Abstract
The paper concerns the study of the Cauchy–Dirichlet problem for a class of hyperbolic second-order operators with double characteristics in presence of transition in a domain of \({\mathbb {R}}^3\). Firstly, we establish some a priori local and global estimates. Then, we obtain some existence results.
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1 Introduction
Let \(\Omega = ]0, + \infty [ \times \Omega _0\), where \(\Omega _0\) is an open set of \(\mathbb {R}^2\) with enough smooth boundary (for example \(\Omega _0\) is of class \(C^m\), with \(m \ge 2\)). Let us set \(S= [0, + \infty [ \times \partial \Omega _0\), where \( \partial \Omega _0\) is the boundary of \(\Omega _0\). Let us consider the following class of hyperbolic second-order operators with double characteristics in presence of transition:
where \(x = (x_0, x_1, x_2 )\), \(\mathrm{Im} \, a_2(x)=(x_0-\alpha (x'))\widetilde{a}_2(x)\), with \(\widetilde{a}_2(x)\) real function, \(D_{x_j}= \dfrac{1}{i} \partial _{x_j}\), \(j=0,1,2\), the coefficients belong in \(C^{\infty }(\widetilde{\Omega })\), \(\widetilde{\Omega } =[0, + \infty [ \times \widetilde{\Omega }_0\), with \(\widetilde{\Omega }_0\) an open set containing strictly \(\Omega _0\), and \(\alpha \) is a real function. Let \(x'=(x_1,x_2)\), \(\xi =(\xi _0, \xi _1, \xi _2)=(\xi _0, \xi ')\), where we set \(\xi '= (\xi _1, \xi _2)\). Let
be the symbol of P, let
be the characteristic set and let
be the fundamental matrix of P at \(\rho \). The spectrum of \(F_p(\rho )\), denoted by \(\mathrm{Spec}(F_p(\rho ))\), has an important rule to study the well-posedness of the Cauchy–Dirichlet problem associated to the operator P. In particular, it results (see [10])
The fundamental matrix of P at \(\rho \) has only pure imaginary eigenvalues with a possible exception of a pair of nonzero real eigenvalues \(\pm \lambda \) (see [9,10,11]). If \(F_p(\rho )\) has a pair of nonzero real eigenvalues, P is called effectively hyperbolic at \(\rho \). If \(F_p(\rho )\) has only pure imaginary eigenvalues and if there are only Jordan blocks of dimension 2 in the Jordan normal form of \(F_p(\rho )\) corresponding to the eigenvalue 0, i.e., \(\mathrm{Ker} F_p(\rho )^2 \cap \mathrm{Im} F_p(\rho )^2 = \{ 0\}\), P is called non-effectively hyperbolic of type 1 at \(\rho \). Instead, if \(F_p(\rho )\) has only pure imaginary eigenvalues and if there is only a Jordan block of dimension 4 and no block of dimension 3 in the Jordan normal form of \(F_p(\rho )\) corresponding to the eigenvalue 0, i.e., \(\mathrm{Ker} F_p(\rho )^2 \cap \mathrm{Im} F_p(\rho )^2\) is 2-dimensional, P is called non-effectively hyperbolic of type 2 at \(\rho \). Furthermore, let
(see [9]). It is easy to deduce
We say that we have a transition exactly when at least two among the above sets are nonempty.
The paper continues the study on the following Cauchy–Dirichlet problem
started in [7]. In fact, in [7], several a priori estimates of local or global nature in Sobolev spaces with general exponent \(s \le 0\) for the class of second-order hyperbolic operators (1) are proved. Here, we establish some existence results for the Cauchy–Dirichlet problem (2). To this aim, we need to obtain other a priori estimates in Sobolev spaces with exponent \(s \le 0\). The proofs of such estimates make use of delicate variational techniques because of the degeneration on the characteristic set and of the transition between \(\Sigma _-\), \(\Sigma _0\) and \(\Sigma _+\). More precisely, the function \(\alpha \) in (1) depends on the variables \(x_1\) and \(x_2\). As a consequence, the coefficient \(x_0 - \alpha (x')\) degenerates on the characteristic set with respect to all the variables. Setting \(\beta = x_0 -\alpha (x')\), if \(| \partial _{x_1} \alpha (x')|<1\), \(\beta =0\) and \(\xi _0= \xi _1=0\), then \(F_p(\rho )\) has two distinct nonzero real eigenvalues. If \(| \partial _{x_1} \alpha (x')| > 1\), \(\beta =0\) and \(\xi _0= \xi _1=0\), \(F_p(\rho )\) has two nonzero imaginary eigenvalues. In conclusion, let \(\overline{\Sigma }\) be the set of points \(\rho =(x_0,x',\xi )\) of \(\Sigma \) such that \(\beta =0\) and \(\xi _0=\xi _1=0\). We have 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 \(| \partial _{x_1} \alpha (x')|=1\). Hence, even if we study the special class of operators (1), the transition from effectively hyperbolic to non-effectively hyperbolic occurs. A class more general of hyperbolic second-order operators with double characteristics is analyzed in [6]. It is worth to underline that the coefficient \(x_0 - \alpha (x')\) does not contain the parameter \(\lambda \) very helpful to prove global estimates near the boundary of \(\Omega \) in [5]. Finally, we remark that the operator (1) contains the first-order terms and the zero-order term, which have an important rule to study the well-posedness of the problem. Instead in [4], the subprincipal term is identically zero; consequently, the Hörmander–Ivrii–Petkov condition is automatically verified.
