DIV–CURL TYPE THEOREM, H-CONVERGENCE AND STOKES FORMULA IN THE HEISENBERG GROUP

In this paper, we prove a div–curl type theorem in the Heisenberg group ℍ1, and then we develop a theory of H-convergence for second order differential operators in divergence form in ℍ1. The div–curl theorem requires an intrinsic notion of the curl operator in ℍ1 (that we denote by curlℍ), that turns out to be a second order differential operator in the left invariant horizontal vector fields. As an evidence of the coherence of this definition, we prove an intrinsic Stokes formula for curlℍ. Eventually, we show that this notion is related to one of the exterior differentials in Rumin's complex on contact manifolds.

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Space of Second-Order Linear Differential Operators as a Module over the Lie Algebra of Vector Fields

Advances in Mathematics ◽

10.1006/aima.1997.1683 ◽

1997 ◽

Vol 132 (2) ◽

pp. 316-333 ◽

Cited By ~ 22

Author(s):

C. Duval ◽

V.Yu. Ovsienko

Keyword(s):

Lie Algebra ◽

Vector Fields ◽

Differential Operators ◽

Second Order ◽

Order Linear ◽

Linear Differential Operators ◽

Linear Differential

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The Expansion Theorems for Sturm-Liouville Operators with an Involution Perturbation

10.20944/preprints202109.0247.v1 ◽

2021 ◽

Author(s):

Abdizhahan Sarsenbi

Keyword(s):

Differential Operator ◽

Differential Operators ◽

Second Order ◽

Order Differential Operator ◽

Spectral Expansions ◽

Sturm Liouville ◽

Complex Valued ◽

Liouville Operators

In this work, we studied the Green’s functions of the second order differential operators with involution. Uniform equiconvergence of spectral expansions related to the second-order differential operators with involution is obtained. Basicity of eigenfunctions of the second-order differential operator operator with complex-valued coefficient is established.

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The Dirichlet Problem for Second-Order Divergence Form Elliptic Operators with Variable Coefficients: The Simple Layer Potential Ansatz

Abstract and Applied Analysis ◽

10.1155/2015/276810 ◽

2015 ◽

Vol 2015 ◽

pp. 1-11 ◽

Cited By ~ 5

Author(s):

Alberto Cialdea ◽

Vita Leonessa ◽

Angelica Malaspina

Keyword(s):

Dirichlet Problem ◽

Differential Operators ◽

Differential Forms ◽

Variable Coefficients ◽

Divergence Form ◽

Boundary Integral ◽

Second Order ◽

Boundary Integral Method ◽

Layer Potential ◽

Partial Differential Operators

We investigate the Dirichlet problem related to linear elliptic second-order partial differential operators with smooth coefficients in divergence form in bounded connected domains ofRm(m≥3) with Lyapunov boundary. In particular, we show how to represent the solution in terms of a simple layer potential. We use an indirect boundary integral method hinging on the theory of reducible operators and the theory of differential forms.

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The polynomial growth solutions to some sub-elliptic equations on the Heisenberg group

Communications in Contemporary Mathematics ◽

10.1142/s0219199717500699 ◽

2019 ◽

Vol 21 (01) ◽

pp. 1750069 ◽

Cited By ~ 1

Author(s):

Hairong Liu ◽

Tian Long ◽

Xiaoping Yang

Keyword(s):

Dirichlet Problem ◽

Heisenberg Group ◽

Elliptic Equations ◽

Polynomial Growth ◽

Bounded Solution ◽

Type Theorem ◽

Divergence Form ◽

Liouville Type ◽

Periodic Coefficients ◽

Liouville Type Theorem

We give an explicit description of polynomial growth solutions to some sub-elliptic operators of divergence form with [Formula: see text]-periodic coefficients on the Heisenberg group, where the periodicity has to be meant with respect to the Heisenberg geometry. We show that the polynomial growth solutions are necessarily polynomials with [Formula: see text]-periodic coefficients. We also prove the Liouville-type theorem for the Dirichlet problem to these sub-elliptic equations on an unbounded domain on the Heisenberg group, show that any bounded solution to the Dirichlet problem must be constant.

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Commutators with fractional differentiation for second-order elliptic operators on ℝn

Communications in Contemporary Mathematics ◽

10.1142/s021919971950010x ◽

2019 ◽

Vol 22 (02) ◽

pp. 1950010

Author(s):

Yanping Chen ◽

Qingquan Deng ◽

Yong Ding

Keyword(s):

Differential Operators ◽

Divergence Form ◽

Second Order ◽

Bounded Mean Oscillation ◽

Fractional Differentiation ◽

Mean Oscillation ◽

Fractional Differential ◽

Fractional Differential Operators ◽

Second Order Elliptic Operators ◽

Complex Coefficients

Let [Formula: see text] be a second-order divergence form elliptic operator and [Formula: see text] an accretive, [Formula: see text] matrix with bounded measurable complex coefficients in [Formula: see text] In this paper, we establish [Formula: see text] theory for the commutators generated by the fractional differential operators related to [Formula: see text] and bounded mean oscillation (BMO)–Sobolev functions.

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Discontinuous Galerkin gradient discretisations for the approximation of second-order differential operators in divergence form

Computational and Applied Mathematics ◽

10.1007/s40314-017-0558-2 ◽

2017 ◽

Vol 37 (4) ◽

pp. 4023-4054 ◽

Cited By ~ 2

Author(s):

Robert Eymard ◽

Cindy Guichard

Keyword(s):

Discontinuous Galerkin ◽

Differential Operators ◽

Divergence Form ◽

Second Order

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Reducibility of differential operators some: examples

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100000608 ◽

1979 ◽

Vol 86 (1) ◽

pp. 29-33 ◽

Cited By ~ 1

Author(s):

B. Fishel ◽

N. Denkel

Keyword(s):

Differential Operator ◽

Differential Operators ◽

Volterra Operator ◽

Second Order ◽

Order Differential Operator ◽

Order Operator ◽

Minimal Operator ◽

Deficiency Indices ◽

First Order ◽

Symmetric Operators

A symmetric operator on a Hilbert space, with deficiency indices (m; m) has self-adjoint extensions. These are ‘highly reducible’. The original operator may be irreducible, (see example (i), below). Can the mechanism whereby reducibility is achieved be understood? The concrete examples most readily studied are those associated with differential operators. It is easy to obtain operators, associated with a formal linear differential operator, having deficiency indices (m; m). What of reducibility? Nothing seems to be known. In the case of the first-order operator we were able, using the Volterra operator, to establish irreducibility of the associated minimal operator. To investigate symmetric operators associated with a second-order differential operator, different methods had to be developed. They apply also to the first-order operator, and we employ them to demonstrate the irreducibility of the associated minimal operator. In the second-order case the minimal operator proves reducible, and we also exhibit examples of reducibility of associated symmetric operators. It would clearly be of interest to elucidate the influence of the boundary conditions on reducibility.

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