta '}' \supseteq \varOmega _{\overline{x}_0, k, \eta }'$. Moreover, it is possible to choose k, $\eta$, $\eta '$ such that assumptions (i) and (ii) are verified in $\varOmega _{\overline{x}_0, k, \eta , \eta '}'$.

The following result holds.

Theorem 5

Let us assume that assumptions (i), (ii) and (iii) hold. Then, for every $\overline{x}_0 > 0$ and for every $\varepsilon > 0$ there exists $k > 0$ such that

$$\begin{aligned}&\Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert \le \varepsilon \left( \Vert \, ^t Pu \Vert + \Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert u \Vert \right) , \nonumber \\&\qquad \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0,k, \eta }, \ D'u \cdot n'|_S =0. \end{aligned}$$

(28)

Moreover, for every $\overline{x}_0 > 0$ there exist $k > 0$ and $c > 0$ such that

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert + \Vert u \Vert \le c \Vert \, ^t Pu \Vert , \nonumber \\&\qquad \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0,k, \eta }, \ D'u \cdot n'|_S =0. \end{aligned}$$

(29)

Proof

If the intersection between $\Gamma$ and the plane $x_0 = \overline{x}_0$ is empty, integrating by parts, as in the proof of Lemma 2, in the following inner products

$$\begin{aligned} \left( e^{\tau (x_0 - \overline{x}_0)} \, ^t Pu, \partial _{x_0} u \right) + \left( e^{\tau (x_0 - \overline{x}_0)} \partial _{x_0} u, \, ^t Pu \right) \end{aligned}$$

we easily obtain that for every $\varepsilon > 0$ there exists $k > 0$ such that

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert + \Vert u \Vert \le \varepsilon \Vert \, ^t Pu \Vert , \\&\ \qquad \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0,k}, \ D'u \cdot n'|_S =0. \end{aligned}$$

If the intersection between $\Gamma$ and the plane $x_0 = \overline{x}_0$ is nonempty, we proceed as follows. We remark that the intersection between $\varOmega _{\overline{x}_0,k, \eta }$ and the plane $x_0 = \overline{x}_0$ has zero measure. Moreover, the intersection between $\varOmega _{\overline{x}_0,k, \eta }$ and the surface S is empty or has zero measure, for k small enough. Let us set

$$\begin{aligned} g_{\overline{x}_0}(x') = \frac{\alpha (x') - \overline{x}_0}{\partial _{x_1} \alpha (x')}, \quad h_{\overline{x}_0}(x')= \left| \frac{\partial ^2_{x_1} \alpha (x') (\alpha (x') - \overline{x}_0)}{(\partial _{x_1} \alpha (x'))^2} \right| . \end{aligned}$$

Let us observe that, for a fixed $\varepsilon >0$, there exists $k>0$ such that $|x_0 - \overline{x}_0| < \varepsilon$, $|\alpha (x') - \overline{x}_0| < \varepsilon$, $|g_{\overline{x}_0}(x')| < \varepsilon$, $|h_{\overline{x}_0}(x')| < \varepsilon$, for every $x \in \varOmega _{\overline{x}_0, k,h}$. Let $A_{\overline{x}_0} u = g_{\overline{x}_0}(x') \partial _{x_1} u + (x_0 - \overline{x}_0) \partial _{x_0} u$. Integrating by parts in the following inner products

$$\begin{aligned} (\, ^tPu, A_{\overline{x}_0} u) + (A_{\overline{x}_0} u, \, ^tPu) \end{aligned}$$

and since the intersections between $\varOmega _{\overline{x}_0,k, \eta }$ and the plane $x_0 = \overline{x}_0$ and the surface S, respectively, have zero measure, we have

$$\begin{aligned}&(h_{\overline{x}_0} (x') \partial _{x_0} u, \partial _{x_0} u) + (h_{\overline{x}_0} (x') \partial _{x_1} u, \partial _{x_1} u) + ((4-h_{\overline{x}_0}(x')) (x_0 - \alpha (x'))^2 \partial _{x_2}u, \partial _{x_2}u) \\&\quad \le 4 |(g_{\overline{x}_0}(x') (x_0 - \alpha (x')) \partial _{x_2} \alpha (x') \partial _{x_2} u, \partial _{x_1} u )| \\&\quad \quad + 4 |((x_0 - \overline{x}_0) (x_0 - \alpha (x')) \partial _{x_2} \alpha (x') \partial _{x_2} u, \partial _{x_0} u )| \\&\quad \quad + 2 |( (x_0 - \alpha (x'))^2 \partial _{x_2} g_{\overline{x}_0}(x') \partial _{x_2} u, \partial _{x_2} u )| + 2 | \mathrm{Re} (\, ^t P_1u, A_{\overline{x}_0}u) | \\&\quad \quad + 2 | \mathrm{Re} (\, ^t P_0u, A_{\overline{x}_0}u) | + | (\, ^t Pu, A_{\overline{x}_0}u) | + |( A_{\overline{x}_0}u, \, ^t Pu) |, \\&\quad \quad \ \forall u \in C^{\infty }_0(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0, k, h}, \ D'u \cdot n'|_S=0. \end{aligned}$$

This implies

$$\begin{aligned}&(4- \varepsilon ) \Vert (x_0 - \alpha (x')) \partial _{x_2}u \Vert ^2 \\&\quad \le c \varepsilon \left( \Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert ^2 + \Vert u \Vert ^2 \right) + \varepsilon \Vert \, ^t Pu \Vert ^2,\\&\qquad \qquad \qquad \forall u \in C^{\infty }_0(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0, k, \eta , \eta '}, \ D'u \cdot n'|_S=0. \end{aligned}$$

Hence, taking $\varepsilon$ small enough, we deduce

$$\begin{aligned}&\Vert (x_0 - \alpha (x')) \partial _{x_2}u \Vert \le \varepsilon \left( \Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert \right) + \varepsilon \Vert \, ^t Pu \Vert ,\nonumber \\&\quad \forall u \in C^{\infty }_0(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0, k, \eta , \eta '}, \ D'u \cdot n'|_S=0. \end{aligned}$$

(30)

For $\varepsilon$ small enough, it follows

$$\begin{aligned}&\Vert (x_0 - \alpha (x')) \partial _{x_2}u \Vert \le \varepsilon \left( \Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert \right) + \varepsilon \Vert \, ^t Pu \Vert ,\nonumber \\&\quad \forall u \in C^{\infty }_0(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0, k, \eta , \eta '}', \ D'u \cdot n'|_S=0. \end{aligned}$$

(31)

Now, we obtain (29) integrating by parts in the following inner products:

$$\begin{aligned} (\, ^tPu, Au) + (Au, \, ^t Pu). \end{aligned}$$

In fact, we have

$$\begin{aligned}&h_1 \Vert \partial _{x_0} u \Vert ^2 + h_1 \Vert \partial _{x_1} u \Vert ^2 + (4-h_2) \Vert (x_0-\alpha (x')) \partial _{x_2} u \Vert ^2 \nonumber \\&\quad \le c \Vert (x_0-\alpha (x')) \partial _{x_2} u \Vert \left( \Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert \right) + c \Vert (x_0-\alpha (x')) \partial _{x_0} u \Vert \nonumber \\&\quad \quad + \Vert (x_0-\alpha (x')) \partial _{x_1} u \Vert +c \Vert (x_0-\alpha (x')) \partial _{x_1} u \Vert \nonumber \\&\quad \quad + c \Vert u \Vert \left( \Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert \right) + c \Vert \, ^t Pu \Vert \left( \Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert \right) , \nonumber \\&\qquad \ \forall u \in C^{\infty }_0(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0, k, \eta , \eta '}', \ \frac{4}{5} \le h < 1, \ D'u \cdot n'|_S=0. \end{aligned}$$

(32)

Making use of (32) and Lemma 1, it follows

$$\begin{aligned}&\quad \Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert + \Vert u \Vert \le c \Vert \, ^t Pu \Vert , \nonumber \\&\forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0,k, \eta ,\eta '}', \ \frac{4}{5} \le h < 1, \ D'u \cdot n'|_S =0. \end{aligned}$$