Several scholars considered the Cauchy problem either for effectively or non-effectively hyperbolic operators with double characteristics (see, for instance, [8, 10,11,12,13,14,15,16]). In [9], another class of hyperbolic second-order operators with double characteristics is analyzed. In particular, the \(C^{\infty }\) well-posedness of the Cauchy problem and Carleman estimates for non-effectively hyperbolic operators have been obtained. In [17], some energy estimates for a different class of hyperbolic second-order operators are established. Moreover, the \(C^{\infty }\) well-posedness of the Cauchy problem for non-effectively hyperbolic operators is studied. We underline that in [9, 17] the Cauchy problem for a class of operators in a form more general then (1) is analyzed, but a priori estimates only when \(\Sigma = \Sigma _- \sqcup \Sigma _0\) are established. Instead, thanks to variational and pseudodifferential techniques different from the ones used in [9, 17], we are able to examine the mixed Cauchy–Dirichlet problem and we prove 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 \(\Sigma = \Sigma _+\). Moreover, in the class of operators (1), studied also in [1,2,3], 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.
We set \(\beta (x)=x_0-\alpha (x')\), \(g(x')= \frac{\alpha (x')}{\partial _{x_1} \alpha (x')}\), \(h(x')=1-\partial _{x_1} g(x')\), in \(\widetilde{\Omega }\),
Moreover, let \(B= (b_{hk})_{h,k=0,1}\) be the quadratic matrix-function whose elements are given by:
where \(\widetilde{a}_0\) and \(\widetilde{a}_1\) are the imaginary parts of \(a_0\) and \(a_1\), respectively.
We suppose
-
(i)
\(g,h \in C^{\infty }(\Omega '_0)\), \(h(x') \in [h_1, h_2]\), \(\forall x' \in \Omega '_0\), with \(0<h_1<h_2 < 4\);
-
(ii)
the matrix-function B is positive definite in \(\Gamma '\), namely there exists \(k>0\) such that \(B(x') \eta \cdot \eta \ge k \Vert \eta \Vert ^2\), \(\forall \eta = (\eta _1, \eta _2) \ne (0,0)\), \(\forall x \in \Gamma '\);
-
(iii)
\(g(x') n_1|_{S} \ge 0\), for every \(x' \in \Omega _0' \cap \partial \Omega _0\).
We remark that if \(\widetilde{a}_0 = \widetilde{a}_1 =0\), on \(\Gamma '\), assumption (ii) is verified.
The main goal of the paper is to prove the following results:
Theorem 1
Let (i), (ii) and (iii) be satisfied. If \(f \in L^{2}_{loc}(\overline{\Omega })\), there exists \(w \in L^2_{loc}(\overline{\Omega })\) such that
where \(\overline{\Omega } = [0, + \infty [ \times \overline{\Omega }_0\).
Theorem 2
Let (i), (ii) and (iii) be satisfied. Let \(f \in H^{r}_{loc}(\overline{\Omega })\), with \(r \ge 2\), the Cauchy–Dirichlet problem
admits a solution \(u \in H^{r}_{loc}(\overline{\Omega } \setminus \partial \Omega _0)\).
Let us consider some operators which satisfy assumptions (i), (ii) and (iii) and for which we have a transition.
Example 1
Let \(\alpha (x')= x_1^3 e^{kx_2}\) be functions in an open set \(\widetilde{\Omega }_0\) of \(\mathbb {R}^2\) contained (0, 0). Let \(P= D_{x_0}^{(2)} - D_{x_1}^{(2)} - (x_0-\alpha (x'))^2 D_{x_2}^{(2)} - i a_0 D_{x_0}\), where \(a_0 >0\). It results \(g(x') = \frac{1}{3}x_1\) and \(h(x') = \frac{2}{3}\), then assumption (i) is verified for every \(\widetilde{\Omega }_0\). Assumption (ii) is satisfied for every \(\widetilde{\Omega }_0 \subseteq ]-\infty , \frac{2}{a_0} ] \times \mathbb {R}\). Moreover, assumption (iii) is fulfilled if \(n_1\) on \(\partial \Omega _0 \cap \Omega _0'\) is positive (for example if \(\widetilde{\Omega }_0\) is a circle of center in (0, 0)). Then, we can choose \(\widetilde{\Omega }_0\) such that \(|\partial _{x_1} \alpha (x')|\) admits values either less than or equal than or greater than 1. As a consequence, it follows \(\Sigma = \Sigma _- \sqcup \Sigma _0 \sqcup \Sigma _+\), with \(\Sigma _-\), \(\Sigma _0\) and \(\Sigma _+\) nonempty.
Example 2
Now, let us consider \(\alpha (x')= (ax_1+bx_2+c)^2\), with \(a,b,c \in \mathbb {R}\), \(a,b \ne 0\), in an open set \(\widetilde{\Omega }_0\) of \(\mathbb {R}^2\) contained (0, 0). Let \(P= D_{x_0}^{(2)} - D_{x_1}^{(2)} - (x_0-\alpha (x'))^2 D_{x_2}^{(2)} +a_0 D_{x_0} - i a_1(x) (x_0-\alpha (x')) (D_{x_1} + D_{x_2})\), where \(a_0 \in \mathbb {R}\) and \(a_1 \in C^{\infty }\). It results \(g(x')= \dfrac{ax_1+bx_2+c}{2 a}\) and \(h(x')= \dfrac{1}{2}\). Hence, assumption (i) is always verified. Moreover, we can choose \(\widetilde{\Omega }_0\) such that assumption (iii) is fulfilled and both \(|\partial _{x_1} \alpha (x')| \le 1\) and \(|\partial _{x_1} \alpha (x')| \ge 1\) hold. Therefore, the existence of a solution is ensured in presence of transition.
The paper is organized as follows. In Sect. 2, some preliminary notations are recalled. In Sect. 3, a priori estimates obtained in [7] are referred. Section 4 is devoted to prove a priori estimates under the assumption \(|\partial _{x_1} \alpha (x')| \le 1\). Instead, Sect. 5 concerns estimates under the assumption \(|\partial _{x_1} \alpha (x')| \ge 1\). In Sect. 6, conclusive estimates in \(L^2\) are proved. In Sect. 7, estimates in Sobolev spaces with \(s<0\) are established making use of the pseudodifferential operator theory. Section 8 concerns the study of some global estimates. Finally, Sects. 9 and 10 deal with the proofs of Theorems 1 and 2, respectively.
