A Hardy type inequality on fractional order Sobolev spaces on the Heisenberg group

2018 ◽  
pp. 917-949
Author(s):  
Adimurthi Adimurthi ◽  
Arka Mallick
Keyword(s):  
Heisenberg Group ◽  
Fractional Order ◽  
Sobolev Spaces ◽  
Type Inequality ◽  
Hardy Type Inequality ◽  
Hardy Type

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 A Higher-Order Hardy-Type Inequality in Anisotropic Sobolev Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Paolo Secchi
Keyword(s):  
Sobolev Spaces ◽  
Hyperbolic Systems ◽  
Type Inequality ◽  
Initial Boundary ◽  
Higher Order ◽  
Hardy Type Inequality ◽  
Characteristic Boundary ◽  
Hardy Type ◽  
Anisotropic Sobolev Spaces ◽  
Initial Boundary Value

We prove a higher-order inequality of Hardy type for functions in anisotropic Sobolev spaces that vanish at the boundary of the space domain. This is an important calculus tool for the study of initial-boundary-value problems of symmetric hyperbolic systems with characteristic boundary.

A Sharp Adams-Type Inequality for Weighted Sobolev Spaces

2020 ◽  
Vol 71 (2) ◽  
pp. 517-538
Author(s):  
João Marcos do Ó ◽  
Abiel Costa Macedo ◽  
José Francisco de Oliveira
Keyword(s):  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Type Inequality ◽  
Higher Order ◽  
Sobolev Type ◽  
Hardy Type Inequality ◽  
Hardy Type ◽  
Classical Work

Abstract In a classical work (Ann. Math.128, (1988) 385–398), D. R. Adams proved a sharp Trudinger–Moser inequality for higher-order derivatives. We derive a sharp Adams-type inequality and Sobolev-type inequalities associated with a class of weighted Sobolev spaces that is related to a Hardy-type inequality.

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.

Degenerate p-Laplacian Operators and Hardy Type Inequalities on H-Type Groups

2010 ◽  
Vol 62 (5) ◽  
pp. 1116-1130 ◽  
Author(s):  
Yongyang Jin ◽  
Genkai Zhang
Keyword(s):  
Vector Fields ◽  
Orthonormal Basis ◽  
Type Inequality ◽  
Laplacian Operator ◽  
Hardy Type Inequality ◽  
Invariant Vector ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Left Invariant Vector ◽  
Laplacian Operators

AbstractLet 𝔾 be a step-two nilpotent group of H-type with Lie algebra 𝔊 = V ⊕ t. We define a class of vector fields X = {Xj} on 𝔾 depending on a real parameter k ≥ 1, and we consider the corresponding p-Laplacian operator Lp,ku = divX(|∇Xu|p−2∇Xu). For k = 1 the vector fields X = {Xj} are the left invariant vector fields corresponding to an orthonormal basis of V; for 𝔾 being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator Lp,k and as an application, we get a Hardy type inequality associated with X.

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