BEST CONSTANTS AND POHOZAEV IDENTITY FOR HARDY–SOBOLEV-TYPE OPERATORS

2013 ◽  
Vol 15 (03) ◽  
pp. 1250050 ◽  
Author(s):  
ADIMURTHI
Keyword(s):  
Hardy Inequality ◽  
Direct Proof ◽  
Sobolev Type ◽  
General Elliptic ◽  
Functional Aspects ◽  
Approximation Lemma ◽  
Definition Of ◽  
Multiparticle Systems ◽  
The Hardy Inequality ◽  
Optimal Inequalities

This paper is threefold. Firstly, we reformulate the definition of the norm induced by the Hardy inequality (see [J. L. Vázquez and N. B. Zographopoulos, Functional aspects of the Hardy inequality. Appearance of a hidden energy, preprint (2011); http://arxiv. org/abs/1102.5661]) to more general elliptic and sub-elliptic Hardy–Sobolev-type operators. Secondly, we derive optimal inequalities (see [C. Cowan, Optimal inequalities for general elliptic operator with improvement, Commun. Pure Appl. Anal.9(1) (2010) 109–140; N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy–Rellich inequalities, Math. Ann.349(1) (2010) 1–57 (electronic)]) for multiparticle systems in ℝN and Heisenberg group. In particular, we provide a direct proof of an optimal inequality with multipolar singularities shown in [R. Bossi, J. Dolbeault and M. J. Esteban, Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators, Commun. Pure Appl. Anal.7(3) (2008) 533–562]. Finally, we prove an approximation lemma which allows to show that the domain of the Dirichlet–Laplace operator is dense in the domain of the corresponding Hardy operators. As a consequence, in some particular cases, we justify the Pohozaev-type identity for such operators.

scholarly journals INEQUALITIES ASSOCIATED WITH DILATIONS

2009 ◽  
Vol 11 (02) ◽  
pp. 265-277 ◽  
Author(s):  
TOHRU OZAWA ◽  
HIRONOBU SASAKI
Keyword(s):  
Hardy Inequality ◽  
Sobolev Type ◽  
Semi Group ◽  
The Hardy Inequality

Some properties of distributions f satisfying x · ∇ f ∈ Lp (ℝn), 1 ≤ p < ∞, are studied. The operator x · ∇ is the generator of a semi-group of dilations. We first give Sobolev type inequalities with respect to the operator x · ∇. Using the inequalities, we also show that if [Formula: see text], x · ∇ f ∈ Lp (ℝn) and |x|n/p|f(x)| vanishes at infinity, then f belongs to Lp (ℝn). One of the Sobolev type inequalities is shown to be equivalent to the Hardy inequality in L2 (ℝn).

scholarly journals Functional aspects of the Hardy inequality: appearance of a hidden energy

2012 ◽  
Vol 12 (3) ◽  
pp. 713-739 ◽  
Author(s):  
Juan Luis Vázquez ◽  
Nikolaos B. Zographopoulos
Keyword(s):  
Hardy Inequality ◽  
Functional Aspects ◽  
The Hardy Inequality

scholarly journals On locally essentially bounded divergence measure fields and sets of locally finite perimeter

2020 ◽  
Vol 13 (2) ◽  
pp. 179-217 ◽  
Author(s):  
Giovanni E. Comi ◽  
Kevin R. Payne
Keyword(s):  
Approximation Theory ◽  
Smooth Boundary ◽  
Direct Proof ◽  
Leibniz Rule ◽  
Divergence Measure ◽  
Locally Finite ◽  
Sets Of Finite Perimeter ◽  
Finite Perimeter ◽  
Approximation Lemma ◽  
Natural Way

AbstractChen, Torres and Ziemer ([9], 2009) proved the validity of generalized Gauss–Green formulas and obtained the existence of interior and exterior normal traces for essentially bounded divergence measure fields on sets of finite perimeter using an approximation theory through sets with a smooth boundary. However, it is known that the proof of a crucial approximation lemma contained a gap. Taking inspiration from a previous work of Chen and Torres ([7], 2005) and exploiting ideas of Vol’pert ([29], 1985) for essentially bounded fields with components of bounded variation, we present here a direct proof of generalized Gauss–Green formulas for essentially bounded divergence measure fields on sets of finite perimeter which includes the existence and essential boundedness of the normal traces. Our approach appears to be simpler since it does not require any special approximation theory for the domains and it relies only on the Leibniz rule for divergence measure fields. This freedom allows one to localize the constructions and to derive more general statements in a natural way.

scholarly journals Hardy-type inequalities for Dunkl operators with applications to many-particle Hardy inequalities

2020 ◽  
pp. 2050024 ◽  
Author(s):  
Andrei Velicu
Keyword(s):  
Hardy Inequality ◽  
Root Systems ◽  
Dunkl Operators ◽  
Hardy Inequalities ◽  
One Dimensional ◽  
Hardy Type Inequalities ◽  
Rellich Inequality ◽  
Nirenberg Inequality ◽  
Special Case ◽  
The Hardy Inequality

In this paper, we study various forms of the Hardy inequality for Dunkl operators, including the classical inequality, [Formula: see text] inequalities, an improved Hardy inequality, as well as the Rellich inequality and a special case of the Caffarelli–Kohn–Nirenberg inequality. As a consequence, one-dimensional many-particle Hardy inequalities for generalized root systems are proved, which in the particular case of root systems [Formula: see text] improve some well-known results.

scholarly journals The Prehistory of the Hardy Inequality

2006 ◽  
Vol 113 (8) ◽  
pp. 715 ◽  
Author(s):  
Alois Kufner ◽  
Lech Maligranda ◽  
Lars-Erik Persson
Keyword(s):  
Hardy Inequality ◽  
The Hardy Inequality

scholarly journals Theory of reflexive-value support in professional training of future educators

2019 ◽  
pp. 110-118
Author(s):  
M. V. Tsiulina
Keyword(s):  
Professional Training ◽  
Soft Skills ◽  
Theoretical Studies ◽  
Economic Changes ◽  
Cognitive Components ◽  
Functional Aspects ◽  
Self Development ◽  
Definition Of ◽  
Key Terms ◽  
Functional Components

The novelty of reflexive-value support in professional training of future educators is caused by social and economic changes, which made it necessary to review the principals of teacher training. The analysis of concepts and notions highlighted the key terms: support, pedagogical support, reflection, values, value reflection. Theoretical studies revealed the major trends in the process of pedagogical support and feature its main peculiarities. The author presents her own definition of «reflexive-value support in professional training of future educators». As the result of the content analysis there was defined the structure of reflexive-value support in professional training of future educators. The structure manifests the essence and the dynamics of the process, its valuable, reflexive, creative, communicative, cognitive components. The functional analysis showed the main functional components: motivation, values, cognition, communication, regulation, reflection, evaluation, self-development and strategies and soft skills for implementing these functional aspects.

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