2 Notations and preliminaries
Let \(\alpha =(\alpha _0, \alpha _1, \alpha _2) \in \mathbb {N}^3_0\). We indicate the derivative of order \(|\alpha |\) by \(\partial ^{\alpha }\), the derivative of order h with respect to \(x_j\) by \(\partial ^{h}_{x_j}\) and the derivative of order h with respect to \(x_j\) and \(x_p\) by \(\partial ^{h}_{x_j, x_p}\).
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. We indicate the external normal versor to the boundary \(\partial \Omega \) by \(n=(n_0,n_1,n_2)\).
Let \(C_0^{\infty }(\overline{\Omega })\) be the space of restrictions of functions belonging to \(C^{\infty }_0(\mathbb {R}^3)\) on \(\overline \Omega \). For each \(K \subseteq \overline{\Omega }\) compact set, let \(C^{\infty }_0 (K)\) be the set of functions \(\varphi \in C^{\infty }_0 (\overline{\Omega })\) having support contained in K. Set \(\Omega _k=[0,k[ \times \Omega _0\), let us introduce
Moreover, let \(C^{*\infty }_0(\overline{\Omega })\) be the space of functions \(u \in C^{\infty }_0(\overline{\Omega })\) such that \(\gamma _1 \partial _{x_0} u(0,x') = \gamma _2 u(0,x')\), where \(\gamma _1, \gamma _2 \in \mathbb {R}\). Consequently, we can introduce \(C^{* \infty }(\widetilde{\Omega })\) and \(C^{* \infty }(\overline{\Omega }_k)\). It is worth to remark that if \(u \in C^{\infty }_0(\Omega )\), then \(u \in C^{*\infty }_0(\overline{\Omega }_0)\). Furthermore, if \(u(x_0,x')=u_1(x') u_2(x_0)\), with \(u_1 \in C^{\infty }_0(\overline{\Omega }_0)\), \(u|_{\partial \Omega _0}=0\) and \(u_2 \in C^{\infty }_0([0,k[)\) then \(u \in C^{*\infty }_0(\overline{\Omega }_k)\).
Let \(S(\mathbb {R}^3)\) be the space of rapidly decreasing functions. Let \(S(\overline{\Omega })\) be the space of restrictions of functions belonging to \(S(\mathbb {R}^3)\) on \(\overline{\Omega }\).
Let \(\Omega = ]0, + \infty [ \times \Omega _0\) and let \(s \in \mathbb {R}\), the norm in \(H^{0,s}\) is given by
where the Fourier transform is done only with respect to the variable \(x'\). Let \(A_s: C^{\infty }_0(\Omega ) \rightarrow C^{\infty }(\Omega )\) be the pseudodifferential operator defined by
For every \(\varphi (x') \in C^{\infty }_0(\Omega _0)\), the operator \(\varphi A_s u\) extends as a linear continuous operator from \(H^{0,r}_{comp.}(\Omega )\) into \(H^{0,r-s}_{loc}(\Omega )\), where \(r,s \in \mathbb {R}\). In particular, in \(\Omega _k =[0,k[ \times \Omega _0\), for \(k>0\), let \(H^{0,s}(\Omega _k)\) be the space of \(u \in H^{0,s}(\Omega _k)\) such that \(\mathrm{supp} \; u \subseteq \Omega _k\). Moreover, if \(\mathrm{supp} \ \varphi \subseteq \Omega _0 \setminus \mathrm{supp} \ u\), then \(\varphi A_s u\) is a regularizing operator with respect to the variable \(x'\). It results
The norms \(\Vert u \Vert _{H^{0,s}(\Omega )}\) and \(\Vert A_s u \Vert _{L^2(\Omega )}\) are equivalent for any \(s \in \mathbb {R}\).
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
Let \(H^{p,s}(\Omega )\) be the space of restrictions of elements of \(H^{p,s}(\mathbb {R}^3)\) on \(\Omega \) endowed with the norm
In the same way, the space \(H^{p,s}(\Omega _k)\) can be introduced.
At last, we consider the transposed operator of the operator P:
3 Some known preliminary results
First of all, we recall a priori estimate for the solution to the problem (2) (see [2], Lemma 3.1).
Lemma 1
Let \(u \in S(\overline{\Omega })\) and let \(p, \alpha _0, \alpha _1, \alpha _2 \in \mathbb {N}_0\). Then
Moreover, we have the following preliminary result (see [7], Lemma 3.2).
Lemma 2
Let \(u \in S(\overline{\Omega })\), it results
The next result holds (see [7], Lemma 3.3).
Lemma 3
For every \(\varepsilon , \delta >0\) there exists \(k>0\) such that, if
it results
We present a priori estimate (see [7], Theorem 3.4).
Theorem 3
Let (i) and (iii) be satisfied. Then, there exist \(k>0\) and \(c>0\) such that
Moreover, we recall the following result (see [7], Theorem 3.5).
Theorem 4
Let (i) and (iii) be satisfied. For every \(\varepsilon >0\) there exist \(k>0\) and a neighborhood \(I_{x'}\) in \(\Omega _0 \cap \Gamma \) such that
Let \(\overline{x}_0 >0\) and let \(k>0\), we denote by \(\Omega _{\overline{x}_0,k}=] \overline{x}_0, \overline{x}_0+ k [ \times \overline{\Omega }_0\). Let us show the following preliminary result (see [7], Lemma 4.1).
