Summary: We consider an entire graph \(S:x_{N+1}=f(x)\), \(x\in\mathbb R^N\) in \(\mathbb R^{N+1}\) of a continuous real function \(f\) over \(\mathbb R^N\) with \(N\geq 1\). Let \(\varOmega\) be an unbounded domain in \(\mathbb R^{N+1}\) with boundary \(\partial\varOmega =S\). Consider nonlinear diffusion equations of the form \(\partial_tU=\varDelta\phi (U)\) containing the heat equation \(\partial_tU=\varDelta U\). Let \(U=U(X,t)=U(x,x_{N+1},t)\) be the solution of either the initial-boundary value problem over \(\varOmega\) where the initial value equals zero and the boundary value equals 1, or the Cauchy problem where the initial datum is the characteristic function of the set \(\mathbb R^{N+1}\backslash\varOmega\). The problem we consider is to characterize \(S\) in such a way that there exists a stationary level surface of \(U\) in \(\varOmega\).
We introduce a new class \(\mathcal{A}\) of entire graphs \(S\) and, by using the sliding method due to Berestycki, Caffarelli, and Nirenberg, we show that \(S\in\mathcal{A}\) must be a hyperplane if there exists a stationary level surface of \(U\) in \(\varOmega\). This is an improvement of the previous result of R. Magnanini and the author [J. Differ. Equations 252, No. 1, 236–257 (2012; Zbl 1243.35096), Theorem 2.3 and Remark 2.4]. Next, we consider the heat equation in particular and we introduce the class \(\mathcal{B}\) of entire graphs \(S\) of functions \(f\) such that \(S\{| f(x)-f(y)|:| x-y|\leq 1\}\) 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 \(\varOmega\). Related to the problem, we consider a class \(\mathcal{W}\) of Weingarten hypersurfaces in \(\mathbb R^{N+1}\) with \(N\geq 1\). 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.
For the entire collection see [Zbl 1257.35001].