An inverse problem for the minimal surface equation

Abstract

We use the method of higher order linearization to study an inverse boundary value problem for the minimal surface equation on a Riemannian manifold $({\mathbb{R}}^n,g)$, where the metric $g$ is conformally Euclidean. In particular we show that with the knowledge of Dirichlet-to-Neumann map associated to the minimal surface equation, one can determine the Taylor series of the conformal factor $c(x)$ at $x_n=0$ up to a multiplicative constant. We show this both in the full data case and in some partial data cases.

22 pages, Modified assumptions in Theorem 1.1 and added partial data results, changed the proof accordingly, removed Propositions 5.2 and 5.3 for being redundant in this modified case of the main theorem, corrected typos