On hyperbolic equations with double characteristics in the presence of transition

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

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

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

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

the symbol of P, by Σ the characteristic set

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

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

Let us note that (see [1])

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

It is easy to verify

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

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

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

see Section 6.

Let us assume that:

1. (i)

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

2. (ii)

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

3. (iii)

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

4. (iv)

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

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

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

Example 1.1

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

Example 1.2

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

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

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

Example 1.3

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

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

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

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

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

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

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

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

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

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

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

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

Lemma 3.1

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

Proof

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

and therefore

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

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

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

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

Theorem 3.1

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

Proof

Let us integrate by parts in the inner product

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

Let us compute separately every inner product:

For the second one, we have

Moreover, as a result

and that implies

Substituting (12) in (11), we obtain

We compute

Moreover, as a result

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

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

Let us consider

from which it follows that

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

Finally, we have

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

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

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

As a consequence, we have the following corollary.

Corollary 3.1

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

Proof

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

Let us, first, prove some preliminary results.

Lemma 4.1

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

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

Proof

Let us consider

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

This implies

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

where

It results

and then

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

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

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

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

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

Lemma 4.2

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

Furthermore, we are able to prove the following.

Lemma 4.3

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

Proof

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

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

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

Lemma 4.4

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

as a result

Proof

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

This implies

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

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

As a consequence, we establish the following result.

Lemma 4.5

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

for every $s<0$, as a result

Proof

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

Let us compute

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

Similarly, we obtain

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

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

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

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

On the other hand, we have

As a consequence, we obtain

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

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

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

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

At last, we remark that

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

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

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

Now, we are able to prove the following theorem.

Theorem 4.1

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

Proof

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

Furthermore, we have

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

Making use of the same technique, we have

and similarly

where L is large enough. Finally, we have

having used Lemma 4.2.

On the other hand we have the result

Let us observe that

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

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

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

and for ε small enough the claim is established. □

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

Theorem 4.2

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

Proof

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

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

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

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

On the other hand, we have

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

Hence,

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

Then as a result

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

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

Therefore, we have

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

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

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

Then, taking into account Lemma 4.5,

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

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

 □

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

Theorem 5.1

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

Proof

Let S be the space

Let T be the linear functional defined as

Making use of Theorems 4.1 and 4.2, we have

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

 □

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

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

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

Let us proceed by induction. We prove

Hence, we compute

from which we have

This implies

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

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

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

and that implies

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

and combining with (55), it follows that

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

and making use of (55), as a result

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

for k small enough.

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

becomes

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

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

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

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

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

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

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

where

is a solution to the problem

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

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

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

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

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

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

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

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

The first author was partially supported by STAR 2014 ‘Variational Analysis and Equilibrium Models in Physical and Socio-Economic Phenomena’ (Grant 14-CSP3-C03-099). The authors cordially thank the referees for their valuable comments and suggestions, which lead to a clearer presentation of this work.

Affiliations

1. Department of Mathematics and Applications “R. Caccioppoli”, University of Naples “Federico II”, via Cintia - Monte S. Angelo, Naples, 80126, Italy

   • Annamaria Barbagallo
   •  & Vincenzo Esposito

Corresponding author

Correspondence to Annamaria Barbagallo.

Competing interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ contributions

All authors contributed equally in this article. They read and approved the final manuscript.

Reprints and Permissions