The $L^p$ Carleman estimate and a partial data inverse problem

TL;DR

This paper develops an explicit Green's function for a conjugated Laplacian to address partial boundary data inverse problems for the Schrödinger equation, extending Carleman estimates to functions supported near part of the boundary.

Contribution

It introduces a new explicit Green's function for the conjugated Laplacian and applies it to solve partial data inverse problems and derive boundary-supported Carleman estimates.

Findings

Constructed an explicit Green's function for the conjugated Laplacian.

Solved a partial data inverse problem for Schrödinger equations with $L^{n/2}$ potentials.

Derived $L^p$ Carleman estimates applicable to functions supported near boundary parts.

Abstract

We construct an explicit Green's function for the conjugated Laplacian $e^{-\omega \cdot x/h}\Delta e^{-\omega \cdot x/h}$, which let 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 Schr\"odinger equation with potential $q \in L^{n/2}$. We also use this Green's function to derive $L^p$ Carleman estimates similar to the ones in Kenig-Ruiz-Sogge \cite{krs}, but for functions with support up to part of the boundary.