Lipschitz stability in an inverse problem for the wave equation

Abstract

We are interested in the inverse problem of the determination of the potential $p(x), x\in\Omega\subset\mathbb{R}^n$ from the measurement of the normal derivative $\partial_\nu u$ on a suitable part $\Gamma_0$ of the boundary of $\Omega$, where $u$ is the solution of the wave equation $\partial_{tt}u(x,t)-\Delta u(x,t)+p(x)u(x,t)=0$ set in $\Omega\times(0,T)$ and given Dirichlet boundary data. More precisely, we will prove local uniqueness and stability for this inverse problem and the main tool will be a global Carleman estimate, result also interesting by itself.