The $$\bar{\partial }$$-Neumann problem on the intersection of two weakly q-convex domains

2021 ◽  
Author(s):  
Shaban Khidr ◽  
Salomon Sambou
Keyword(s):  
Neumann Problem ◽  
Convex Domains

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 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

Lp Neumann problem for some Schrödinger equations in (semi-)convex domains

2019 ◽  
Vol 22 (02) ◽  
pp. 1950007
Author(s):  
Sibei Yang ◽  
Dachun Yang
Keyword(s):  
Neumann Problem ◽  
Convex Domain ◽  
Boundary Data ◽  
Muckenhoupt Weight ◽  
Weight Class ◽  
Convex Domains ◽  
Reverse Hölder Class ◽  
Special Cases ◽  
The Neumann Problem ◽  
Reverse Hölder

Let [Formula: see text], [Formula: see text] be a bounded (semi-)convex domain in [Formula: see text] and the non-negative potential [Formula: see text] belong to the reverse Hölder class [Formula: see text]. Assume that [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the Muckenhoupt weight class on [Formula: see text], the boundary of [Formula: see text]. In this paper, the authors show that, for any [Formula: see text], the Neumann problem for the Schrödinger equation [Formula: see text] in [Formula: see text] with boundary data in (weighted) [Formula: see text] is uniquely solvable. The obtained results in this paper essentially improve the known results which are special cases of the results obtained by Shen [Indiana Univ. Math. J. 43 (1994) 143–176] and Tao and Wang [Canad. J. Math. 56 (2004) 655–672], via extending the range [Formula: see text] of [Formula: see text] into [Formula: see text].

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].

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