An observability estimate in \(L_2(\Omega)\times H^{-1}(\Omega)\) for second-order hyperbolic equations with variable coefficients (Q2712208)

The results are on the hyperbolic system NEWLINE\[NEWLINE \begin{align*}{ y_{tt} + {\cal A}y &= 0, \quad (0 \le t < \infty, \ x \in \Omega),\cr y &= 0,\quad (0 \le t < \infty,\ x \in \Gamma \setminus \Gamma_1),}\end{align*}NEWLINE\]NEWLINE where \(\Omega\) is a bounded \(n\)-dimensional domain with smooth boundary \(\Gamma\) and \(\Gamma_1 \subseteq \Gamma.\) Here, \(\mathcal A\) is a smooth second order elliptic self-adjoint differential operator, and the solutions belong to the class NEWLINE\[NEWLINE \begin{align*}{ (y, y_t) &\in C(0, T; L^2(\Omega) \times H^{-1} (\Omega)) , \cr y|_{(0, T) \times \Gamma_1} &\in L^2((0, T) \times \Gamma_1) , \quad {\partial y \over\partial \nu_{\cal A}}\bigg|_{(0, T) \times \Gamma_1} \in H^{-1}((0, T) \times \Gamma_1) . }\end{align*} NEWLINE\]NEWLINE Let \({\mathcal A}_0 = {\mathcal A}\) restricted to the domain \(H^2(\Omega) \cap H_0^1(\Omega).\) The energy of a solution \(y\) is \({\mathcal E}_y(t) = \|y(t)\|^2_{L^2(\Omega)} + \|{\mathcal A}_0^{-1/2}y_t(t)\|^2_{L^2(\Omega)}\) (which is equivalent to the square of the norm of \((y(t), y_t(t))\) in \(L^2(\Omega) \times H^{-1}(\Omega)).\) The main result in this paper (the observability estimate) is a bound for the integral of \({\mathcal E}_y(t)\) in \(0 \leq t \leq T\) in terms of suitable boundary traces of \(y\) and \(y_t\) on \((0, T) \times \Gamma_1\) and lower order terms. The hypotheses include a Riemann geometric assumption involving the coefficients of \({\mathcal A}\) and the domain \(\Omega\) and a related assumption on \(\Gamma_1.\) There is a corresponding estimate for the nonhomogeneous equation \(y_{tt} + {\mathcal A}y = f.\) NEWLINENEWLINENEWLINEThe authors compare their theorem with existing results for this problem.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00050].