Determining the potential in a wave equation without a geometric condition. Extension to the heat equation (Q2817014)

The paper deals with the initial-boundary value problem for the wave equation NEWLINE\[NEWLINE\begin{cases} \partial_t^2u-\Delta u+q(x)u=0,\text{ in }Q=\Omega\times (0,\tau),\\ u=0,\text{ on }\Sigma=\Gamma\times (0,\tau),\\ u(\cdot,0)=u_0,\quad \partial_tu(\cdot,0)=0, \end{cases}\tag{P}NEWLINE\]NEWLINE where \(\Omega\) is a \(C^3\)-smooth bounded domain of \(\mathbb{R}^n\), \(n\geq 2\), with boundary \(\Gamma\). Using a spectral decomposition method and an interpolation inequality established by L. Robbiano, the authors prove a logarithmic stability estimate for the inverse problem of determining the potential \(q\) in (P) from boundary measurements. An inverse coefficient problem for the heat equation is also investigated.