Inverse/observability estimates for second-order hyperbolic equations with variable coefficients.(English)Zbl 0931.35022

The authors consider the Dirichlet and respectively the Neumann mixed second-order hyperbolic problems in the unknown $$w(t,x)$$ and their dual homogeneous problems in $$\psi(t,x)$$, as it follows: $\begin{cases} w_{tt}+{\mathcal A}w= F_1(w)\text{ in }\Omega,\quad & \psi_{tt}+{\mathcal A}_\psi=F(\psi)\text{ in }Q,\\ w(0,\cdot)= w_0,\;w_t(0, \cdot)= w_1\text{ in }\Omega, \quad & \psi(T, \cdot)= \psi_0,\;\psi_t(T,\cdot)= \psi_1\text{ in }\Omega,\\ w|_{\Sigma_0} =0;\;w |_{\Sigma_1}=u,\quad & \psi|_\Sigma=0, \end{cases} \tag{1}$
$\begin{cases} w_{tt} +{\mathcal A}_w=F_1(w);\quad & \psi_{tt}+{\mathcal A}_\psi=F(\psi)\text{ in }Q;\\ w(0,\cdot)=w_0,\;w_t(0,\cdot)=w_1;\quad & \psi(T, \cdot)= \psi_0,\;\psi_t(T, \cdot)= \psi_1\text{ in }\Omega; \\ w|_{\Sigma_0}=0; \quad & \psi |_{ \Sigma_0} =0\text{ in }\Sigma_0; \\ {\partial w\over\partial \nu_{\mathcal A}} |_{\Sigma_1}=u; \quad & \left[{\partial\psi \over\partial \nu_{\mathcal A}}+ B \psi \right]_{\Sigma_1}= 0\text{ in }\Sigma_1; \end{cases} \tag{2}$ where the boundary $$\Gamma= \partial\Omega$$ is of class $$C^2$$, $$\Gamma=\Gamma_0 \cup \Gamma_1$$, $${\mathcal A}$$ is the second-order differential operator ${\mathcal A}w \equiv -\sum^n_{i,j=1}{\partial\over\partial x_i}\left(a_{ij}(x){\partial w \over\partial x_j} \right)$ satisfying the uniform ellipticity condition, $$F$$ is a suitable first-order differential operator depending on the original operator $$F_1$$ and $$u\in L_2(0,T;L_2(\Gamma_1))$$.
The first goal of this paper is to establish an a priori inequality of the homogeneous Dirichlet $$\psi$$-problem (1): There exists a constant $$T_0>0$$, depending upon the triple $$(\Omega,\Gamma_0, \Gamma_1)$$ and the coefficients $$a_{ij}$$ such that for all $$T>T_0$$, there is a constant $$c_T>0$$ for which $\int^T_0 \int_{\Gamma_1} \left( {\partial \psi\over \partial\nu_{\mathcal A}} \right)^2 d\Sigma_1\geq c_T\bigl \|(\psi_0, \psi_1) \bigr\|^2_{H^1_0 (\Omega) \times L_2(\Omega)}.\tag{3}$ The inequality (3) is the continuous observability inequality for the $$\psi$$-problem (1) and it is, by duality, equivalent to the exact controllability property of the nonhomogeneous $$w$$-problem (1) at time $$T$$, on the space $$L_2(\Omega)\times H^{-1}(\Omega)$$, within the class of $$L_2 (0,T; L_2(\Gamma_1))$$-controls.
The second goal is to establish the continuous observability inequality for the $$\psi$$-problem (2): There exists a constant $$T_0>0$$, depending upon the triple $$(\Omega, \Gamma_0,\Gamma_1)$$ and the coefficients $$a_{ij}$$, such that for all $$T>T_0$$ there is a constant $$c_T>0$$ for which $\int^T_0 \int_{\Gamma_1}\psi^2_t d\Sigma_1\geq c_T \bigl\|(\psi_0,\psi_1) \bigr \|^2_{H^1_{\Gamma_0}(\Omega)\times L_2(\Omega)},$ where $$H^1_{\Gamma_0} (\Omega)= \{f\in H^1(\Omega);f|_{\Gamma_0}=0\}$$.
The method used has three main steps: the first step employs a Riemann geometry approach to reduce the original variable coefficient principal part problem in $$\Omega\subset\mathbb{R}^n$$ to a problem on an appropriate Riemann manifold; the second step employs explicit Carleman estimates at the differential level to take care of the variable first-order terms; the third step employs the micro-local analysis yielding a sharp trace estimate, in the Neumann case.

MSC:

 35B45 A priori estimates in context of PDEs 35L20 Initial-boundary value problems for second-order hyperbolic equations 93B05 Controllability 35L70 Second-order nonlinear hyperbolic equations
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References:

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