Stability of memory reconstruction from the Dirichlet-to-Neumann operator
The article is devoted to determining the memory function \( k(x,t) \) in the integro-differential equation \[ u_{tt} (x,t) - \Delta u(x,t) - \int\limits_{-\infty}^{t} u(x, \tau) k(x, t - \tau) d\tau = 0 \tag{1} \] with the boundary condition \[ u(x,t) {\mid}_{\partial \Omega \times \mathbb R} = g(x,t). \tag{2} \] Here \( x \in \Omega \subset {\mathbb {R}}^n\), \(n \geq 3\), \(t >0\), \(\Omega\) is a bounded domain with a smooth boundary and the memory function \( k(x,t)\) belongs to the class \[ W(K) := \bigl\{k(x,t) \in C^2 \bigl(\overline{\Omega} \times [0, \infty) \bigr) \mid |k |_{C^2 (\overline{\Omega} \times [0, \infty))} \leq K \bigr\} \] for some \( K > 0 \), \( k(x,t) = 0 \) for \( t < 0 \). The authors consider the inverse problem of the reconstruction of \(k(x,t) \) from the Dirichlet-to-Neumann operator \( H\: H^{1}_{\gamma} (\partial \Omega \times {\mathbb R} ) \longrightarrow H^{0}_{\gamma} (\partial \Omega \times {\mathbb R} )\), where \( H\bigl( g(x,t)\bigr) = {\partial}_{\nu} u(x,t) \) is the normal derivative of the solution to the direct problem (1), (2). The main results of the article state that, for different \( k_1, k_2 \in W(K) \), there exists a positive \( {\gamma}_0 = {\gamma}_0 (K, \Omega) \) such that, for \( \gamma > {\gamma}_0 \), the corresponding Dirichlet-to-Neumann operators \( H_1 \), \(H_2 \) act correctly and satisfy the estimate \( |k_1 - k_2 |_{\gamma , 0} \leq \omega (|H_1 - H_2 |), \) where \( \omega (\varepsilon) \sim C(\Omega , K, \gamma , n) \cdot \bigl(C(\Omega)/(\ln {\varepsilon}^{-1})\bigr)^{1/(n+2)} \) as \( \varepsilon \rightarrow 0 \). Here \( |. |_{\gamma , s} \) is the norm in the space \( H^{s}_{\gamma} (\partial \Omega \times {\mathbb R} )\) that consists of the functions \( u(x,t) \) such that \( e^{- \gamma t} u(x,t) \in H^s (\partial \Omega \times {\mathbb R} )\).
scientific article; zbMATH DE number 1099582