Theory of self-adjoint operators generated by strongly singular, second-order expression of divergence type

1983 ◽  
Vol 16 (3) ◽  
pp. 225-227
Author(s):  
Yu. B. Orochko
Keyword(s):  
Second Order ◽  
Divergence Type ◽  
Adjoint Operators

scholarly journals Positive functionals and oscillation criteria for second order differential systems

1979 ◽  
Vol 22 (3) ◽  
pp. 277-290 ◽  
Author(s):  
Garret J. Etgen ◽  
Roger T. Lewis
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Second Order ◽  
Linear Operators ◽  
The Self ◽  
Differential Systems ◽  
Bounded Linear ◽  
Positive Functionals ◽  
Image Position ◽  
Adjoint Operators

Let ℋ be a Hilbert space, let ℬ = (ℋ, ℋ) be the B*-algebra of bounded linear operators from ℋ to ℋ with the uniform operator topology, and let ℒ be the subset of ℬ consisting of the self-adjoint operators. This article is concerned with the second order self-adjoint differential equation

scholarly journals Singular solutions for second-order non-divergence type elliptic inequalities in punctured balls

2014 ◽  
Vol 123 (1) ◽  
pp. 251-279 ◽  
Author(s):  
Marius Ghergu ◽  
Vitali Liskevich ◽  
Zeev Sobol
Keyword(s):  
Second Order ◽  
Singular Solutions ◽  
Divergence Type ◽  
Elliptic Inequalities

Second-Order Differential Operators on Arbitrary Open Sets

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Differential Operators ◽  
Neumann Boundary ◽  
Second Order ◽  
Sesquilinear Forms ◽  
Sectorial Operators ◽  
Mechanical Interpretation ◽  
Adjoint Operators ◽  
Bounded Below ◽  
Distributional Inequality

In this chapter, three different methods are described for obtaining nice operators generated in some L2 space by second-order differential expressions and either Dirichlet or Neumann boundary conditions. The first is based on sesquilinear forms and the determination of m-sectorial operators by Kato’s First Representation Theorem; the second produces an m-accretive realization by a technique due to Kato using his distributional inequality; the third has its roots in the work of Levinson and Titchmarsh and gives operators T that are such that iT is m-accretive. The class of such operators includes the self-adjoint operators, even ones that are not bounded below. The essential self-adjointness of Schrödinger operators whose potentials have strong local singularities are considered, and the quantum-mechanical interpretation of essential self-adjointness is discussed.

scholarly journals A CHARACTERIZATION OF SELF-ADJOINT OPERATORS DETERMINED BY THE WEAK FORMULATION OF SECOND-ORDER SINGULAR DIFFERENTIAL EXPRESSIONS

2009 ◽  
Vol 51 (2) ◽  
pp. 385-404 ◽  
Author(s):  
MOHAMED EL-GEBEILY ◽  
DONAL O'REGAN
Keyword(s):  
Complete Characterization ◽  
Second Order ◽  
Inner Product ◽  
Type I ◽  
Weak Formulation ◽  
Friedrichs Extension ◽  
Sesquilinear Form ◽  
Adjoint Operators ◽  
Special Case

AbstractIn this paper we describe a special class of self-adjoint operators associated with the singular self-adjoint second-order differential expression ℓ. This class is defined by the requirement that the sesquilinear form q(u, v) obtained from ℓ by integration by parts once agrees with the inner product 〈ℓu, v〉. We call this class Type I operators. The Friedrichs Extension is a special case of these operators. A complete characterization of these operators is given, for the various values of the deficiency index, in terms of their domains and the boundary conditions they satisfy (separated or coupled).

scholarly journals Study of a second-order nonlinear elliptic problem generated by a divergence type operator on a compact Riemannian manifold

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 4811-4820
Author(s):  
A. Abnoune ◽  
E. Azroul ◽  
M.T.K. Abbassi
Keyword(s):  
Riemannian Manifold ◽  
Elliptic Problem ◽  
Compact Riemannian Manifold ◽  
Second Order ◽  
Type Operator ◽  
Nonlinear Elliptic Problem ◽  
Nonlinear Elliptic ◽  
Divergence Type

In this paper, we will study a second-order nonlinear elliptic problem generated by an operator of divergence type (or type leray-Lion) : (P1){ A(u) = f in M u = 0 on ? (1) on (M,g) a compact Riemannian manifold et ? its border.

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