Stationary level surfaces and Liouville-type theorems characterizing hyperplanes

TL;DR

This paper characterizes entire graphs in Euclidean space that admit stationary level surfaces of nonlinear diffusion equations, proving they must be hyperplanes under various geometric and analytic conditions, thus extending classical Liouville-type theorems.

Contribution

The paper introduces new classes of entire graphs and uses the sliding method and viscosity solutions to show such graphs must be hyperplanes if stationary level surfaces exist, improving previous results.

Findings

Stationary level surfaces imply the graph is a hyperplane for class .

Graphs in class  with bounded differences are hyperplanes if stationary isothermic surfaces exist.

Weingarten hypersurfaces satisfying geometric conditions are hyperplanes under stationary surface assumptions.

Abstract

We consider an entire graph $S$ in $\mathbb R^{N+1}$ of a continuous real function $f$ over $\mathbb R^{N}$ with $N\ge 1$. Let $\Omega$ be an unbounded domain in $\mathbb R^{N+1}$ with boundary $S$. Consider nonlinear diffusion equations of the form $\partial_t U= \Delta \phi(U)$ containing the heat equation. Let $U$ be the solution of either the initial-boundary value problem over $\Omega$ where the initial value equals zero and the boundary value equals 1, or the Cauchy problem where the initial data is the characteristic function of the set $\mathbb R^{N+1}\setminus \Omega$. The problem we consider is to characterize $S$ in such a way that there exists a stationary level surface of $U$ in $\Omega$. We introduce a new class $\mathcal A$ of entire graphs $S$ and, by using the sliding method, we show that $S\in\mathcal A$ must be a hyperplane if there exists a stationary level surface of $U$ in $\Omega$. This is an improvement of the previous result. Next, we consider the heat equation in particular and we introduce the class $\mathcal B$ of entire graphs $S$ of functions $f$ such that each ${|f(x)-f(y)|: |x-y| \le 1}$ is bounded. With the help of the theory of viscosity solutions, we show that $S \in \mathcal B$ must be a hyperplane if there exists a stationary isothermic surface of $U$ in $\Omega$. This is a considerable improvement of the previous result. Related to the problem, we consider a class $\mathcal W$ of Weingarten hypersurfaces in $\mathbb R^{N+1}$ with $N \ge 1$. Then we show that, if $S$ belongs to $\mathcal W$ in the viscosity sense and $S$ satisfies some natural geometric condition, then $S \in \mathcal B$ must be a hyperplane. This is also a considerable improvement of the previous result.