(33)

If $\mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0,k, \eta } {\setminus } \varOmega _{\overline{x}_0,k, \eta ,\eta '}'$, we integrate by parts in the inner products

$$\begin{aligned}&(\, ^t Pu, (x_0-\overline{x}_0) \partial _{x_0}u) + ((x_0-\overline{x}_0) \partial _{x_0}u, \, ^t Pu) \\&\quad \le \Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert ^2 \\&\qquad + 2((x_0- \overline{x}_0) (x_0 - \alpha (x')) \partial _{x_2} u, \partial _{x_2} u) \\&\qquad + 4 ((x_0- \overline{x}_0) (x_0 - \alpha (x')) \partial _{x_2} \alpha (x') \partial _{x_2} u, \partial _{x_0} u) \\&\qquad + ( \, ^t (P-P_2)u, (x_0 - \overline{x}_0) \partial _{x_0} u) \\&\qquad + ( (x_0 - \overline{x}_0) \partial _{x_0} u, \, ^t (P-P_2)u). \end{aligned}$$

Then, for k small enough and taking into account Lemma 1, it follows

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert ^2 + \Vert u \Vert ^2 \\&\quad \le \varepsilon \Vert \, ^t Pu \Vert ^2 + \varepsilon \left( \Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert ^2 + \Vert u \Vert ^2 \right) \\&\quad \quad + \varepsilon \Vert (x_0 - \alpha (x'))^{\frac{1}{2}} \partial _{x_2} u \Vert ^2. \end{aligned}$$

Hence, for $\varepsilon$ small enough, it results

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert ^2 + \Vert u \Vert ^2 \nonumber \\&\quad \le \varepsilon \left( \Vert \, ^t Pu \Vert ^2 + \Vert (x_0 - \alpha (x'))^{\frac{1}{2}} \partial _{x_2} u \Vert ^2 \right) . \end{aligned}$$

(34)

We estimate $\Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert$, as done in (18). Computing the inner products

$$\begin{aligned} (\, ^t Pu, \partial _{x_0} u ) + ( \partial _{x_0} u , \, ^t Pu), \quad \forall u \in C^{\infty }_0(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0,k, \eta } {\setminus } \varOmega _{\overline{x}_0,k, \eta }', \end{aligned}$$

and proceeding with the same technique, we deduce

$$\begin{aligned}&\Vert (x_0 - \alpha (x'))^{\frac{1}{2}} \partial _{x_2} u \Vert ^2 + \int _{\Gamma } \left[ (\partial _{x_0} u)^2 + 2 \partial _{x_1} \alpha (x') \partial _{x_0} u \partial _{x_1} u + (\partial _{x_1} u)^2 \right] d \sigma \nonumber \\&\quad \le \Vert \, ^t Pu \Vert ^2 + | ((x_0 - \alpha (x')) \partial _{x_2} u, \partial _{x_0} u) |. \end{aligned}$$

(35)

Making use of (34) and (35), we have

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert ^2 + \Vert u \Vert ^2 \le \varepsilon \Vert \, ^t Pu \Vert ^2, \nonumber \\&\qquad \qquad \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0,k, \eta } {\setminus } \varOmega _{\overline{x}_0,k, \eta }'. \end{aligned}$$

(36)

Since $\Gamma \cap \varOmega _{\overline{x}_0}$ has zero measure, without lost generality, we consider $u \in C_0^{\infty }(\overline{\varOmega })$ with $\mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0,k} {\setminus } (\Gamma \cap \varOmega _{\overline{x}_0})$, where $\varOmega _{\overline{x}_0} = \{ x \in \overline{\varOmega }: \ x_0 = \overline{x}_0 \}$. Now, let $\varphi \in C_0^{\infty }(\overline{\varOmega })$ such that $\varphi \equiv 1$ on $\varOmega _{\overline{x}_0,k, \eta }'$ and $\varphi \equiv 0$ on $\varOmega _{\overline{x}_0,k, \eta } {\setminus } \varOmega _{\overline{x}_0,k, \eta , \eta '}'$. If $u \in C_0^{\infty }(\overline{\varOmega })$ such that $\mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0,k, \eta }$ and $D'u \cdot n'|_S=0$, we can apply (31) to $\varphi u$ obtaining

$$\begin{aligned} \Vert (x_0- \alpha (x')) \partial _{x_2} \varphi u \Vert \le \varepsilon ( \Vert \partial _{x_0} \varphi u \Vert + \Vert \partial _{x_1} \varphi u \Vert ) + \varepsilon \Vert \, ^t P \varphi u \Vert , \end{aligned}$$

(37)

then it follows

$$\begin{aligned} \Vert \partial _{x_0} \varphi u \Vert + \Vert \partial _{x_1} \varphi u \Vert + \Vert (x_0- \alpha (x')) \partial _{x_2} \varphi u \Vert + \Vert \varphi u \Vert \le c \Vert \, ^t P \varphi u \Vert . \end{aligned}$$

(38)

On the other hand, by (36), it results

$$\begin{aligned}&\Vert \partial _{x_0} (1- \varphi ) u \Vert + \Vert \partial _{x_1} (1- \varphi ) u \Vert + \Vert (x_0- \alpha (x')) \partial _{x_2} (1- \varphi ) u \Vert \nonumber \\&\quad + \Vert (1- \varphi ) u \Vert \le c \Vert \, ^t P (1- \varphi ) u \Vert . \end{aligned}$$

(39)

Taking into account (37), (39) and for $\varepsilon$ small enough, we have

$$\begin{aligned} \Vert (x_0- \alpha (x')) \partial _{x_2} u \Vert \le \varepsilon ( \Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert \, ^t P u \Vert ). \end{aligned}$$

As a consequence, (28) holds. Moreover, by (38) and (39) and for $\varepsilon$ small enough, we obtain

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0- \alpha (x')) \partial _{x_2} u \Vert + \Vert u \Vert \le \varepsilon \Vert \, ^t P u \Vert , \\&\qquad \forall u \in C^{\infty }_0(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0,k, \eta }, \ D'u \cdot n'|_S=0, \end{aligned}$$

and, hence, (29) is also proved. $\square$

The following results holds.

Theorem 6

For every $\overline{x}_0>0$ and $\varepsilon >0$ there exists $k >0$ such that for every $\eta \in \left] \frac{4}{5}, 1 \right[$ it results

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert + \Vert u \Vert \le c \Vert \, ^t Pu \Vert , \nonumber \\&\quad \ \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0,k} {\setminus } \varOmega _{\overline{x}_0,k, \eta }, \ D'u \cdot n'|_S =0. \end{aligned}$$

(40)

Proof

Itegrating by parts in the following inner products

$$\begin{aligned} (\, ^t Pu, (x_0 - \overline{x}_0) \partial _{x_0}u) + ((x_0 - \overline{x}_0) \partial _{x_0}u, \, ^t Pu), \end{aligned}$$

we obtain

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \frac{1}{2} \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert ^2 + \frac{1}{2} \left( (x_0 - \alpha (x'))^2 \partial _{x_2} u, \partial _{x_2} u \right) \\&\qquad + 2 \left( (x_0 - \overline{x}_0) (x_0 - \alpha (x')) \partial _{x_2} u, \partial _{x_2} u \right) \\&\quad \le \varepsilon \left( \Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \Vert (x_0 - \alpha (x')) \partial _{x_0} u \Vert ^2 + c \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert ^2 + \Vert \, ^t Pu \Vert ^2 \right) . \end{aligned}$$

For $c |x_0 - \overline{x}_0| < \varepsilon$, taking into account that

$$\begin{aligned} \frac{1}{2} \left( (x_0 - \alpha (x'))^2 \partial _{x_2} u, \partial _{x_2} u \right) + 2 \left( (x_0 - \overline{x}_0) (x_0 - \alpha (x')) \partial _{x_2} u, \partial _{x_2} u \right) \ge 0, \end{aligned}$$

in $\varOmega _{\overline{x}_0, k} {\setminus } \varOmega _{\overline{x}_0,k,\eta }$, for $\frac{4}{5}< \eta < 1$, by using Lemma 1 and for $\varepsilon$ small enough, there exists $k>0$ such that

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert + \Vert u \Vert \le \varepsilon \Vert \, ^t Pu \Vert , \\&\forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0,k} {\setminus } \varOmega _{\overline{x}_0,k, \eta , \eta '}', \ D'u \cdot n'|_S =0. \end{aligned}$$

As a consequence, the claim is achieved. $\square$

Hence, we obtain the following theorem.