Lemma 4
Let \(u \in S(\Omega )\) such that \(\partial _{x_0} u|_{\Omega _0} =0\), let \(p, \alpha _0, \alpha _1, \alpha _2 \in \mathbb {N}\) and \(\overline{x}_0 >0\). It results
We consider another preliminary lemma (see [7], Lemma 4.2).
Lemma 5
For every \(\varepsilon , \delta > 0\) and \(\overline{x}_0 >0\), there exists \(k>0\) such that, setting
it results
We recall the following preliminary result (see [7], Lemma 4.3).
Lemma 6
Let (i), (ii) and (iii) be satisfied. Let \(\overline{x}_0 >0\), for every \(\varepsilon >0\) there exists \(k, \delta >0\) such that, setting
it results
At last, we present the following result (see [7], Theorem 4.4).
Theorem 5
Let (i), (ii) and (iii) be satisfied. Let \(\overline{x}_0>0\). There exist \(k>0\) and \(c>0\) such that
4 Estimates under the assumption \(|\partial _{x_1} \alpha (x') | \le 1\)
Let \(\overline{x}_0 \ge 0\), let us denote by
The following result holds.
Theorem 6
Let (i), (ii) and (iii) be satisfied. Let us assume that there exist two positive numbers \(k'\) and \(\delta \) such that \(|\partial _{x_1}\alpha (x')| \le 1\) on \(\Omega _0 \cap J_{k', \delta , 0}\). Then, for every \(\varepsilon >0\) there exists \(0 < k \le k'\) such that
Proof
Let us consider the following inner products
By means of integrations by parts, for every \(u \in C_{0}^{\infty }(\overline{\Omega })\) such that \(\mathrm{supp} \, u \subseteq \Omega _{k'} \cap J_{k', \delta , 0}\) and \(u|_S=0\), we have
From which it follows
We denote by
Since \((x_0-\alpha (x')) \left( \frac{5}{2} x_0 - \frac{1}{2} \alpha (x') \right) >0\), in \(\Omega _{k'} \setminus \Omega _{k', \frac{1}{5}}\), by (7) one has
where we denoted by \((\cdot , \cdot )_{\Omega _{k^{'}, \frac{1}{5}}}\) the inner product on \(\Omega _{k^{'}, \frac{1}{5}}\). Furthermore, it results
In \(\Omega _{k', \frac{1}{5}}\), we consider the following inner products
If u is identically zero on \(\Gamma _{\eta }\), where \(\Gamma _{\eta }\) is the surface \(x_0 = \eta \alpha (x')\), with \(0 < \eta \le \frac{1}{5}\), integrating by parts, we have
By (8) and (9), if \(| \partial _{x_1} \alpha (x')| \le 1\), on \(\Omega _0 \cap J_{k', \delta , 0}\), and \(k'\) is small enough, the claim follows assuming that u is identically zero on \(\Gamma _{\eta }\).
Let \(u \in C_{0}^{\infty }(\overline{\Omega })\) such that \(\mathrm{supp} \, u \subseteq \Omega _{k'}\) and \(u|_S=0\). Let \(\chi \) be a function of class \(C^{\infty }\) such that \(\chi (t)=1\), for \(t \ge \eta \), and \(\chi (t)=0\), for \(0 \le t \le \frac{\eta }{2}\). Rewriting (9) for \(u \chi \left( \frac{x_0}{\alpha (x')} \right) \) and adding (8), there exists \(0 < k \le k'\) such that
Making use of the previous inequality and Lemma 3 with k small enough, the claim is achieved. \(\square \)
We set
with \(\overline{x}_0 >0\) and \(k>0\), and we prove the following result.
Theorem 7
Let (i), (ii) and (iii) be satisfied. Let us assume that there exist two positive numbers \(k'\) an \(\delta \) such that \(|\partial _{x_1}\alpha (x')| \le 1\), on \(\Omega _{\overline{x}_0} \cap J_{k', \delta , \overline{x}_0}\), where \(\Omega _{\overline{x}_0}\) is the part of the plane \(x_0= \overline{x}_0\) in \(\Omega _{\overline{x}_0,k}\). Then, for every \(\varepsilon >0\) there exists \(0< k \le k'\) such that
Proof
Let \(u \in C^{\infty }_0(\overline{\Omega })\) such that \(\mathrm{supp} \, u \subseteq \Omega _{\overline{x}_0, k'} \cap J_{k', \delta , \overline{x}_0}\) and \(u|_S=0\), integrating by parts in the following inner products
we obtain
Taking into account that \(\frac{1}{2} \Vert (x_0 - \alpha (x')) \partial _{x_2} u \Vert ^2 + 2 ((x_0 - \overline{x}_0) (x_0-\alpha (x')) \partial _{x_2} u, \partial _{x_2} u) < 0\), if \(x_0 \le \frac{1}{5} \alpha (x') + \frac{4}{5} \overline{x}_0\) or \(x_0 \ge \alpha (x')\), it results
where \(\Omega _{\overline{x}_0, k', \eta } = \left\{ x \in \Omega _{\overline{x}_0,k'}: \ \eta \alpha (x') + (1- \eta ) \overline{x}_0 \le x_0 \le \alpha (x') \right\} \), with \(0 < \eta \le \frac{1}{5}\).
In \(\Omega _{\overline{x}_0,k', \eta }\), we consider the following inner products
Proceeding as done above, we obtain
where \(\Gamma _{\eta , (1-\eta ) \overline{x}_0}\) is the surface \(x_0 = \eta \alpha (x') + (1- \eta ) \overline{x}_0\), with \(0< \eta \le \frac{1}{5}\). Making use of (11) and (12), we deduce the claim assuming that the gradient of u with respect to \(x_0\) and \(x_1\) is zero on \(\Gamma _{\eta , (1-\eta ) \overline{x}_0}\).
