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1.
Peter Franek 《Advances in Applied Clifford Algebras》2007,17(3):469-480
In this paper, we show the existence of a sequence of invariant differential operators on a particular homogeneous model G/P of a Cartan geometry. The first operator in this sequence is locally the Dirac operator in 2 Clifford variables, D = (D
1, D
2), where D
i
= ∑
j
e
j
. ∂
ij
. It follows from the construction that this operator is invariant with respect to the action of the group G. There are 2 other G-invariant differential operators following it so that the sequence of operators is exact. We compute the local expression
of these operators and show that it coincides with the operators described in [2, 5, 6] by the tools of Clifford analysis.
However, it follows from our approach that the operators are invariant.
The work presented here was supported by the grants GAUK 447/2004 and GA ČR 201/05/H005. 相似文献
2.
In this paper an extension of Clifford analysis to superspace is given, inspired by the abstract framework of radial algebra.
This framework leads to the introduction of the so-called super-dimension, an important parameter appearing in several formulae.
Also the relevant differential operators are introduced and their basic properties are proven.
The first author is a Research Assistant of the Research Foundation – Flanders (FWO – Vlaanderen). 相似文献
3.
Fred Brackx Jarolím Bureš Hennie De Schepper David Eelbode Frank Sommen Vladímir Souček 《Complex Analysis and Operator Theory》2007,1(3):341-365
Hermitean Clifford analysis focusses on h–monogenic functions taking values in a complex Clifford algebra or in a complex spinor space. Monogenicity is expressed here
by means of two complex mutually adjoint Dirac operators, which are invariant under the action of a Clifford realisation of
the unitary group. In this contribution we present a deeper insight in the transition from the orthogonal setting to the Hermitean
one. Starting from the orthogonal Clifford setting, by simply introducing a so-called complex structure J ∈ SO(2n;
), the fundamental elements of the Hermitean setting arise in a quite natural way. Indeed, the corresponding projection operators
1/2 (1 ± iJ) project the initial basis (eα, α = 1, . . . , 2n) onto the Witt basis and moreover give rise to a direct sum decomposition of
into two components, where the SO(2n;
)-elements leaving those two subspaces invariant, commute with the complex structure J. They generate a subgroup which is doubly covered by a subgroup of Spin(2n;
), denoted Spin
J
(2n;
), being isomorphic with the unitary group U(n;
). Finally the two Hermitean Dirac operators are shown to originate as generalized gradients when projecting the gradient
on the invariant subspaces mentioned, which actually implies their invariance under the action of Spin
J
(2n;
). The eventual goal is to extend the complex structure J to the whole Clifford algebra
, in order to conceptually unravel the true meaning of Hermitean monogenicity and its connections to orthogonal monogenicity.
During the final redaction of this paper, we received the sad news that our friend, colleague and co-author Jarolím Bureš
died on October 1, 2006
Submitted: October 16, 2006. Accepted: December 29, 2006. 相似文献
4.
We develop a constructive framework to define difference approximations of Dirac operators which factorize the discrete Laplacian.
This resulting notion of discrete monogenic functions is compared with the notion of discrete holomorphic functions on quad-graphs.
In the end Dirac operators on quad-graphs are constructed. 相似文献
5.
A regular extension phenomenon of functions defined on Euclidean space with values in a Clifford algebra was studied by Le
Hung Son in the 90’s using methods of Clifford analysis, a function theory which, is centred around the notion of a monogenic
function, i.e. a null solution of the firstorder, vector-valued Dirac operator in .
The isotonic Clifford analysis is a refinement of the latter, which arises for even dimension. As such it also may be regarded
as an elegant generalization to complex Clifford algebra-valued functions of both holomorphic functions of several complex
variables and two-sided biregular function theories.
The aim of this article is to present a Hartogs theorem on isotonic extendability of functions on a suitable domain of . As an application, the extension problem for holomorphic functions and so for the two-sided biregular ones is discussed.
相似文献
6.
