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Comparison principles and Liouville theorems for prescribed mean curvature equations in unbounded domains. (English) Zbl 0705.49022

The purpose is to give some comparison principles and Liouville theorems for prescribed mean curvature equations in unbounded domains.
Let \(\Omega \subset {\mathbb{R}}^ 2\) (bounded or unbounded), for every \(R>0\), let \(B_ R=\{x\in {\mathbb{R}}^ n| | x| <R\}\), \({\tilde \Omega}{}_ R=\Omega \cap B_ R\), for \(u\in C^ 1(\Omega)\), Tu denotes the vector (Du/\(\sqrt{1+| Du|^ 2})\), where Du is the gradient vector of u.
Theorem 2.3. Let \(\partial \Omega =\Sigma^{\alpha}+\Sigma^{\beta}\) be a decomposition of \(\partial \Omega\) such that \(\Sigma^{\beta}\) is of class \(C^ 1\) and for every u,v\(\in C^ 2(\Omega)\cap C^ 1(\Omega \cup \Sigma^{\beta})\cap C^ 0({\bar \Omega})\), let \(M(R)=\max_{{\tilde \Omega}_ R}| u-v|\). Suppose that
(i) div Tu\(=div Tv\) in \(\Omega\)
(ii) \(u=v\) on \(\Sigma^{\alpha}\)
(iii) \(Tu\cdot \nu =Tv\cdot \nu\) on \(\Sigma^{\beta}\) where \(\nu\) is the outer normal of \(\Sigma^{\beta}\)
(iv) \(M(R)=O(\sqrt{\log R})\) as \(R\to \infty.\)
Then if \(\partial \Omega =\Sigma^{\beta}\), we have \(u(x)\equiv v(x)+text{constant}\), otherwise \(u(x)=v(x).\)
Other results are similar as above. The main idea of our theorems depends on the following inequality only: \[ (*)\quad (Tu-Tv)\cdot (Du-Dv)\geq | Tu-Tv|^ 2. \] Consider the equation in divergence form: div A(x,w,Dw)\(=f(x,w,Dw)\). If there exists a positive constant \(\lambda\) such that \[ | A(x,w,Dw)-A(x,Dw',Dw')| \leq \lambda (Dw-Dw')\cdot (A(x,w,Dw)-A(x,w',Dw')) \] and \((Dw-Dw')\cdot (A(x,w,Dw)-A(x,w',Dw'))=0\) if and only if \(Dw=Dw'\). In this case we have similar results.
The reader may find the inequality (*) in a paper of L.-F. Tam [“On the uniqueness of capillary surfaces”, Variational methods for free surface interfaces, Proc. Conf., Menlo Park/Calif. 1985, 99-108 (1987)] where C. Wong is another spelling of the author’s name.
Reviewer: Jenn-Fang Hwang

MSC:

49Q05 Minimal surfaces and optimization
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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References:

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