Let u be a function belonging to \(C_{0}^{\infty }(\overline{\Omega })\) such that \(\mathrm{supp} \, u \subseteq \Omega _{\overline{x}_0}\) and \(u|_S=0\). Let \(\chi \) be a function of class \(C^{\infty }\) such that \(\chi (t)=1\), if \(|t| \ge \eta \), and \(\chi (t)=0\), if \(|t| < \frac{\eta }{2}\). Rewriting (12) for \(u \chi \left( \dfrac{x_0}{\alpha (x')} \right) \) and adding (11), there exists \(0 < k \le k'\) such that
Finally, the claim follows from the previous inequality and by using Lemma 5 for k small enough. \(\square \)
5 Estimates under the assumption \(|\partial _{x_1} \alpha (x')| \ge 1\)
For every \(\overline{x}_0 \ge 0\), we set
The next result holds.
Theorem 8
Let (i), (ii) and (iii) be satisfied. Let us assume that \(|\partial _{x_1}\alpha (x')| \ge 1\), on \(\Gamma _{\overline{x}_0}\). Then, there exist \(k>0\) and \(c>0\) such that
Moreover, for every \(\varepsilon > 0\) there exists \(k>0\) such that
Proof
Let d > 0 and let us set
where \(g_d(x') = \dfrac{\alpha (x') +d}{\partial _{x_1} \alpha (x')}\), and consider the sum of the inner products
For every \(u \in C^{\infty }_0(\overline{\Omega })\) such that \(u|_S=0\), it results:
Let us integrate by parts in the first inner products of the principal part in (15)
Moreover, integrating by parts in the second inner products in (15), we have
Since \(u|_S=0\), it results
Making use of the assumption (iii), it follows
Denoting the tangential derivative of u along the section of S of the equal height by \(\dfrac{\partial u}{\partial \tau }\), we obtain
where we took into account that \(\dfrac{\partial u}{\partial \tau }=0\), since \(u=0\) on S.
Adding (16) and (17) and making use of (18), (19) and (20), we have
By assumption (i), there exist two positive numbers k and \(\delta \) such that, for \(d > \frac{1}{h_1} |g(x')|\), where \(x' \in \Omega _0 \cap J_{k, \delta , \overline{x}_0}\), it results \((\overline{x}_0 +d)^2 - (g_d(x'))^2 \ge 0\) and, hence,
By using (21), we deduce
Now, we consider the first-order terms. Integrating by parts, it results
Since \(\alpha (x')\), \(g_d(x')\) and \(\beta (x')\) vanish on \(\Omega _0 \cap \Gamma \), for every \(\delta >0\) there exist a neighborhood \(I_{x'}\) in \(\Omega _0 \cap \Gamma \) and \(k>0\) such that
Let \(\varphi \in C^{\infty }_0(\overline{\Omega })\) such that \(\varphi \equiv 1\), on \([0,k'[ \times I'_{x'}\), with \(I'_{x'} \subseteq I_{x'}\) and \(k'<k\), \(0 \le \varphi (x) \le 1\) and \(\mathrm{supp} \, \varphi \subseteq [0, k[ \times I_{x'}\). Without lost generality, we can consider \([0, k'[ \times I'_{x'}\) such that \(|x_0 - \alpha (x')| \ge \dfrac{\varepsilon }{2}\), for every \(x \in \Omega _k \setminus ([0, k'[ \times I'_{x'})\). Using (22) and the previous remarks, it follows
Taking into account Lemma 1, we get
Making use of Lemmas 1 and 3, for \(\delta \) and k small enough, we obtain
Let \(\chi \in C_0^{\infty }(\mathbb {R})\) such that \(\chi (t)=1\), if \(|t| \le \frac{1}{2}\), and \(\chi (t)=0\), if \(|t| >1\). We rewrite (25) for \(u \chi \left( \frac{x_0 - \alpha (x')}{\delta } \right) \) and apply Lemma 5 to \(u \left[ 1- \chi \left( \frac{x_0 - \alpha (x')}{\delta } \right) \right] \). Adding the obtained estimates, for \(\delta \) small enough and k suitable and small, we reach (13).
Instead, in order to get (14), let γ > 0 and let us consider the operator
where
Integrating by parts in the inner products \((\, ^t Pu, A_{\overline{x}_0, \gamma }u) + (A_{\overline{x}_0, \gamma }u, \, ^t Pu)\), using the same arguments as done and since \(g_{\overline{x}_0, \gamma }(x')\) has the same sign of \(g(x')\) on \(S \cap I_{k, \delta }\), we deduce
For \(\delta \) small enough and since \(|\partial _{x_1} \alpha (x')| >1\), on \(\Gamma _{\overline{x}_0}\), it results
As a consequence, for \(\varepsilon \) small enough and k suitable and small, we have
Rewriting the previous inequality for \(u \chi \left( \frac{x_0 - \alpha (x')}{\delta } \right) \) and applying Lemma 6 to \(u \left[ 1- \chi \left( \frac{x_0 - \alpha (x')}{\delta } \right) \right] \), as done above, (14) follows for \(\overline{x}_0 >0\).
On the other hand, if \(\overline{x}_0 =0\), considering the inner products
and proceeding as before, we obtain (13) and, then, (14) for \(\gamma \) small enough. \(\square \)
6 Conclusive a priori estimates
Let us assume that \(|\partial _{x_1} \alpha (x') |=1\) in some points of the plane \(x_0 = \overline{x}_0\), with \(\overline{x}_0 > 0\). Let \(\Omega _{\overline{x}_0}\) be the intersection between the plane \(x_0 = \overline{x}_0\) and \(\Omega \). Let \(\Gamma _{\overline{x}_0} = \Gamma \cap \Omega _{\overline{x}_0}\). Let \(\Gamma _{\overline{x}_0}'\) be the set of the points of \(\Gamma _{\overline{x}_0}\) where \(\partial _{x_1} \alpha (x_1)=1\) and, finally, let \(I_{\overline{x}_0}\) be a neighborhood of \(\overline{x}_0\) in \(\Gamma _{\overline{x}_0}'\) on \(\Omega _{\overline{x}_0}\) such that \(\partial _{x_1} \alpha (x_1) \lessgtr 1\) outside \(I_{\overline{x}_0}\). The following result holds.
