Stationary Level Surfaces and Liouville-Type Theorems Characterizing Hyperplanes

Abstract

We consider an entire graph S:x _N+1=f(x), x∈ℝ^N in ℝ^N+1 of a continuous real function f over ℝ^N with N≥1. Let Ω be an unbounded domain in ℝ^N+1 with boundary ∂Ω=S. Consider nonlinear diffusion equations of the form ∂ _t U=Δϕ(U) containing the heat equation ∂ _t U=ΔU. Let U=U(X,t)=U(x,x _N+1,t) 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 datum is the characteristic function of the set ℝ^N+1∖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 of entire graphs S and, by using the sliding method due to Berestycki, Caffarelli, and Nirenberg, we show that must be a hyperplane if there exists a stationary level surface of U in Ω. This is an improvement of the previous result (Magnanini and Sakaguchi in J. Differ. Equ. 252:236–257, 2012, Theorem 2.3 and Remark 2.4). Next, we consider the heat equation in particular and we introduce the class of entire graphs S of functions f such that {|f(x)−f(y)|:|x−y|≤1} is bounded. With the help of the theory of viscosity solutions, we show that must be a hyperplane if there exists a stationary isothermic surface of U in Ω. This is a considerable improvement of the previous result (Magnanini and Sakaguchi in J. Differ. Equ. 248:1112–1119, 2010, Theorem 1.1, case (ii)).

Related to the problem, we consider a class of Weingarten hypersurfaces in ℝ^N+1 with N≥1. Then we show that, if S belongs to in the viscosity sense and S satisfies some natural geometric condition, then must be a hyperplane. This is also a considerable improvement of the previous result (Sakaguchi in Discrete Contin. Dyn. Syst., Ser. S 4:887–895, 2011, Theorem 1.1).