An inverse problem for the minimal surface equation

TL;DR

This paper investigates an inverse boundary value problem for the minimal surface equation on a conformally Euclidean manifold, demonstrating that the Dirichlet-to-Neumann map determines the Taylor series of the conformal factor at a boundary point.

Contribution

It introduces a higher order linearization method to recover the Taylor series of the conformal factor from boundary measurements, including partial data cases.

Findings

The Dirichlet-to-Neumann map determines the Taylor series of the conformal factor at the boundary.

The method applies both in full data and partial data scenarios.

The approach advances inverse boundary value problem techniques for nonlinear PDEs.

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.