Theorem 9
Let (i), (ii) and (iii) be satisfied. If on the plane \(x_0 = \overline{x}_0>0\) there exist points in which \(|\partial _{x_1} \alpha (x')|=1\), then there exist \(k>0\) and \(c>0\) such that
Moreover, for every \(\varepsilon >0\) there exists \(k>0\) such that
Proof
Let \(\Omega _{\overline{x}_0} \cap \Gamma \), let \(\overline{x}' \in \Omega _0\) such that \(| \partial _{x_1} \alpha (\overline{x}')|=1\). We set
Evidently, it results \(|\partial _{x_1} \alpha (x')| \le 1\) on the curve \(x_0 - \overline{x}_0 = \gamma (x')\) and \(x \in J_{k, \delta , \overline{x}_0}\), with suitable k and \(\delta \). Therefore, there exists \(\eta \) such that \(|\partial _{x_1} \alpha (x')| \le 1\) if \(|x_0 - \overline{x}_0| \le \eta \gamma (x')\) and \(x \in J_{k, \delta , \overline{x}_0}\). Whereas \(|\partial _{x_1} \alpha (x')| \ge 1\) on \(\Omega _{\overline{x}_0}\) if \(|x_0 - \overline{x}_0| \ge \eta \gamma (x')\). Let \(\chi \in C^{\infty }(\mathbb {R})\) such that \(\chi (t)=0\) if \(t \le \frac{\eta }{2}\) and \(\chi (t) =1\) if \(t \ge \eta \). For every \(u \in C^{\infty }_0(\overline{\Omega })\) such that \(\mathrm{supp} \, u \subseteq J_{k, \delta , \overline{x}_0}\) and \(u|_S=0\), we rewrite (13) and (14) for \(\chi \left( \frac{x_0 - \overline{x}_0}{\gamma (x')} \right) u\) and (10) for \(\left( 1- \chi \left( \frac{x_0 - \overline{x}_0}{\gamma (x')} \right) \right) u\). Adding such inequalities, for k small enough, we have
and
From (29), (30) and Lemma 5, it follows
and
\(\square \)
With the same techniques used in Theorem 6 if \(\overline{x}_0 =0\) and Theorems 8 and 9 if \(\overline{x}_0 >0\), we obtain the next result.
Theorem 10
Let (i), (ii) and (iii) be satisfied. If on the plane \(x_0 = \overline{x}_0>0\) there exist points in which \(|\partial _{x_1} \alpha (x')|=1\), then there exist \(k>0\) and \(c>0\) such that
Moreover, for every \(\varepsilon >0\) there exists \(k>0\) such that
7 Estimates in Sobolev spaces with \(s<0\)
Let \(\Omega _{\overline{x}_0}'\) be the intersection between \(\Omega '\) and the plane \(x_0 = \overline{x}_0\). Let \(\Gamma _{\overline{x}_0}'\) be the set of points belonging into \(\Gamma _{\overline{x}_0}= \Gamma \cap \Omega _{\overline{x}_0}'\) such that \(|\partial _{x_1} \alpha (x') |=1\). Moreover, let \(J_{\overline{x}_0}\) be the intersection between a neighborhood of \(\Gamma _{\overline{x}_0}'\) and \(\Omega _{\overline{x}_0}'\). We are able to prove the following estimate in Sobolev spaces with \(s<0\).
Theorem 11
Let (i), (ii) and (iii) be satisfied. Then, for every \(\overline{x}_0 \ge 0\) and for every \(s<0\) there exist \(k>0\) and \(c>0\) such that
Proof
Firstly, let \(x_0>0\). Let \(\varphi \in C^{\infty }_0(\mathbb {R}^2)\) such that \(\mathrm{supp} \, \varphi \subseteq \Omega _{0}'\), \(\varphi \equiv 1\) on \(\Omega _0\), with \(\Omega _0 \subset \Omega _0'\). For every \(u \in C^{\infty }_0(\overline{\Omega }_k)\) such that \(\mathrm{supp} \, u \subseteq \Omega _{\overline{x}_0,k}= [\overline{x}_0, \overline{x}_0+k[ \times \Omega _0\), we set \(v_s= \varphi (x') A_s u\). Making use of Theorem 10, it follows
We have
where \(R= [ \varphi , A_s] u\) is a regularizing pseudodifferential operator.
By using (35) and Lemma 4, we obtain
Furthermore, it results
where \(R_1\) and \(R_2\) are regularizing pseudodifferential operators.
Finally, we get
where \(R_3\) and \(R_4\) are regularizing pseudodifferential operators, \(B_{s-1}\) and \(B'_{s}\) are pseudodifferential operators of order \(s-1\) and s, respectively. Adding (36), (37), (38) and using Lemma 4, it follows
Moreover, it results
where \(R_5\) and \(R_6\) are regularizing operators.
The commutator \([\, ^t P, A_s]\) is given by
We consider the principal part:
where \(B_{s+1}\) and \(B_s\) are pseudodifferential operators of order \(s+1\) and s, respectively. The symbol of \(B_{s+1}\) is given by
Then, \(B_{s+1} u = (x_0-\alpha (x'))\varphi (x') \partial _{x_2} B_s'u\), where \(B_s'\) is a pseudodifferential operator of order s. Moreover, taking into account Theorem 10, we deduce
being \([\, ^t P, B'_s]\) a pseudodifferential operator of order \(s-1\) and its principal symbol \(b'(x, \xi )\) of the same type of \(b(x, \xi )\). Hence, making use of Lemma 4, it results
We consider the first-order part of the commutator
where \(B_{s-1}\) and \(B_{s}\) are pseudodifferential operators of order \(s-1\) and s, respectively.