Theorem 7

Let us suppose that assumptions (i), (ii) and (iii) hold. Then, for every $\overline{x}_0>0$ and for every $\varepsilon >0$ there exists $k >0$ such that

$$\begin{aligned}&\Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert \le \varepsilon \left( \Vert \, ^t Pu \Vert + \Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert u \Vert \right) , \nonumber \\&\qquad \quad \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0,k}, \ D'u \cdot n'|_S =0. \end{aligned}$$

(41)

Moreover, for every $\overline{x}_0 >0$ there exist $k>0$ and $c>0$ such that

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert + \Vert u \Vert \le c \Vert \, ^t Pu \Vert , \nonumber \\&\qquad \ \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0,k}, \ D'u \cdot n'|_S =0. \end{aligned}$$

(42)

Proof

Let $\varphi \in C_0^{\infty }(\overline{\varOmega })$ such that $\varphi \equiv 1$ on $\varOmega _{\overline{x}_0, k, \eta _1} \cap \mathrm{supp} \, u$, and let $\psi \in C_0^{\infty }(\overline{\varOmega })$ such that $\psi \equiv 1$ on $\varOmega _{\overline{x}_0, k} {\setminus } \varOmega _{\overline{x}_0, k, \eta _1}$ and $\psi \equiv 0$ on $\varOmega _{\overline{x}_0, k} {\setminus } \varOmega _{\overline{x}_0, k, \eta _2}$, with $\frac{4}{5}< \eta _1< \eta _2 < 1$.

Applying Theorem 5 to $\varphi u$, Theorem 6 to $\psi u$ and adding the obtained inequalities, the claims are achieved $\square$

With analogous proof of Theorem 5, we are able to establish the following result.

Theorem 8

Let us suppose that assumptions (i), (ii) and (iii) hold. It results that for every $\overline{x}_0>0$ and for every $\varepsilon >0$ there exists $k >0$ such that

$$\begin{aligned}&\Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert \le \varepsilon \left( \Vert \, ^t Pu \Vert + \Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert u \Vert \right) , \\&\qquad \ \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0,k, \eta }, \ L'u \cdot n'|_S =0. \end{aligned}$$

Moreover, for every $\overline{x}_0 >0$ there exist $k>0$ and $c>0$ such that

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert + \Vert u \Vert \le c \Vert \, ^t Pu \Vert ,\\&\qquad \ \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0,k, \eta }, \ L'u \cdot n'|_S =0. \end{aligned}$$

Now, we prove a useful estimate.

Theorem 9

For every $\overline{x}_0>0$ and $\varepsilon >0$ there exists $k>0$ such that, for every $\eta \in \left] \frac{4}{5}, 1 \right[$, it results

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0 -\alpha (x')) \partial _{x_2} u \Vert + \Vert u \Vert \le c \Vert \, {{}^{t}}P u \Vert , \\&\forall u \in C^{\infty }_0 (\varOmega _{\overline{x}_0,k}): \ \mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0,k} {\setminus } \varOmega _{\overline{x}_0,k, \eta }, \ L'u \cdot n'|_S =0. \end{aligned}$$

Proof

Integrating by parts, we obtain

$$\begin{aligned}&(\, ^tPu, (x_0- \overline{x}_0) \partial _{x_0} u) + ((x_0- \overline{x}_0) \partial _{x_0} u, \, ^t Pu) \nonumber \\&\quad = \Vert \partial _{x_0} u \Vert ^2 + \frac{1}{2} \Vert \partial _{x_1} u \Vert ^2 + \frac{1}{2} \Vert (x_0-\alpha (x')) \partial _{x_2} u \Vert ^2 \nonumber \\&\qquad + \frac{1}{2} ((x_0 - \alpha (x'))^2 \partial _{x_2} u, \partial _{x_2} u) \nonumber \\&\qquad + 2 ( (x_0 - \overline{x}_0) (x_0-\alpha (x')) \partial _{x_2} u, \partial _{x_2} u ) \nonumber \\&\qquad + 4 ( (x_0 - \overline{x}_0) (x_0-\alpha (x')) \partial _{x_2} \alpha (x') \partial _{x_2} u, \partial _{x_0} u ) \nonumber \\&\qquad + 2 \int _S (\partial _{x_1} u \cdot n_1+ \beta ^2(x) \partial _{x_2} u \cdot n_2) (x_0 - \overline{x}_0) \partial _{x_0} u d \sigma . \end{aligned}$$

(43)

On the other hand, as done in (25), we have

$$\begin{aligned}&\int _S ( \widetilde{a}_1(x) n_1+ \beta (x) \widetilde{a}_2(x) n_2) u x_0 \partial _{x_0} u d \sigma \nonumber \\&\quad = \frac{1}{2} \left( x_0 \partial _{x_1} \widetilde{a}_1(x), \partial _{x_0} u^2 \right) - \left( \widetilde{a}_1(x) u, \partial _{x_1} u \right) - \left( x_0 \partial _{x_0} \widetilde{a}_1(x), \partial _{x_1} u \right) \nonumber \\&\qquad + \frac{1}{2} (x_0 \partial _{x_2}((x_0 - \alpha (x')) \widetilde{a}_2(x)), \partial _{x_0} u^2) - (( x_0 - \alpha (x')) \widetilde{a}_2(x) u, \partial _{x_2} u) \nonumber \\&\qquad - (x_0 \partial _{x_0} ((x_0- \alpha (x')) \widetilde{a}_2(x) ) u, \partial _{x_2} u ). \end{aligned}$$

(44)

Making use of (43) and (44) and taking into acount

$$\begin{aligned} \frac{1}{2} \left( (x_0 - \alpha (x'))^2 \partial _{x_2} u, \partial _{x_2} u \right) + 2 \left( (x_0 - \overline{x}_0) (x_0- \alpha (x')) \partial _{x_2} u, \partial _{x_2} u \right) \ge 0, \end{aligned}$$

for $\eta \in \left] \frac{4}{5}, 1 \right[$, we deduce

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert ^2 + \Vert \partial _{x_1} u \Vert ^2 + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert ^2 \\&\quad \le \big ( \Vert (x_0-\overline{x}_0)^{\frac{1}{2}} \partial _{x_0} u \Vert ^2 + \Vert (x_0-\overline{x}_0)^{\frac{1}{2}} \partial _{x_1} u \Vert ^2 + \Vert (x_0-\overline{x}_0)^{\frac{1}{2}} (x_0 - \alpha (x')) \partial _{x_2} u \Vert ^2 \\&\qquad + \Vert (x_0-\overline{x}_0)^{\frac{1}{2}} u \Vert ^2 + \Vert (x_0-\overline{x}_0)^{\frac{1}{2}} \, ^t Pu \Vert ^2 \big ) . \end{aligned}$$

Then, by using Lemma 1, the claim follows taking $|x_0 - \overline{x}_0|>k$, with k small enough. $\square$

Proceeding as in the proof of Theorem 7, we obtain the following theorem.

Theorem 10

Let us suppose that assumptions (i), (ii) and (iii) hold. Then, for every $\overline{x}_0>0$ and for every $\varepsilon >0$ there exists $k >0$ such that

$$\begin{aligned}&\Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert \le \varepsilon \left( \Vert \, ^t Pu \Vert + \Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert u \Vert \right) , \\&\qquad \quad \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0,k}, \ L'u \cdot n'|_S =0. \end{aligned}$$

Moreover, for every $\overline{x}_0 >0$ there exist $k>0$ and $c>0$ such that

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert + \Vert u \Vert \le c \Vert \, ^t Pu \Vert , \\&\qquad \ \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _{\overline{x}_0,k}, \ L'u \cdot n'|_S =0. \end{aligned}$$

As a consequence, the next result holds.

