平稳水平曲面与表征超平面的Liouville型定理
Stationary level surfaces and Liouville-type theorems characterizing hyperplanes
摘要：
本文考虑了在$\mathbb R^{N}$上的连续实函数$f$的$\mathbb R^{N+1}$中的整图$S$，其中$N\ge 1$ .设$Ω$是$\mathbb R^{N+1}$中具有边界$S$的无界域.考虑包含热方程的$\partial_t U= Δϕ(U)$形式的非线性扩散方程。设$U$是$Ω$上的初边值问题的解，其中初值为零，边值为1，或者是Cauchy问题的解，其中初值是集合$\mathbb R^{N+1}\setminus Ω$的特征函数。我们考虑的问题是以这样一种方式描述$S$，即在$Ω$中存在$U$的平稳水平表面。引入了一类新的整图$S$的$\mathcal A$，并利用滑动方法证明了如果$S$中存在$U$的平稳水平曲面，则$S\in\mathcal A$必为超平面.这是对先前结果的改进。接下来，我们考虑热方程，特别是我们介绍了类$\mathcal B$的整个图$S$的功能$f$，使每个${|f(x)-f(y)|: |x-y| \le 1}$有界。利用粘性溶液理论，证明了若$Ω$中存在$U$的定态等温面，则$S \in \mathcal B$必为超平面.这是对先前结果的一个相当大的改进。与此相关，我们考虑了具有$N \ge 1$的$\mathbb R^{N+1}$中的一类Weingarten超曲面$\mathcal W$.证明了若$S$在粘性意义下属于$\mathcal W$，且$S$满足某些自然几何条件，则$S \in \mathcal B$必是超平面.这也是对以前结果的一个相当大的改进。
英文摘要：
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 $Ω$ be an unbounded domain in $\mathbb R^{N+1}$ with boundary $S$. Consider nonlinear diffusion equations of the form $\partial_t U= Δϕ(U)$ containing the heat equation. Let $U$ be the solution of either the initial-boundary value problem over $Ω$ 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 $\mathbb R^{N+1}\setminus Ω$. The problem we consider is to characterize $S$ in such a way that there exists a stationary level surface of $U$ in $Ω$. 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 $Ω$. 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 $Ω$. 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.
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