2012/02/16 by Sakaguchi, Shigeru
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1202.3528
We consider an entire graph S in \mathbb RN+1 of a continuous real function f over \mathbb RN with N≥ 1. Let Ω be an unbounded domain in \mathbb RN+1 with boundary S. Consider nonlinear diffusion equations of the form ∂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 RN+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 \mathcal A of entire graphs S and, by using the sliding method, we show that S∈\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| ≤ 1 is bounded. With the help of the theory of viscosity solutions, we show that S ∈ \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 RN+1 with N ≥ 1. Then we show that, if S belongs to \mathcal W in the viscosity sense and S satisfies some natural geometric condition, then S ∈ \mathcal B must be a hyperplane. This is also a considerable improvement of the previous result.