Theorem 11

Let us suppose that assumptions (i), (ii) and (iii) hold. Then, for every $\overline{x}_0>0$ and for every $\varepsilon >0$ there exists $k >0$ such that

$$\begin{aligned}&\Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert \le \varepsilon \left( \Vert \, ^t Pu \Vert + \Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert u \Vert \right) , \\&\qquad \ \forall u \in C_0^{\infty }(\overline{\widetilde{\varOmega }}): \ \mathrm{supp} \, u \subseteq \widetilde{\varOmega }_{\overline{x}_0,k}= [\overline{x}_0, \overline{x}_0+k[ \times \widetilde{\varOmega }. \end{aligned}$$

Moreover, for every $\overline{x}_0 >0$ there exist $k>0$ and $c>0$ such that

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert + \Vert \partial _{x_1} u \Vert + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert + \Vert u \Vert \le c \Vert \, ^t Pu \Vert , \\&\qquad \qquad \qquad \forall u \in C_0^{\infty }(\overline{\widetilde{\varOmega }}): \ \mathrm{supp} \, u \subseteq \widetilde{\varOmega }_{\overline{x}_0,k}. \end{aligned}$$

5 A priori estimates in Sobolev spaces with $s<0$

First of all, let us obtain a priori estimate in Sobolev spaces with $s < 0$ by using the theory of pseudodifferental operators.

Theorem 12

Let us suppose that assumptions (i), (ii) and (iii) hold. For every $\overline{x}_0 \ge 0$ and for every $s < 0$ there exist $k > 0$ and $c > 0$ such that

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert _{H^{0,s}} + \Vert \partial _{x_1} u \Vert _{H^{0, s}} + \Vert (x_0 -\alpha (x')) \partial _{x_2} u \Vert _{H^{0,s}} + \Vert u \Vert _{H^{0,s}} \le c \Vert \, ^t Pu \Vert _{H^{0,s}},\nonumber \\&\qquad \ \forall u \in C^{\infty }_0(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq [\overline{x}_0, \overline{x}_0+k[ \times \varOmega _0 = \varOmega _{\overline{x}_0,k}, \ D'u \cdot n'|_S=0. \end{aligned}$$

(45)

Proof

Let $u \in C^{\infty }_0(\overline{\widetilde{\varOmega }}_{\overline{x}_0,k})$, let $\varphi \in C^{\infty }_0(\overline{\widetilde{\varOmega }})$ such that $\mathrm{supp} \, \varphi \subseteq \widetilde{\varOmega }_{\overline{x}_0,k}$, $D' \varphi \cdot n'|_S=0$ and $\varphi \equiv 1$ on the support of the projection of u on the plane $x_0= \overline{x}_0$. Let us set $v_s= \varphi (x') A_s u$. Applying the claims of Theorems 4 and 11 if we have $\overline{x}_0 = 0$ or $\overline{x}_0 \ne 0$, respectively, it results

$$\begin{aligned} \Vert \partial _{x_0} v_s \Vert + \Vert \partial _{x_1} v_s \Vert + \Vert (x_0 -\alpha (x')) \partial _{x_2} v_s \Vert + \Vert v_s \Vert \le c \Vert \, ^t P v_s \Vert . \end{aligned}$$

(46)

We remark that

$$\begin{aligned}&\Vert \partial _{x_0} v_s \Vert = \Vert \partial _{x_0} \varphi (x') A_s u \Vert \nonumber \\&\quad = \Vert \varphi (x') A_s \partial _{x_0} u \Vert \nonumber \\&\quad = \Vert A_s \varphi (x') \partial _{x_0} u + [ \varphi , A_s] \partial _{x_0} u \Vert \nonumber \\&\quad \ge \Vert A_s \partial _{x_0} u \Vert - \Vert R \partial _{x_0} u \Vert \nonumber \\&\quad \ge \Vert \partial _{x_0} u \Vert _{H^{0,s}} - \Vert R \partial _{x_0} u \Vert , \end{aligned}$$

(47)

where $R= [ \varphi , A_s] u$ is a regularizing operator with respect to the variable $x'$.

On the other hand, making use of Lemma 1 and taking into account that

$$\begin{aligned}&\partial _{x_0}^2 u = - \, ^tPu + \partial _{x_1}^2 u + (x_0 - \alpha (x'))^2 \partial _{x_2}^2 u + 4 (x_0 - \alpha (x')) \partial _{x_2} \alpha \, \partial _{x_2} u \\&\quad - \frac{1}{i} \sum _{j=0}^2 a(x) \partial _{x_j} u - \frac{1}{i} \sum _{j=0}^2 \partial _{x_j} a(x) u - 2(\partial _{x_2} \alpha )^2 u +b(x) u, \end{aligned}$$

we obtain

$$\begin{aligned} \Vert R \partial _{x_0} u \Vert\le & {} \Vert R (x_0- \overline{x}_0) \partial ^2_{x_0} u \Vert \nonumber \\\le & {} c \Vert (x_0- \overline{x}_0) \, ^t Pu \Vert _{H^{0,s}} + c \Vert (x_0- \overline{x}_0) \partial _{x_0} u \Vert _{H^{0,s}} \nonumber \\&+ c \Vert (x_0- \overline{x}_0) u \Vert _{H^{0,s}}. \end{aligned}$$

(48)

Making use of (47) and (48), it follows

$$\begin{aligned}&\Vert \partial _{x_0} v_s \Vert \nonumber \\&\quad \ge \Vert \partial _{x_0} u \Vert _{H^{0,s}} - c \Vert (x_0- \overline{x}_0) \, ^t Pu \Vert _{H^{0,s}} - c \Vert (x_0- \overline{x}_0) \partial _{x_0} u \Vert _{H^{0,s}} \nonumber \\&\quad \quad - c \Vert (x_0- \overline{x}_0) u \Vert _{H^{0,s}}. \end{aligned}$$

(49)

Then choosing k small enough and $|x_0 - \overline{x}_0| <k$, it results

$$\begin{aligned} \Vert \partial _{x_0} v_s \Vert \ge \Vert \partial _{x_0} u \Vert _{H^{0,s}} - c \Vert \, ^t Pu \Vert _{H^{0,s}}. \end{aligned}$$

Proceeding with the same technique, we easily obtain that

$$\begin{aligned} \Vert \partial _{x_1} v_s \Vert\ge & {} \Vert \partial _{x_1} u \Vert _{H^{0,s}} - c \Vert u \Vert _{H^{0,s}} \nonumber \\\ge & {} \Vert \partial _{x_1} u \Vert _{H^{0,s}} - c \Vert (x_0-\overline{x}_0) \partial _{x_0} u \Vert _{H^{0,s}}, \end{aligned}$$

(50)

where we applied Lemma 1. Adding (49) and (50), for $|x_0 - \overline{x}_0|<k$ with k small enough, we deduce

$$\begin{aligned} \Vert \partial _{x_0} v_s \Vert + \Vert \partial _{x_1} v_s \Vert \ge \Vert \partial _{x_0} u \Vert _{H^{0,s}} + \Vert \partial _{x_1} u \Vert _{H^{0,s}} - c \Vert \, ^t Pu \Vert _{H^{0,s}}. \end{aligned}$$

(51)

With analogous computations, we have

$$\begin{aligned} \Vert (x_0 - \alpha (x')) \partial _{x_2} v_s \Vert \ge \Vert A_s (x_0 - \alpha (x')) \varphi (x') \partial _{x_2} u \Vert - \Vert R u \Vert - \Vert B_{s-1} \partial _{x_1} u \Vert , \end{aligned}$$