By using Lemma 4, we have
where \(B^{(i)}_s\) are pseudodifferential operators of order s. Hence, it results
Taking into account (43) and (44), it follows
We estimate the zero-order part:
Making use of (42), (45), (46) and for \(| x_0 - \overline{x}_0 | \le k < \varepsilon \), we obtain
Taking into account (40), (47) and Lemma 4, denoted the generic regularizing operator by R, it follows
By using (34), (39), (48) and Lemma 4, it results
For \(\varepsilon \) small enough and making use of Lemma 4, we have
For \(|x_0 - \overline{x}_0|\) small enough and using Lemma 4, we deduce
Since the function \(\varphi \) is the same for every functions u, then c does not depend on u but depends on the distance between \(\partial \widetilde{\Omega }'_0\) and \(\partial \widetilde{\Omega }_0\) and k is small enough.
Now, if \(x_0 =0\), for every \(u \in C^{\infty }_0(\overline{\Omega })\) such that \(\mathrm{supp} \, u \subseteq [0,k[ \times \Omega _{0}\), we set \(v_s= \varphi (x') A_s u\). Making use of Theorem 10, it results
Proceeding as done above, we obtain the analogous inequality of (39):
where we used Lemma 1 instead of Lemma 4. Considering \(\Vert \, ^t P v_s \Vert \) and proceeding again as done before and taking into account Theorem 10 and Lemma 1, we have
Moreover, using (50), (51) and (52), we obtain
\(\square \)
8 Global estimates
In this section, we obtain fundamental global estimates in order to prove the existence of a solution to the Cauchy–Dirichlet problem (2).
Theorem 12
Let (i), (ii) and (iii) be satisfied. Then, for every \(k>0\) and \(s<0\) there exists \(c>0\) such that
Moreover, for \(s=0\) and for every \(k>0\) there exists \(c>0\) such that
Finally, for every \(k>0\) and \(s<0\) there exists \(c>0\) such that
where \([u]= {\left\{ \begin{array}{ll} \begin{array}{ll} u, &{} \quad \mathrm{in} \ \Omega _k= [0,k[ \times \Omega _0 \\ 0, &{} \quad \mathrm{in} \ \Omega _k= [0,k[ \times (\mathbb {R}^2 \setminus \Omega _0) \end{array} \end{array}\right. }\).
Proof
Let \(k > 0\), let us set \(\Omega _k = [0, k[ \times \Omega _0\). For the compactness of \([0, k] \times \overline{\Omega }_0\), there exists a finite number of subsets \(\{ \Omega _1, \Omega _2, \ldots , \Omega _p \}\) of \(\Omega _k\), given by
with \(h_0 =0\), \(h_p=h\), \(h_{i-1}< h_i'< h_i\), for every \(i=1,\ldots ,p\), and such that (33) holds in every \(\Omega _i\), for \(i=1,\ldots ,p\).
Let \(u \in C_0^{\infty }(\Omega _k)\), let \(\varphi \in C_0^{\infty }([0,h_1[)\), with \(\varphi \equiv 1\) on \([0,h_1'[\) and \(0 \le \varphi \le 1\) in \([0, h_1[\). Rewriting (33) for \(\varphi u\), it results
where \(\varphi _1 \in C_0^{\infty }(\Omega _0)\) such that \(\mathrm{supp} \, \varphi _1 \subseteq [h_1',h_2[\), \(\varphi _1 \equiv 1\) in \([h_1',h_2'] \times \Omega _0\).
We can deduce that
where \(\varphi _0 = \varphi \) and \(\varphi _i \in C_0^{\infty } ([0,k[)\) such that \(\mathrm{supp} \, \varphi _i \subseteq [h_i',h_{i+1}[\), for every \(i = 1, \ldots , p\).
On the other hand, we have
Using (33), (57) and proceeding by recurrence on i, we easily obtain
for \(i = 1, \ldots , p\). Taking into account the previous inequality, we have
For the arbitrariness of k, (58) holds for every \(u \in C_0^{\infty }(\overline{\Omega })\). The proof of (54) is thereby completed.
Furthermore, taking into account (31), we obtain (55).
Finally, we prove (56). Let \(u \in C^{\infty }_0(\overline{\Omega })\) such that \(\mathrm{supp} \, u \subseteq [0, k[ \times \overline{\Omega }_0\) and \(u|_S=0\). Let \(\{ u_n \}\) be a sequence in \(C^{\infty }_0(\overline{\Omega })\) such that \(\mathrm{supp} \, u_n \subseteq [0, k[ \times \overline{\Omega }_0\) and \(u_n \rightarrow u\) in \(H^{2,1}\). We have that \(u_n \rightarrow u\) and \(Pu_n \rightarrow P[u]\) in \(H^{0,s}\), for every \(s<0\). Hence, rewriting (54) for \(u_n\), for every \(n \in \mathbb {N}\), and passing to the limit as \(n \rightarrow + \infty \), we obtain (56). \(\square \)
9 Proof of Theorem 1
Let V be the subspace of \(L^2(\Omega _k)\), where \(\Omega _k= ]0,k[ \times \Omega _0\), made up of functions \(\psi = \, ^tPu\), with \(u \in C^{\infty }_0(\overline{\Omega })\) such that \(\mathrm{supp} \, u \subseteq [0,k[ \times \overline{\Omega }_0\) and \(u|_S=0\). Let us consider the functional
It results
Making use of (55), we have
where \(c'=c \Vert f \Vert _{L^2(\Omega _k)}\). Therefore, it is possible to extend T as a linear continuous functional into \(L^{2}(\Omega _k)\). Making use of a representation theorem, there exists \(w \in L^2(\Omega _k)\) such that
In particular, we have
Hence, w is a solution in the sense of distributions to the equation
For the arbitrariness of k and since \(f \in L^2_{loc}(\overline{\Omega })\), Theorem 1 is proved.
