2018
DOI: 10.1093/imrn/rnx296
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Classical Neumann Problems for Hessian Equations and Alexandrov–Fenchel’s Inequalities

Abstract: Recently, the first named author together with Xinan Ma [11], have proved the existence of the Neumann problems for Hessian equations. In this paper, we proceed further to study classical Neumann problems for Hessian equations. We prove here the existence of classical Neumann problems under the uniformly convex domain in R n . As an application, we use the solution of the classical Neumann problem to give a new proof of a family of Alexandrov-Fenchel inequalities arising from convex geometry. This geometric ap… Show more

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Cited by 19 publications
(22 citation statements)
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“…So, we cannot obtain a uniform bound for the solutions of (3), and cannot use the method of continuity directly to get the existence. As in [16,20], we consider the k-admissible solution u ε of the equation…”
Section: Remark 14 For the Classical Neumann Problem Of Hessian Quomentioning
confidence: 99%
See 3 more Smart Citations
“…So, we cannot obtain a uniform bound for the solutions of (3), and cannot use the method of continuity directly to get the existence. As in [16,20], we consider the k-admissible solution u ε of the equation…”
Section: Remark 14 For the Classical Neumann Problem Of Hessian Quomentioning
confidence: 99%
“…2050018-3 is solvable in strictly (k − 1)-convex domains, and see [19] for the proof. For general cases, the problem is open.…”
Section: Remark 16mentioning
confidence: 99%
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“…For related results on the Neumann or oblique derivative problem for some class of fully nonlinear elliptic equations can be found in Urbas [33] and [34]. For the Neumann problem of k-Hessian equations, Trudinger [31] established the existence theorem when the domain is a ball, and Ma-Qiu [27] and Qiu-Xia [28] solved the strictly convex domain case. D.K.…”
Section: Introductionmentioning
confidence: 99%