(52)

where $B_{s-1}$ is a pseudodifferential operator of order $s-1$. As a consequence, $B_{s-1} \partial _{x_2}$ is a pseudodifferential operator of order s. By using the continuity property of pseudodifferential operators, it results

$$\begin{aligned} \Vert (x_0 - \alpha (x')) \partial _{x_2} v_s \Vert \ge \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert _{H^{0,s}} - c \Vert (x_0 - \overline{x}_0) \partial _{x_1} u \Vert _{H^{0,s}}. \end{aligned}$$

(53)

Adding (51) and (53), for $|x_0- \overline{x}_0|<k$ with k small enough, it follows

$$\begin{aligned}&\Vert \partial _{x_0} v_s \Vert + \Vert \partial _{x_1} v_s \Vert + \Vert (x_0 - \alpha (x')) \partial _{x_2} v_s \Vert \nonumber \\&\quad \ge \Vert \partial _{x_0} u \Vert _{H^{0,s}} + \Vert \partial _{x_1} u \Vert _{H^{0,s}} + \Vert (x_0- \alpha (x')) \partial _{x_2} u \Vert _{H^{0,s}} - c \Vert \, ^t Pu \Vert _{H^{0,s}}. \end{aligned}$$

(54)

Finally we have

$$\begin{aligned} \Vert \, ^t Pv_s \Vert\le & {} \Vert A_s \, ^t P u \Vert + \Vert R' \, ^t P u \Vert + \Vert \varphi (x') [\, ^t P, A_s] u \Vert + \Vert R u \Vert \nonumber \\\le & {} \Vert \, ^t P u \Vert _{H^{0,s}} + \Vert (x_0- \overline{x}_0) \partial _{x_0} u \Vert _{H^{0,s}} + \Vert \varphi (x') [\, ^t P, A_s] u \Vert , \end{aligned}$$

(55)

where $R'$ and R are regularizing operators. Moreover, it results

$$\begin{aligned} \varphi (x') [\, ^t P, A_s] u = \varphi (x') [\, ^t P_2, A_s] u + \varphi (x') [\, ^t P_1, A_s] u + \varphi (x') [\, ^t P_0, A_s] u. \end{aligned}$$

(56)

Evidently, we have

$$\begin{aligned}{}[\, ^t P_2, A_s] u = B_{s+1} u + B_s u, \end{aligned}$$

where $B_{s+1}$ and $B_s$ are pseudodifferential operators of order $s+1$ and s, respectively. The principal symbol of $B_{s+1}$ is

$$\begin{aligned} b(x, \xi ')= & {} - \frac{1}{i} \left( 2 (x_0-\alpha (x')) (- \partial _{x_1} \alpha (x')) \xi ^2_2 \right) \varphi (x') \partial _{\xi _1}(1+ |\xi '|^2)^{\frac{s}{2}} \\&- \frac{1}{i} \left( 2 (x_0-\alpha (x')) (- \partial _{x_2} \alpha (x')) \xi ^2_2 \right) \varphi (x') \partial _{\xi _2}(1+ |\xi '|^2)^{\frac{s}{2}} \end{aligned}$$

As a consequence, $B_{s+1} u = (x_0-\alpha (x')) \partial _{x_2} B_s''u+B_s' {u}$, where $B_s''$ and $B_s'$ are pseudodifferential operators of order s.

Making use of (41), we obtain

$$\begin{aligned} \Vert B_{s+1} u \Vert= & {} \Vert (x_0- \alpha (x')) \partial _{x_2} B''_su + B_{s}' {u} \Vert \\\le & {} \varepsilon \left( \Vert \, ^t P B_s'' u \Vert + \Vert \partial _{x_0} B_s''u \Vert + \Vert \partial _{x_1} B_s'' u \Vert + \Vert B_s' u \Vert \right) +\Vert B_{s}' {u}\Vert . \end{aligned}$$

Then, it follows

$$\begin{aligned}&\Vert \varphi (x') [^t P_2, A_s] u \Vert \le \varepsilon \big ( \Vert \, ^t P B_s'' u \Vert + \Vert \partial _{x_0} B_s''u \Vert \nonumber \\&\qquad + \Vert \partial _{x_1} B_s'' u \Vert + \Vert B_s'' u \Vert \big ) + \Vert B_s u \Vert + \Vert B_s' u \Vert \nonumber \\&\qquad \le \varepsilon \big ( \Vert \, ^t P u \Vert _{H^{0,s}} + \Vert \, [^t P, B_s''] u \Vert + \Vert \partial _{x_0}u \Vert _{H^{0,s}} + \Vert [\partial _{x_0}, B_s''] u \Vert + \Vert \partial _{x_1} u \Vert _{H^{0,s}} \nonumber \\&\qquad \Vert [\partial _{x_1}, B_s''] u \Vert + \Vert u \Vert _{H^{0,s}} \big ) + \Vert B_s (x_0 - \overline{x}_0) \partial _{x_0} u \Vert + \Vert B_s' (x_0 - \overline{x}_0) \partial _{x_0} u \Vert \nonumber \\&\qquad \le \varepsilon \big ( \Vert \, ^t P u \Vert _{H^{0,s}} + \Vert \partial _{x_0}u \Vert _{H^{0,s}} + \Vert \partial _{x_1} u \Vert _{H^{0,s}} + \Vert u \Vert _{H^{0,s}} \big ), \end{aligned}$$

(57)

where we considered $|x_0-\overline{x}_0|< k < \varepsilon$.

On the other hand, it results

$$\begin{aligned} \varphi (x') [\, ^t P_1, A_s] u = B_{s-1} \partial _{x_0} u + B_{s} u + B_{s-1} u. \end{aligned}$$

Taking into account Lemma 1, we have

$$\begin{aligned}&\Vert \varphi (x') [\, ^t P_1, A_s ] \Vert \le \Vert B_{s-1} \partial _{x_0} u \Vert + \Vert B_{s} u \Vert + \Vert B_{s-1} u \Vert \nonumber \\&\qquad \le c \Vert (x_0 - \overline{x}_0) B_{s-1} \partial _{x_0}^2 u \Vert + c \Vert u \Vert _{H^{0,s}} \nonumber \\&\qquad \le c \Vert (x_0 - \overline{x}_0) B_{s-1} P u \Vert + c \Vert (x_0 - \overline{x}_0) B_{s} \partial _{x_1} u \Vert \nonumber \\&\qquad + c \Vert (x_0 - \overline{x}_0) B_{s}' (x_0 - \alpha (x')) \partial _{x_2} u \Vert \nonumber \\&\qquad + c \Vert (x_0 - \overline{x}_0) B_{s}'' u \Vert + c \Vert (x_0 - \overline{x}_0) \partial _{x_0} u \Vert _{H^{0,s}} \nonumber \\&\qquad \le c \Vert (x_0 - \overline{x}_0) P u \Vert _{H^{0,s}} + c \Vert (x_0 - \overline{x}_0) \partial _{x_1} u \Vert _{H^{0,s}} \nonumber \\&\qquad + c \Vert (x_0 - \overline{x}_0) (x_0 - \alpha (x')) \partial _{x_2} u \Vert _{H^{0,s}} \nonumber \\&\qquad + c \Vert (x_0 - \overline{x}_0) u \Vert _{H^{0,s}} + c \Vert (x_0 - \overline{x}_0) \partial _{x_0} u \Vert _{H^{0,s}} \nonumber \\&\qquad \le \varepsilon ( \Vert P u \Vert _{H^{0,s}} + \Vert \partial _{x_0} u \Vert _{H^{0,s}} + \Vert \partial _{x_1} u \Vert _{H^{0,s}} \nonumber \\&\qquad + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert _{H^{0,s}} + \Vert u \Vert _{H^{0,s}} ), \end{aligned}$$

(58)

where $B_s$, $B_s'$, $B_s''$ are pseudodifferential operators of order s, $B_{s-1}$ is a pseudodifferential operator of order $s-1$ and we supposed that $0< |x_0- \overline{x}_0| < \frac{\varepsilon }{c}$.

