scholarly journals Classical Neumann Problems for Hessian Equations and Alexandrov–Fenchel’s Inequalities

2018 ◽  
Vol 2019 (20) ◽  
pp. 6285-6303 ◽  
Author(s):  
Guohuan Qiu ◽  
Chao Xia
Keyword(s):  
Neumann Problem ◽  
Convex Geometry ◽  
Uniformly Convex ◽  
Convex Domains ◽  
Neumann Problems ◽  
Geometric Application ◽  
Hessian Equations

Abstract Recently, the 1st named author together, with Xinan Ma [12], has 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 for uniformly convex domains in $\mathbb {R}^{n}$. 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 application is motivated by Reilly [18].

scholarly journals Poincaré Inequalities and Neumann Problems for the p-Laplacian

2018 ◽  
Vol 61 (4) ◽  
pp. 738-753 ◽  
Author(s):  
David Cruz-Uribe ◽  
Scott Rodney ◽  
Emily Rosta
Keyword(s):  
Quadratic Form ◽  
Neumann Problem ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Poincaré Inequalities ◽  
Existence Of Weak Solutions ◽  
Neumann Problems ◽  
Poincare Inequalities ◽  
Associated Matrix ◽  
Weighted Poincaré Inequalities

AbstractWe prove an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a Neumann problem related to a degenerate p-Laplacian. The Poincaré inequalities are formulated in the context of degenerate Sobolev spaces defined in terms of a quadratic form, and the associated matrix is the source of the degeneracy in the p-Laplacian.

scholarly journals Weak Solutions and Energy Estimates for a Class of Nonlinear Elliptic Neumann Problems

2013 ◽  
Vol 13 (2) ◽  
Author(s):  
Gabriele Bonanno ◽  
Giovanni Molica Bisci ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Neumann Problem ◽  
Laplace Operator ◽  
Weak Solutions ◽  
Existence Results ◽  
Energy Estimates ◽  
Lack Of Compactness ◽  
Neumann Problems ◽  
Estimates Of Solutions ◽  
Nonlinear Elliptic

AbstractWe establish existence results and energy estimates of solutions for a homogeneous Neumann problem involving the p-Laplace operator. The case of large dimensions, corresponding to the lack of compactness of W

Multiple Solutions for Nonlinear Neumann Problems with Asymmetric Reaction, via Morse Theory

2011 ◽  
Vol 11 (4) ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Variational Methods ◽  
Neumann Problem ◽  
Morse Theory ◽  
Multiple Solutions ◽  
Smooth Solutions ◽  
Asymmetric Reaction ◽  
Neumann Problems ◽  
Nonlinear Neumann Problem ◽  
Asymmetric Behaviour ◽  
Nonlinear Neumann Problems

AbstractWe consider a nonlinear Neumann problem driven by the p-Laplacian and with a reaction which exhibits an asymmetric behaviour near +∞ and near −∞. Namely, it is (p − 1)- superlinear near +∞ (but need not satisfy the Ambrosetti-Rabinowitz condition) and it is (p − 1)-linear near −∞. Combining variational methods with Morse theory, we show that the problem has at least three nontrivial smooth solutions.

Global solvability and regularity for ∂¯ on an annulus between two weakly convex domains which satisfy property (P)

2019 ◽  
Vol 12 (03) ◽  
pp. 1950041
Author(s):  
Sayed Saber
Keyword(s):  
Neumann Problem ◽  
Complex Manifold ◽  
Global Solvability ◽  
Closed Formula ◽  
Convex Domains ◽  
Weakly Convex ◽  
Canonical Solution ◽  
Dimension Formula ◽  
Satisfy Property

Let [Formula: see text] be a complex manifold of dimension [Formula: see text] and let [Formula: see text]. Let [Formula: see text] be a weakly [Formula: see text]-convex and [Formula: see text] be a weakly [Formula: see text]-convex in [Formula: see text] with smooth boundaries such that [Formula: see text]. Assume that [Formula: see text] and [Formula: see text] satisfy property [Formula: see text]. Then the compactness estimate for [Formula: see text]-forms [Formula: see text] holds for the [Formula: see text]-Neumann problem on the annulus domain [Formula: see text]. Furthermore, if [Formula: see text] is [Formula: see text]-closed [Formula: see text]-form, which is [Formula: see text] on [Formula: see text] and which is cohomologous to zero on [Formula: see text], the canonical solution [Formula: see text] of the equation [Formula: see text] is smooth on [Formula: see text].

scholarly journals Compactness of the ∂-Neumann Problem on Convex Domains

1998 ◽  
Vol 159 (2) ◽  
pp. 629-641 ◽  
Author(s):  
Siqi Fu ◽  
Emil J Straube
Keyword(s):  
Neumann Problem ◽  
Convex Domains ◽  
The Neumann Problem

scholarly journals Multiple solutions for some Neumann problems in exterior domains

2006 ◽  
Vol 73 (3) ◽  
pp. 353-364 ◽  
Author(s):  
Tsing-San Hsu ◽  
Huei-Li Lin
Keyword(s):  
Neumann Problem ◽  
Multiple Solutions ◽  
The Other ◽  
Exterior Domains ◽  
Neumann Problems ◽  
The Neumann Problem

In this paper, we show that if q(x) satisfies suitable conditions, then the Neumann problem -Δu+u = q(x)ⅠuⅠp−2u in Ω has at least two solutions of which one is positive and the other changes sign.

scholarly journals Duality by reproducing kernels

2003 ◽  
Vol 2003 (6) ◽  
pp. 327-395 ◽  
Author(s):  
A. Shlapunov ◽  
N. Tarkhanov
Keyword(s):  
Differential Operator ◽  
Neumann Problem ◽  
Compact Manifold ◽  
Reproducing Kernels ◽  
Elliptic Differential Operator ◽  
Regularity Theorem ◽  
Neumann Problems ◽  
Smooth Compact Manifold ◽  
Convexity Properties ◽  
The Neumann Problem

LetAbe a determined or overdetermined elliptic differential operator on a smooth compact manifoldX. Write𝒮A(𝒟)for the space of solutions of the systemAu=0in a domain𝒟⋐X. Using reproducing kernels related to various Hilbert structures on subspaces of𝒮A(𝒟), we show explicit identifications of the dual spaces. To prove the regularity of reproducing kernels up to the boundary of𝒟, we specify them as resolution operators of abstract Neumann problems. The matter thus reduces to a regularity theorem for the Neumann problem, a well-known example being the∂¯-Neumann problem. The duality itself takes place only for those domains𝒟which possess certain convexity properties with respect toA.

scholarly journals The Neumann problem in Lp on Lipschitz and convex domains

2008 ◽  
Vol 255 (7) ◽  
pp. 1817-1830 ◽  
Author(s):  
Aekyoung Shin Kim ◽  
Zhongwei Shen
Keyword(s):  
Neumann Problem ◽  
Convex Domains ◽  
The Neumann Problem
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