Stationary Level Surfaces and Liouville-Type Theorems Characterizing Hyperplanes
Stationary Level Surfaces and Liouville-Type Theorems Characterizing Hyperplanes
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静止水平面和表征超平面的刘维尔型定理
DOI：
10.1007/978-88-470-2841-8_17
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
Shigeru Sakaguchi
中科院分区：
文献类型：
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作者：
Shigeru Sakaguchi
考虑N≥ 1的连续真实的函数f的整图S：xN + 1= f（x），x∈ <$N.设Ω是N+ 1中的无界区域，其边界为S.考虑非线性扩散方程的形式为Δ t U= Δ t（U），其中包含热方程Δ t U= ΔU。设U= U（X，t）= U（x，xN + 1，t）是Ω上初值为零、边值为1的初边值问题的解，或者是初值为集合<$N+ 1 <$Ω的特征函数的柯西问题的解.我们考虑的问题是以这样一种方式来刻画S，即在Ω中存在U的平稳水平曲面。本文引入了一类新的整图S，并利用Berestycki，Caffarelli和Nirenberg的滑动方法证明了：如果在Ω中存在U的平稳水平曲面，则S必是超平面.这是对先前结果的改进（Magnanini和Sakaguchi in J. Differ.当量ag 252：236-257，2012，Theorem 2.3和Remark 2.4）。接下来，我们特别考虑热方程，并引入函数f的整图类S，使得{|f（x）-f（y）|:|x− y| 1}是有界的。利用粘性解理论，证明了如果U在Ω中存在一个定常等温面，则U必是一个超平面。这是对先前结果的相当大的改进（Magnanini和Sakaguchi in J. Differ.当量ag 248：1112-1119，2010，Theorem 1.1，case（ii））。与此相关，我们考虑了N≥ 1的N+ 1中的一类Weingarten超曲面.证明了若S是粘性的，且S满足某些自然几何条件，则S必是超平面。这也是一个相当大的改进以前的结果（坂口离散连续。动力学系统：S4：887-895，2011，Theorem 1.1）。
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).