It is easy to obtain

$$\begin{aligned} \Vert \varphi (x') [\, ^t P_0, A_s ] u \Vert\le & {} c \Vert u \Vert _{H^{0,s}} \nonumber \\\le & {} c \Vert (x_0 - \overline{x}_0) \partial _{x_0} u \Vert _{H^{0,s}} \nonumber \\\le & {} \varepsilon \Vert \partial _{x_0} u \Vert ^2_{H^{0,s}}. \end{aligned}$$

(59)

By using (57), (58), (59), it follows

$$\begin{aligned}&\Vert \varphi (x') [\, ^t P, A_s ] u \Vert \\&\quad \le \varepsilon \big ( \Vert \, ^t P u \Vert _{H^{0,s}} + \Vert \partial _{x_0} u \Vert _{H^{0,s}} + \Vert \partial _{x_1} u \Vert _{H^{0,s}} + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert _{H^{0,s}} + \Vert u \Vert _{H^{0,s}} \big ). \end{aligned}$$

Making use of (46), (54) and the previous estimate, the claim is achieved. $\square$

With analogous techniques used to prove Theorem 12 but making use of Theorems 3 and 10 instead of Theorems 4 and 11, respectively, we can establish the following relevant estimate.

Theorem 13

Let us suppose that assumptions (i), (ii) and (iii) hold. For every $\overline{x}_0 \ge 0$ and for every $s < 0$ there exist $k > 0$ and $c > 0$ such that

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert _{H^{0,s}} + \Vert \partial _{x_1} u \Vert _{H^{0, s}} + \Vert (x_0 -\alpha (x')) \partial _{x_2} u \Vert _{H^{0,s}} + \Vert u \Vert _{H^{0,s}} \le c \Vert \, ^t Pu \Vert _{H^{0,s}}, \\&\quad \quad \ \forall u \in C^{\infty }_0(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq [\overline{x}_0, \overline{x}_0+k[ \times \varOmega _0 = \varOmega _{\overline{x}_0,k}, \ L'u \cdot n'|_S=0. \end{aligned}$$

6 Global estimates

Now, we obtain a global estimate very useful in order to prove the existence of a solution to the Cauchy–Neumann problem (1).

Theorem 14

Let us suppose that assumptions (i), (ii) and (iii) hold. For every $h > 0$ and $s \le 0$ there exists $c>0$ such that

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert _{H^{0,s}} + \Vert \partial _{x_1} u \Vert _{H^{0,s}} + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert _{H^{0,s}} + \Vert u \Vert _{H^{0,s}} \le c \Vert \, ^t Pu \Vert _{H^{0,-s}}, \\&\qquad \qquad \qquad \quad \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _h, \ D'u \cdot n'|_S =0. \end{aligned}$$

Proof

Let $h > 0$, setting $\varOmega _h = [0, h[ \times \varOmega _0$, for the compactness of $[0, h] \times \overline{\varOmega }_0$, there exists a finite number of subsets $\{ \varOmega _1, \varOmega _2, \ldots , \varOmega _p \}$ of $\varOmega _h$, given by

$$\begin{aligned} \varOmega _1 =[0,k_1[ \times \varOmega _0, \ \varOmega _2 =[k_1',k_2[ \times \varOmega _0, \ldots , \ \varOmega _p =[k_{p-1}',k_p[ \times \varOmega _0, \end{aligned}$$

with $k_0 =0$, $k_p=h$, $k_{i-1}< k_i'< k_i$, for every $i=1,\ldots ,p$, and such that (45) holds in every $\varOmega _i$, for $i=1,\ldots ,p$.

Let $u \in C_0^{\infty }(\varOmega _h)$, with $D'u \cdot n'|_S = 0$, let $\varphi \in C_0^{\infty }([0,k_1[)$, with $\varphi \equiv 1$ on $[0,k_1'[$ and $0 \le \varphi \le 1$ in $[0, k_1[$. Rewriting (45) for $\varphi u$, it results

$$\begin{aligned}&\Vert \partial _{x_0} \varphi u \Vert _{H^{0,s}} + \Vert \partial _{x_1} \varphi u \Vert _{H^{0,s}} + \Vert (x_0- \alpha (x')) \partial _{x_2} \varphi u \Vert _{H^{0,s}} + \Vert \varphi u \Vert _{H^{0,s}} \\&\quad \le c \Vert P \varphi u \Vert _{H^{0,s}} \\&\quad \le c \Vert P u \Vert _{H^{0,s}} + c \Vert [P, \varphi ] u \Vert _{H^{0,s}} \\&\quad \le c \Vert P u \Vert _{H^{0,s}} + c \Vert \partial _{x_0} \varphi \partial _{x_0} u \Vert _{H^{0,s}} + c \Vert (\partial _{x_0}^2 \varphi ) u \Vert _{H^{0,s}} \\&\quad \le c \Vert P u \Vert _{H^{0,s}} + c \Vert \partial _{x_0} u \Vert _{H^{0,s}([k_1',k_1[ \times \varOmega _0)} + c \Vert u \Vert _{H^{0,s}([k_1',k_1[ \times \varOmega _0)} \\&\quad \le c \Vert P u \Vert _{H^{0,s}} + c \Vert \partial _{x_0} u \Vert _{H^{0,s}([k_1',k_2'[ \times \varOmega _0)} + c \Vert u \Vert _{H^{0,s}([k_1',k_2'[ \times \varOmega _0)} \\&\quad \le c \Vert P u \Vert _{H^{0,s}} + c \Vert \partial _{x_0} \varphi _1 u \Vert _{H^{0,s}([k_1',k_2[ \times \varOmega _0)} + c \Vert \varphi _1 u \Vert _{H^{0,s}([k_1',k_2[ \times \varOmega _0)}, \end{aligned}$$

where $\varphi _1 \in C_0^{\infty }(\varOmega _0)$ such that $\mathrm{supp} \, \varphi _1 \subseteq [k_1',k_2[$, $\varphi _1 \equiv 1$ in $[k_1',k_2'] \times \varOmega _0$.

We can deduce that

$$\begin{aligned}&\Vert \partial _{x_0} \varphi _{i-1} u \Vert _{H^{0,s}} + \Vert \partial _{x_1} \varphi _{i-1} u \Vert _{H^{0,s}} + \Vert (x_0- \alpha (x')) \partial _{x_2} \varphi _{i-1} u \Vert _{H^{0,s}} + \Vert \varphi _{i-1} u \Vert _{H^{0,s}} \\&\quad \le c \Vert P u \Vert _{H^{0,s}} + c \Vert \partial _{x_0} \varphi _i u \Vert _{H^{0,s}([k_i',k_{i+1}[ \times \varOmega _0)} + c \Vert \varphi _i u \Vert _{H^{0,s}([k_i',k_{i+1}[ \times \varOmega _0)}, \end{aligned}$$

where $\varphi _0 = \varphi$ and $\varphi _i \in C_0^{\infty } ([0,h[)$ such that $\mathrm{supp} \, \varphi _i \subseteq [k_i',k_{i+1}[$, for every $i = 1, \ldots , p$.

