Stability estimates for the inverse problem of finding the absorption constant (Q1874761)

This paper considers the problem of reconstructing the spacewise dependent coefficient in an equation of hyperbolic type. Let \(u\) and \(\sigma\) satisfy \[ \begin{gathered} u_{tt}- u_{xx}- u_{yy}+ \sigma(x,y) u_t= \delta(y,t),\;u|_{t< 0}\equiv 0,\;(x,y,t)\in \mathbb{R}^3,\\ \text{supp}(\sigma)\subset\{(x, y)\in \mathbb{R}^2\mid y> 0\},\end{gathered}\tag{1} \] where \(\delta\) is the Dirac delta function. In a particular case of the paper, the inverse problem requires finding the absorption coefficient \(\sigma\) in a domain \(D\subset B(0, 2,1):= \{(x,y)\in \mathbb{R}^2\mid x^2+ (y- 2)^2< 1\}\) with smooth boundary, from the traces of the solution \(u\) of the problem (1) and its first-order partial derivatives with respect to \(x\), \(y\) and \(t\) given on \(S:= \{(x,y,t)\in \mathbb{R}^3\mid (x,y)\in\partial D, |y|< t< 5+|y|\}\), i.e. \[ D^k u|_S= f^k(x, y,t),\quad k= (k_1, k_2, k_3),\;|k|\leq 1.\tag{2} \] By defining the sets \[ P:= \{(x, y)\in \mathbb{R}^2\mid [x^2+ (y- 2)^2]^{1/2}+|y|< 9\}, \] \[ \Sigma(\sigma):= \{\sigma(x,y)\in H^8(P)\mid\|\partial^k\sigma/\partial^{k_1} x\partial^{k_2} y\|_{C(P)}\leq \sigma,\;k= (k_1,k_2), |k|\leq 6\} \] then it is shown that the following estimate holds: There exists \(0< \sigma\leq 1\) and \(C= C(\text{dist}(D,\partial B(0,2)))> 0\) such that \[ \int_D (\sigma_1- \sigma_2)^2 dD\leq C \sum_{|k|\leq 1} \int_S (f^k_1- f^k_2)^2 dS dt,\quad \forall\sigma_1,\sigma_2\in \Sigma(\sigma), \] where \(f^k_1\) and \(f^k_2\) are the data of the inverse problem (1) and (2) corresponding to \(\sigma_1\) and \(\sigma_2\), respectively.