scholarly journals A Hardy type inequality in the half-space on Rn and Heisenberg group

2008 ◽  
Vol 347 (2) ◽  
pp. 645-651 ◽  
Author(s):  
Jing-Wen Luan ◽  
Qiao-Hua Yang
Keyword(s):  
Heisenberg Group ◽  
Half Space ◽  
Type Inequality ◽  
Hardy Type Inequality ◽  
Hardy Type

Weighted Hardy–Littlewood–Sobolev inequalities on the upper half space

2016 ◽  
Vol 18 (05) ◽  
pp. 1550067 ◽  
Author(s):  
Jingbo Dou
Keyword(s):  
Half Space ◽  
Type Inequality ◽  
Boundary Term ◽  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Biharmonic Operator ◽  
Trace Inequality ◽  
Hardy Type Inequality ◽  
Hardy Type

In this paper, we establish a weighted Hardy–Littlewood–Sobolev (HLS) inequality on the upper half space using a weighted Hardy type inequality on the upper half space with boundary term, and discuss the existence of extremal functions based on symmetrization argument. As an application, we can show a weighted Sobolev–Hardy trace inequality with [Formula: see text]-biharmonic operator.

scholarly journals Fractional Korn and Hardy-type inequalities for vector fields in half space

2019 ◽  
Vol 21 (07) ◽  
pp. 1850055 ◽  
Author(s):  
Tadele Mengesha
Keyword(s):  
Half Space ◽  
Hardy Inequality ◽  
Vector Fields ◽  
A Priori ◽  
Type Inequality ◽  
Fractional Sobolev Spaces ◽  
Hardy Type Inequality ◽  
Hardy Type Inequalities ◽  
Korn’S Inequality ◽  
Hardy Type

We prove a fractional Hardy-type inequality for vector fields over the half space based on a modified fractional semi-norm. A priori, the modified semi-norm is not known to be equivalent to the standard fractional semi-norm and in fact gives a smaller norm, in general. As such, the inequality we prove improves the classical fractional Hardy inequality for vector fields. We will use the inequality to establish the equivalence of a space of functions (of interest) defined over the half space with the classical fractional Sobolev spaces, which amounts to prove a fractional version of the classical Korn’s inequality.

scholarly journals Pseudo-monotonicity and degenerated or singular elliptic operators

1998 ◽  
Vol 58 (2) ◽  
pp. 213-221 ◽  
Author(s):  
P. Drábek ◽  
A. Kufner ◽  
V. Mustonen
Keyword(s):  
Sobolev Spaces ◽  
Differential Operators ◽  
Weighted Sobolev Spaces ◽  
Type Inequality ◽  
Elliptic Operators ◽  
Hardy Type Inequality ◽  
Hardy Type ◽  
Elliptic Differential Operators

Using the compactness of an imbedding for weighted Sobolev spaces (that is, a Hardy-type inequality), it is shown how the assumption of monotonicity can be weakened still guaranteeing the pseudo-monotonicity of certain nonlinear degenerated or singular elliptic differential operators. The result extends analogous assertions for elliptic operators.

scholarly journals Eigencurves of the p(·)-Biharmonic operator with a Hardy-type term

2020 ◽  
Vol 6 (2) ◽  
pp. 198-209
Author(s):  
Mohamed Laghzal ◽  
Abdelouahed El Khalil ◽  
My Driss Morchid Alaoui ◽  
Abdelfattah Touzani
Keyword(s):  
Dirichlet Problem ◽  
Variational Approach ◽  
Type Inequality ◽  
Biharmonic Operator ◽  
Nondecreasing Sequence ◽  
Hardy Type Inequality ◽  
Principal Curve ◽  
Hardy Type ◽  
Homogeneous Dirichlet Problem

AbstractThis paper is devoted to the study of the homogeneous Dirichlet problem for a singular nonlinear equation which involves the p(·)-biharmonic operator and a Hardy-type term that depend on the solution and with a parameter λ. By using a variational approach and min-max argument based on Ljusternik-Schnirelmann theory on C1-manifolds [13], we prove that the considered problem admits at least one nondecreasing sequence of positive eigencurves with a characterization of the principal curve μ1(λ) and also show that, the smallest curve μ1(λ) is positive for all 0 ≤ λ < CH, with CH is the optimal constant of Hardy type inequality.

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