On the other hand, we have

$$\begin{aligned}&\Vert \partial _{x_0} \varphi _{p-1} u \Vert _{H^{0,s}} + \Vert \partial _{x_1} \varphi _{p-1} u \Vert _{H^{0,s}} + \Vert (x_0- \alpha (x')) \partial _{x_2} \varphi _{p-1} u \Vert _{H^{0,s}} + \Vert \varphi _{p-1} u \Vert _{H^{0,s}} \nonumber \\&\quad \le c \Vert P u \Vert _{H^{0,s}} + c \Vert \partial _{x_0} \varphi _p u \Vert _{H^{0,s}(\varOmega _p)} + c \Vert \varphi _p u \Vert _{H^{0,s}(\varOmega _p)} \nonumber \\&\quad \le c \Vert P u \Vert _{H^{0,s}} + c \left( \Vert \partial _{x_0} u \Vert _{H^{0,s}(\varOmega _p)} + \Vert u \Vert _{H^{0,s}(\varOmega _p)} \right) \nonumber \\&\quad \le c \Vert P u \Vert _{H^{0,s}}. \end{aligned}$$

(60)

By using (47), (60) and proceeding by recurrence on i, we easily obtain

$$\begin{aligned}&\Vert \partial _{x_0} \varphi _{i} u \Vert _{H^{0,s}} + \Vert \partial _{x_1} \varphi _{i} u \Vert _{H^{0,s}} + \Vert (x_0- \alpha (x')) \partial _{x_2} \varphi _{i} u \Vert _{H^{0,s}} \\&\quad + \Vert \varphi _{i} u \Vert _{H^{0,s}} \le c \Vert P u \Vert _{H^{0,s}}, \end{aligned}$$

for $i = 1, \ldots , p$. Taking into account the previous inequality, we have

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert _{H^{0,s}} + \Vert \partial _{x_1} u \Vert _{H^{0,s}} + \Vert (x_0- \alpha (x')) \partial _{x_2} u \Vert _{H^{0,s}} + \Vert u \Vert _{H^{0,s}} \le c \Vert P u \Vert _{H^{0,s}}, \nonumber \\&\qquad \qquad \qquad \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _h, \ D'u \cdot n'|_S =0. \end{aligned}$$

(61)

For the arbitrariness of h, (61) holds for every $u \in C_0^{\infty }(\overline{\varOmega })$ such that $D'u \cdot n'|_S=0$. The proof is thereby completed. $\square$

Proceeding analogously as in the proof of Theorem 14 but by using Theorems 3 and 10 instead of Theorems 2 and 7, respectively, we obtain a global estimate for the Cauchy–Robin problem (2).

Theorem 15

Let us suppose that assumptions (i), (ii) and (iii) hold. For every $h > 0$ and $s \le 0$ there exists $c>0$ such that

$$\begin{aligned}&\Vert \partial _{x_0} u \Vert _{H^{0,s}} + \Vert \partial _{x_1} u \Vert _{H^{0,s}} + \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert _{H^{0,s}} + \Vert u \Vert _{H^{0,s}} \le c \Vert \, ^t Pu \Vert _{H^{0,-s}}, \\&\qquad \qquad \qquad \quad \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _h, \ L'u \cdot n'|_S =0. \end{aligned}$$

7 Existence and regularity results

This section is devoted to establish existence and regularity results for the Cauchy–Neumann problem (1) and the Cauchy–Robin problem (2).

Theorem 16

Let $f \in H^r_{loc}(\overline{\varOmega })$, with $r \ge 2$. Then, for every $h>0$ there exists $v \in H^{0,s}(\varOmega _h)$, with $0 \le s \le r$ such that

$$\begin{aligned} (v, \, ^tPu) = (f,u), \quad \forall u \in C^{\infty }_0(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _h, \ D' u \cdot n'|_S=0. \end{aligned}$$

Proof

Let $B'$ be the subspace of distributions of $H^{0,-s}(\varOmega _h)$ defined on test functions $\varphi \in C^{\infty }_0([0,h[ \times \overline{\varOmega }_0)$ such that $D'\varphi \cdot u'|_S = 0$. Let B be contained in $B'$. Let us define a linear continuous functional in B, as follows

$$\begin{aligned} F(\psi )=F(\, ^t Pu)=(f,u), \quad \forall \psi \in B. \end{aligned}$$

Taking into account Theorem 12, it results

$$\begin{aligned} |F(\psi ) | = |(f,u)| \, \le \Vert f \Vert _{H^{0,s}} \Vert u \Vert _{H^{0,-s}} \, \le c \Vert f \Vert _{H^{0,s}} \Vert \, ^t P u \Vert _{H^{0,-s}}\, \le c' \Vert \psi \Vert _{H^{0,-s}}. \end{aligned}$$

As a consequence, we can extend F in $H^{0,-s}(\varOmega _h) \cap B'$. Making use of a representation theorem, there exists $v \in H^{0,s}(\varOmega _h) \cap B'^*$ such that

$$\begin{aligned} F(w)= (v, w), \quad \forall w \in H^{0,-s}(\varOmega _h). \end{aligned}$$

Then, it results

$$\begin{aligned} F(\psi ){=} (v, \psi ) =(v, \, ^t Pu) {=} (f,u), \quad \forall u \in C^{\infty }_0(\overline{\varOmega }):\mathrm{supp} \, u \subseteq \varOmega _h, D' u \cdot n'|_S=0. \end{aligned}$$

$\square$

We proved that for every $h > 0$ there exists $v \in H^{0,s}(\varOmega _h)$, with $0 \le s \le r$, such that

$$\begin{aligned} P v = f, \quad \hbox {in the sense of distributions}. \end{aligned}$$

Hence, v verifies the following equality:

$$\begin{aligned} \partial _{x_0}^2 v + a_0(x) \partial _{x_0} v + b(x) v = f+ \left( P- \partial _{x_0}^2 -a_0(x) \partial _{x_0} - b(x) \right) v. \end{aligned}$$

Since $f+ \left( P- \partial _{x_0}^2 -a_0(x) \partial _{x_0} - b(x) \right) v \in H^{0,r-2}(\varOmega _h)$, we have $v \in H^{2,r-2}(\varOmega _h)$. Therefore, proceeding by induction, we deduce

$$\begin{aligned} v\in H^{s,r-s}(\varOmega _h), \quad \forall s \le r. \end{aligned}$$

As a consequence, it follows

$$\begin{aligned} v\in H^{r}(\varOmega _h). \end{aligned}$$

Then, there exists $v \in H^r(\varOmega _h)$, with $r \ge 2$, such that

$$\begin{aligned} (v, \, ^tPu) = (f, u), \quad \forall u \in C^{\infty }_0(\varOmega _h): \ D' u \cdot n'|_S=0. \end{aligned}$$

(62)

Taking into account (62) and integrating by parts, we obtain

$$\begin{aligned} (Pv, u) = (f, u), \quad \forall u \in C^{\infty }_0(\mathrm{int} \, \varOmega _h). \end{aligned}$$

(63)

Hence, it results

$$\begin{aligned} Pv=f, \quad \mathrm{a.e. \ in} \ \varOmega _h. \end{aligned}$$

Integrating again by parts in (62), for every $u \in C_0^{\infty }(\varOmega _h)$ such that $u|_{\varOmega _0}=0$, $\partial _{x_0} u|_{\varOmega _0}=0$, $D'u \cdot n'|_S=0$, we have

$$\begin{aligned}&(Pv, u) - \int _S \left( \partial _{x_1} v \cdot n_1 + (x_0 - \alpha (x'))^2 \partial _{x_2} v \cdot n_2 - a_1(x) n_1 v - a_2(x) n_2 v \right) u d \sigma \nonumber \\&\quad \quad = (f, u), \end{aligned}$$

(64)

which implies

$$\begin{aligned} \int _S (L'v \cdot n') u d\sigma =0. \end{aligned}$$

Then, it follows

$$\begin{aligned} L'v \cdot n'|_S =0. \end{aligned}$$

Finally, integrating again by parts in (62), for every $u \in C_0^{\infty }(\varOmega _h)$ such that $u|_S = 0$, $\partial _{x_i} u|_S = 0$, with $i = 1,2$, and supposing that either $\partial _{x_0} u|_{\varOmega _0} = 0$ or $u|_{\varOmega _0} = 0$, we get

$$\begin{aligned} (Pv, u) - \int _{\varOmega _0} v \partial _{x_0} u dx_1 dx_2 = (f, u). \end{aligned}$$

(65)

As a consequence, it results

$$\begin{aligned} v|_{\varOmega _0} =0. \end{aligned}$$

Moreover, we have

$$\begin{aligned} (Pv, u) - \int _{\varOmega _0} \partial _{x_0} v \cdot u dx_1 dx_2 = (f, u). \end{aligned}$$

(66)

Hence, we obtain

$$\begin{aligned} \partial _{x_0} v|_{\varOmega _0} =0. \end{aligned}$$

Making use of (63), (64), (65) and (66), it follows that there exists $v \in H^r(\varOmega _h)$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} Pv= f, \quad \mathrm{in} \ \varOmega _h,\\ v|_{\varOmega _0}=0, \ \frac{d v}{dn}|_{\varOmega _0}=0, \ L'v \cdot n'|_S=0. \end{array}\right. } \end{aligned}$$

Instead if B is the space of functions $\psi = \, ^t Pu$, with $u \in C_0^{\infty }(\overline{\varOmega })$ such that $\mathrm{supp} \, u \subseteq \overline{\varOmega }_h$ and $L'u \cdot n'|_S=0$, proceeding as done before, we obtain the claim.

