New global stability estimates for the Calderón problem in two dimensions (Q2841761)

Let \(D \subset \mathbb{R}^2\) be an open bounded domain with boundary \(\partial D \in C^2\). The author considers the problem of determining the potential \(v\) in the equation \((-\Delta + v) u = 0\) in \(D\) from the Dirichlet-to-Neumann map \(\Phi: H^{1/2}(\partial D) \to H^{-1/2}(\partial D)\) defined by \(\Phi(f) = \partial u/\partial \nu|_{\partial D}\), where \(u\) is the \(H^1\) solution to the above equation satisfying the Dirichlet condition \(u|_{\partial D} = f\). The author proves that if \(0\) is not a Dirichlet eigenvalue for the operator \(-\Delta + v_i\) in \(D\), \(v_i \in W^{m,1}(\mathbb{R}^2)\) for some \(m > 2\), supp \(v_i \subset D\), and \(\Phi_i\) are corresponding Dirichlet-to-Neumann operators, \(i =1,2\), then NEWLINENEWLINE\[NEWLINE\|v_2 - v_1\|_{L^\infty(D)} \leq c \big(\log (3 + \|\Phi_2 - \Phi_1\|^{-1})\big)^{-\alpha},NEWLINE\]NEWLINENEWLINENEWLINE where \(\alpha = m-2\) and \(\|\Phi_2 - \Phi_1\| = \|\Phi_2 - \Phi_1\|_{H^{1/2} \to H^{-1/2}}\).