A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in R^n＋1, and such that its mean curvature is constant, is a sphere. Here we study the p...A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in R^n＋1, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane Xn＋1 =constant in case M satisfies： for any two points （X^1, ^↑Xn＋1）, （X^1, Xn＋1） on M, with Xn＋1 〉^↑Xn＋1, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional conditions. Some variations of the Hopf Lemma are also presented. Several open problems are described. Part I dealt with corresponding one dimensional problems.展开更多
In this paper, we obtain the strong comparison principle and Hopf Lemma for locally Lipschitz viscosity solutions to a class of nonlinear degenerate elliptic operators of the form △2ψ ＋ L（x,△ ψ）, including the ...In this paper, we obtain the strong comparison principle and Hopf Lemma for locally Lipschitz viscosity solutions to a class of nonlinear degenerate elliptic operators of the form △2ψ ＋ L（x,△ ψ）, including the conformal hessian operator.展开更多
文摘A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in R^n＋1, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane Xn＋1 =constant in case M satisfies： for any two points （X^1, ^↑Xn＋1）, （X^1, Xn＋1） on M, with Xn＋1 〉^↑Xn＋1, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional conditions. Some variations of the Hopf Lemma are also presented. Several open problems are described. Part I dealt with corresponding one dimensional problems.
基金partially supported by NSF grant DMS-1501004partially supported by NSFC(11701027)
文摘In this paper, we obtain the strong comparison principle and Hopf Lemma for locally Lipschitz viscosity solutions to a class of nonlinear degenerate elliptic operators of the form △2ψ ＋ L（x,△ ψ）, including the conformal hessian operator.
