Self-adjointness for general elliptic operators with Sobolev-type coefficients

1985 ◽  
pp. 132-141
Author(s):  
Dung Xuan Nguyen
Keyword(s):  
Elliptic Operators ◽  
Sobolev Type ◽  
General Elliptic

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 Properness and topological degree for general elliptic operators

2003 ◽  
Vol 2003 (3) ◽  
pp. 129-181 ◽  
Author(s):  
V. Volpert ◽  
A. Volpert
Keyword(s):  
Topological Degree ◽  
Unbounded Domains ◽  
Fredholm Property ◽  
Elliptic Operators ◽  
Linear Operators ◽  
Index Zero ◽  
Hölder Spaces ◽  
Nonlinear Operators ◽  
General Elliptic ◽  
Holder Spaces

The paper is devoted to general elliptic operators in Hölder spaces in bounded or unbounded domains. We discuss the Fredholm property of linear operators and properness of nonlinear operators. We construct a topological degree for Fredholm and proper operators of index zero.

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