Stationary isothermic surfaces and some characterizations of the hyperplane in the N-dimensional Euclidean space

TL;DR

This paper characterizes stationary isothermic surfaces in N-dimensional Euclidean space, showing they must be hyperplanes under certain conditions, linking heat flow properties to geometric characterizations.

Contribution

It provides a new characterization of hyperplanes as stationary isothermic surfaces in heat flow problems, extending classical Liouville and Bernstein theorems.

Findings

Stationary isothermic surfaces are hyperplanes under general conditions.

Connection established between heat flow boundary conditions and geometric surface properties.

Extension of classical elliptic PDE theorems to geometric characterizations.

Abstract

We consider an entire graph S of a continuous real function over (N-1)-dimensional Euclidean space with N larger than or equal to 3. Let D be a domain in N-dimensional Euclidean space with S as a boundary. Consider in D the heat flow with initial temperature 0 and boundary temperature 1. The problem we consider is to characterize S in such a way that there exists a stationary isothermic surface in D. We show that S must be a hyperplane under some general conditions on S. This is related to Liouville or Bernstein-type theorems for some elliptic Monge-Amp\`ere-type equation.