Stationary level surfaces and Liouville-type theorems characterizing hyperplanes

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.