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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: $$\cases w_{tt}+{\cal A}w= F_1(w)\text{ in }\Omega,\quad & \psi_{tt}+{\cal A}_\psi=F(\psi)\text{ in }Q,\\ w(0,\cdot)= w_0,\ w_t(0, \cdot)= w_1\text{ in }\Omega, \quad & \psi(T, \cdot)= \psi_0,\ \psi_t(T,\cdot)= \psi_1\text{ in }\Omega,\\ w|_{\Sigma_0} =0;\ w |_{\Sigma_1}=u,\quad & \psi|_\Sigma=0, \endcases \tag 1$$ $$\cases w_{tt} +{\cal A}_w=F_1(w);\quad & \psi_{tt}+{\cal A}_\psi=F(\psi)\text{ in }Q;\\ w(0,\cdot)=w_0,\ w_t(0,\cdot)=w_1;\quad & \psi(T, \cdot)= \psi_0,\ \psi_t(T, \cdot)= \psi_1\text{ in }\Omega; \\ w|_{\Sigma_0}=0; \quad & \psi |_{ \Sigma_0} =0\text{ in }\Sigma_0; \\ {\partial w\over\partial \nu_{\cal A}} |_{\Sigma_1}=u; \quad & \left[{\partial\psi \over\partial \nu_{\cal A}}+ B \psi \right]_{\Sigma_1}= 0\text{ in }\Sigma_1; \endcases \tag 2$$ where the boundary $\Gamma= \partial\Omega$ is of class $C^2$, $\Gamma=\Gamma_0 \cup \Gamma_1$, ${\cal A}$ is the second-order differential operator $${\cal A}w \equiv -\sum^n_{i,j=1}{\partial\over\partial x_i}\left(a_{ij}(x){\partial w \over\partial x_j} \right)$$ 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 $$\int^T_0 \int_{\Gamma_1} \left( {\partial \psi\over \partial\nu_{\cal A}} \right)^2 d\Sigma_1\ge c_T\bigl \|(\psi_0, \psi_1) \bigr\|^2_{H^1_0 (\Omega) \times L_2(\Omega)}.\tag 3$$ 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 $$\int^T_0 \int_{\Gamma_1}\psi^2_t d\Sigma_1\ge c_T \bigl\|(\psi_0,\psi_1) \bigr \|^2_{H^1_{\Gamma_0}(\Omega)\times L_2(\Omega)},$$ 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 $\Omega\subset\bbfR^n$ 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.

35B45A priori estimates for solutions of PDE
35L20Second order hyperbolic equations, boundary value problems
35L70Nonlinear second-order hyperbolic equations
Full Text: DOI
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