Partial data inverse problem with $L^{n/2}$ potentials

Abstract:

We construct an explicit Green’s function for the conjugated Laplacian $e^{-\omega \cdot x/h}\Delta e^{-\omega \cdot x/h}$, which lets us control our solutions on roughly half of the boundary. We apply the Green’s function to solve a partial data inverse problem for the Schrödinger equation with potential $q \in L^{n/2}$. Separately, we also use this Green’s function to derive $L^p$ Carleman estimates similar to the ones in Kenig-Ruiz-Sogge [Duke Math. J. 55 (1987), pp. 329–347], but for functions with support up to part of the boundary. Unlike many previous results, we did not obtain the partial data result from the boundary Carleman estimate—rather, both results stem from the same explicit construction of the Green’s function. This explicit Green’s function has potential future applications in obtaining direct numerical reconstruction algorithms for partial data Calderón problems which is presently only accessible with full data [Inverse Problems 27 (2011)].