Moreover, with analogous proof of Theorem 16 but applying Theorem 13 instead of Theorem 12 and considering as $B'$ the subspace of distributions of $H^{0,-s}(\varOmega _h)$ defined on test functions $\varphi \in C^{\infty }([0,h[ \times \overline{\varOmega }_0)$ such that $L'\varphi \cdot n'|_S=0$, the following results holds.

Theorem 17

Let $f \in H^r_{loc}(\overline{\varOmega })$, with $r \ge 2$. Then, for every $h >0$ there exists $v \in H^{0,s} (\varOmega _h)$, with $0 \le s \le r$ such that

$$\begin{aligned} (v, \, ^t P u) = (f, u), \quad \forall u \in C_0^{\infty }(\overline{\varOmega }): \ \mathrm{supp} \, u \subseteq \varOmega _h, \ L'u \cdot n'|_S=0. \end{aligned}$$

(67)

Integrating by parts (67), as done before, it follows that there exists $v \in H^r(\varOmega _h)$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} Pv= f, \quad \mathrm{in} \ \varOmega _h,\\ v|_{\varOmega _0}=0, \ \frac{d v}{dn}|_{\varOmega _0}=0, \ D'v \cdot n'|_S=0. \end{array}\right. } \end{aligned}$$

with $f \in H^r_{loc}(\overline{\varOmega })$.

8 Uniqueness of the solution

In order to establish the uniqueness of a solution to the problem (1), we prove, as a first step, the existence of a solution to the following problems

$$\begin{aligned} {\left\{ \begin{array}{ll} \, ^t Pw= f, \quad \mathrm{in} \ \varOmega _h= ]0,h[ \times \varOmega _0 \\ w(h,x')=0, \ \partial _{x_0} w(h,x')=0, \ D'w \cdot n'|_S=0 \end{array}\right. } \end{aligned}$$

(68)

and

$$\begin{aligned} {\left\{ \begin{array}{ll} \, ^t Pw= f, \quad \mathrm{in} \ \varOmega _h= ]0,h[ \times \varOmega _0 \\ w(h,x')=0, \ \partial _{x_0} w(h,x')=0, \ L'w \cdot n'|_S=0 \end{array}\right. } \end{aligned}$$

(69)

with $f \in H^r(\varOmega _h)$. To this aim, we can proceed in analogous way as done in the proofs of the theorems in Sects. 4, 5, 6, 7 considering, for every $\overline{x}_0 \in ]0, h[$,

$$\begin{aligned} \varOmega _{\overline{x}_0,k}= & {} \left\{ x \in \overline{\varOmega }: x_0 \in ]\overline{x}_0-k, \overline{x}_0[ \times \varOmega _0 \right\} , \\ \varOmega _{\overline{x}_0,k, \eta }= & {} \left\{ x \in \varOmega _{\overline{x}_0,k}: \left( \frac{1}{5} + \eta \right) \alpha (x') + \left( \frac{4}{5} - \eta \right) \overline{x}_0 \le x_0 \le \eta \overline{x}_0 + (1-\eta ) \alpha (x') \right\} , \\ \varOmega _{\overline{x}_0,k, \eta }'= & {} \left\{ x \in \varOmega _{\overline{x}_0,k, \eta }: \left| \partial _{x_1} \alpha (x') \right| \ge 1 \right\} , \\ \varOmega _{\overline{x}_0,k, \eta , \eta '}'= & {} \left\{ x \in \varOmega _{\overline{x}_0,k, \eta }: \left| \partial _{x_1} \alpha (x') \right| \ge 1 - \eta ' \right\} , \end{aligned}$$

where $\frac{4}{5}< \eta < 1$ and $0< k < h$, and the operator $\, ^t P$ instead of the operator P. With these modifications and under assumptions (i), (ii) and (iii), we obtain that there exist solutions to problems (68) and (69), with $f \in H^r_{loc} (\varOmega _0)$. As a consequence, there exists a solution $w \in C^{\infty }(\varOmega _h)$ to the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \, ^t Pw= 0, \quad \mathrm{in} \ \varOmega _h, \\ w(h,x')=0, \ \partial _{x_0} w(h,x')=\varphi (x'), \ D'w \cdot n'|_S=0, \end{array}\right. } \end{aligned}$$

(70)

and exists a solution $w \in C^{\infty }(\varOmega _h)$ to the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \, ^t Pw= 0, \quad \mathrm{in} \ \varOmega _h, \\ w(h,x')=0, \ \partial _{x_0} w(h,x')=\varphi (x'), \ L'w \cdot n'|_S=0 \end{array}\right. } \end{aligned}$$

(71)

with $\varphi \in C_0^{\infty }(\varOmega _0)$.

Now, if $v \in H^r_{loc} (\varOmega _{h'})$, with $r \ge 2$, is a solution to the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} Pv= 0, \quad \mathrm{in} \ \varOmega _{h'}= ]0,h'[ \times \varOmega _0, \ \mathrm{with} \ h' \ge h, \\ v(0,x')=0, \ \partial _{x_0} v(0,x')=0, \ L'v \cdot n'|_S=0, \end{array}\right. } \end{aligned}$$

and w is a solution to (70), it results

$$\begin{aligned} 0 = (v, \, ^tPw) = (Pv, w) + \int _{\varOmega _0} v(h, x') \varphi (x') dx' = \int _{\varOmega _0} v(h, x') \varphi (x') dx'. \end{aligned}$$

For the arbitrary of $\varphi$, it follows that $v(h,x') = 0$. Hence, we get that $v = 0$ in $\varOmega _{h'} = ]0,h'[ \times \varOmega _0$. Moreover, for the arbitrary of $h'$, it results that, under assumptions (i) and (ii), the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} Pv= f, \quad \mathrm{in} \ \varOmega = ]0,+ \infty [ \times \varOmega _0, \\ v|_{\varOmega _0}=0, \ \partial _{x_0} v|_{\varOmega _0}=0, \ L'v \cdot n'|_S=0, \end{array}\right. } \end{aligned}$$

with $f \in H^2_{loc} (\overline{\varOmega })$, admits a unique solution $v \in H^r_{loc} (\overline{\varOmega })$, with $r \ge 2$. Instead, let $v \in H^r_{loc}(\overline{\varOmega })$ be a solution to the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} Pv= 0, \quad \mathrm{in} \ \varOmega = ]0,h'[ \times \varOmega _0, \ \mathrm{with} \ h' \ge h \\ v(0,x')=0, \ \partial _{x_0} v(0,x')=0, \ D'v \cdot n'|_S=0, \end{array}\right. } \end{aligned}$$

and w is a solution to (71), it follows

$$\begin{aligned} 0 = (v, \, ^tPw) = (Pv, w) + \int _{\varOmega _0} v(h, x') \varphi (x') dx' = \int _{\varOmega _0} v(h, x') \varphi (x') dx'. \end{aligned}$$

Therefore, under assumptions (i), (ii) and (iii), also the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} Pv= f, \quad \mathrm{in} \ \varOmega = ]0,+ \infty [ \times \varOmega _0, \\ v|_{\varOmega _0}=0, \ \partial _{x_0} v|_{\varOmega _0}=0, \ D'v \cdot n'|_S=0, \end{array}\right. } \end{aligned}$$

with $f \in H^r_{loc}(\overline{\varOmega })$, admits a unique solution.