10 Proof of Theorem 2
Let us denote by W the subspace of \(\mathcal {D}'([0,k[ \times \Omega _0)\) containing extensions of linear continuous functionals to functions \(\varphi \in C^{\infty }_0(\overline{\Omega }_k)\) such that \(\varphi |_S=0\), where \(\overline{\Omega }_k= [0,k[ \times \overline{\Omega }_0\). It results that \(P[u] \in W\), where \(u \in C^{\infty }_0(\overline{\Omega }_k)\) such that \(u|_S=0\) and \(u=0\) in \([0,k[ \times (\mathbb {R}^2 \setminus \overline{\Omega }_0)\). Moreover, we have
Therefore, the distributions \(^tP[u]\) and \(^t Pu\) are equal in W. Let T be the functional defined into the subspace of W containing the distributions \(\psi = \, ^t P[u]\), for every \(u \in C^{\infty }([0,k[ \times \overline{\Omega }_0)\) such that \(u|_S=0\), given by
Making use of (56), it follows
with \(s \le r\). Then, T can be extended in the subspace \(W'\) of W containing the distributions of W with finite \(H^{0,-s}(\overline{\Omega }_k)\)-norm. As a consequence, there exists \(w \in W'^*\), where \(W'^*\) is the topological dual of \(W'\), such that
On the other hand, it results \(w \in H^{0,s}(\Omega _k)\) and since
it follows for every \(\{ \varphi _n \} \subseteq C^{\infty }_0([0,+ \infty [ \times \overline{\Omega }_0)\) such that \(\varphi _n|_S=0\), \(\forall n \in \mathbb {N}\), and \(\varphi _n \rightharpoonup w\) in \(W'^*\),
we deduce that \(w|_S=0\) (see also below).
Taking into account (61) and (60), we get
From (61), we have
and
Indeed, set \(Lw= Pw+ \partial _{x_0}^2 w - \frac{1}{i} a_0(x) \partial _{x_0} w - b(x) w\), it results
with \(w \in \mathcal {D}'(\Omega _k) \cap H^{0,r}(\Omega _k)\) and \(f-Lw \in L^2(\Omega _k)\). From (62), it follows that w is a solution to a second-order differential equation with zero-order term belonging to \(L^2(\Omega _k)\). Hence, we have \(w \in H^{2,0}(\Omega _k) \cap H^{0,r}(\Omega _k)\). On the other hand, (62) implies
with \(\alpha _1+\alpha _2 \le s-r+2\). Therefore, we obtain
Proceeding by induction in the previous equality, assuming \(u \in H^{2, p-1}\), with \(1 \le p \le r-2\) and taking into account (63), it results
Subsequently, by the equality
with \(0 \le p-2+ \alpha _1 + \alpha _2 \le r-2\), and proceeding by induction on p, it follows
From (61), we deduce
Then, we obtain
Now, making use of (61), we show that the boundary conditions on \(\Omega _0\) are satisfied. Let \(u(x_0,x')=u_0(x_0)u_1(x')\) such that \(u_0 \in C^{\infty }_0([0,k_1[)\), \(u_0(0)=1\), \(\partial _{x_0} u_0(0)=0\) and \(u_1 \in C^{\infty }_0(\Omega _0)\). Integrating by parts in (61), we have
It follows
It implies
Instead, if \(u(x_0,x')= u_0(x_0)u_1(x')\), with \(u_0 \in C^{\infty }_0([0,k_1[)\), \(u_0(0)=0\), \(\partial _{x_0} u_0(0)=1\) and \(u_1 \in C^{\infty }_0(\Omega _0)\), integrating by parts, we obtain
Hence, it results
Then, we have proved that the Cauchy problem
admits a solution \(w \in H^r(\overline{\Omega }_k \setminus \partial \Omega _0)\), for every \(k>0\), under assumptions (i), (ii) and (iii) and if \(f \in H^r(\overline{\Omega }_k)\). Finally, we justify that \(w|_S=0\), as written above. In fact, integrating by parts in (61), we get
It follows
Fixed an arbitrary test function \(\phi \) on S, it is possible to determine u such that \(n_1 \partial _{x_1} u + n_2 (x_0- \alpha (x'))^2 \partial _{x_2} u|_S = \phi (x_0,x')\). Then, we obtain
which implies
In the following, a brief proof of the previous claim is given. Parameterizing the surface S in the following way:
with \(x_0 \in [0,k[\) and \(s \in [0, L(\partial \Omega _0)]\), being s the arc length of \(\partial \Omega _0\), we have
where n is the external normal vector to the surface S. Hence, in order to obtain (64), we need that \(\dfrac{du}{dn}|_S = \phi (x_0,s)\), where \(\phi \) is an arbitrary function belonging to \(C^{\infty }_0( [0,k[ \times ]0, L(\partial \Omega _0) [)\). As a consequence, we have proved the existence of a solution \(w \in H^r(\Omega _k \setminus \partial \Omega _0)\) to the following Cauchy–Dirichlet problem
where \(f \in H^r(\overline{\Omega }_k)\). Since \(f \in H^{r}_{loc}(\overline{\Omega })\) and for the arbitrariness of k, Theorem 2 is obtained.
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Barbagallo, A., Esposito, V. Existence results for the mixed Cauchy–Dirichlet problem for a class of hyperbolic operators. Annali di Matematica 200, 2235–2262 (2021). https://doi.org/10.1007/s10231-021-01078-6
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DOI: https://doi.org/10.1007/s10231-021-01078-6