Lukáš Krump 《Advances in Applied Clifford Algebras》2009,19(2):365-374
The Dirac operator in several operators is an analogue of the - operator in theory of several complex variables. The Hartog’s type phenomena are encoded in a complex of invariant differential
operators starting with the Dirac operator, which is an analogue of the Dolbeault complex. In the paper, a construction of
the complex is given for the Dirac operator in 4 variables in dimension 6 (i.e. in the non-stable range). A peculiar feature
of the complex is that it contains a third order operator. The methods used in the construction are based on the Penrose transform
developed by R. Baston and M. Eastwood.
The work presented here is a part of the research project MSM 0021620839 and was supported also by the grant GA ČR 201/05/2117. 相似文献
7.
Let Fr (0 < r < m + 1) be a smooth r-vector valued function in a suitable open domain of
satisfying
in Ω, where ∂ is the Dirac operator in
. Then it is proved that there exists H
r
, an r-vector valued harmonic function in Ω, such that F
r
=
. Two proofs of this structure theorem are given, one based on properties of harmonic differential forms and one relying upon
primitivation of monogenic functions. 相似文献
8.
Liviu I. Nicolaescu 《Aequationes Mathematicae》1994,48(2-3):143-162
Summary Every generalized laplacianL defined on a manifoldM determines a sheaf of L-harmonic sections namely the sheaf of local solutions ofLu = 0. We study the converse problem: to what extent this sheaf determines the operator. Our main result states that the sheaf ofL-harmonic sections determines the operator up to a conformal factor. Moreover, when the operator is a covariant laplacian and the dimension ofM is greater than 2, the sheaf determinesL up to a multiplicative constant. An interesting consequence is the following: if two Riemann metrics on a smooth manifold of dimension greater than 2 have the same sheaves of harmonic functions then they are homothetic. 相似文献
9.
Orthogonal decompositions of Sobolev spaces in Clifford analysis 总被引:2,自引:0,他引:2
The space L
2(G;ℂ
m
) of Clifford-algebra-valued functions in bounded domains G of ℝ
m
is decomposed into the orthogonal sum of the subspace of poly-left-monogenic functions of arbitrary order k≥1 and its orthogonal complement and as well into the orthogonal sum of the subspace of polyharmonic functions of arbitrary
order k≥1 and its orthogonal complement. The complementary subspaces are given explicitly. In the particular case m=2, complex functions are involved. Although this case has to be treated separately, the results are as before. The proofs
are based on proper higher-order Cauchy–Pompeiu formulas and Green functions for powers of the Laplacian.
Received: July 4, 2000; in final form: January 7, 2001?Published online: December 19, 2001 相似文献
10.
In this paper we introduce the quaternionic Witt basis in
. We then define a notion of quaternionic hermitian vector derivative which leads to hermitian monogenic functions. We study
the resolutions associated to quaternionic hermitian systems in the 4 and 8 dimensional cases. We finally prove Martinelli–Bochner
type formulae.
Communicated by Daniel Alpay.
Received: October 11, 2006; Accepted: October 27, 2006. 相似文献
11.
It has been conjectured that for 2-dimensional planar domains the lifetime of a conditioned Brownian motion is maximized for
two boundary points. The present example shows that such a claim is in general not true on manifolds.
Received: 3 April 2007 相似文献
12.
Sophia Demoulini David M. A. Stuart 《Calculus of Variations and Partial Differential Equations》2007,30(4):523-546
We prove existence and regularity of critical points of arbitrary degree for a generalised harmonic map problem, in which
there is an additional nonlocal polyconvex term in the energy, heuristically of the same order as the Dirichlet term. The
proof of regularity hinges upon a special nonlinear structure in the Euler–Lagrange equation similar to that possessed by
the harmonic map equation. The functional is of a type appearing in certain models of the quantum Hall effect describing nonlocal
Skyrmions. 相似文献
13.
We study pseudo-differential operators on a cylinder
where B has conical singularities. Configurations of that kind are the local model of corner singularities with cross section B. Operators in our calculus are assumed to have symbols a which are meromorphic in the complex covariable with values in the
algebra of all cone operators on B. We show an explicit formula for solutions of the homogeneous equation if a is independent of the axial variable
Each non-bijectivity point of the symbol in the complex plane corresponds to a finite-dimensional space of solutions. Moreover,
we give a relative index formula. 相似文献
14.
Igor Leite Freire Antonio Cândido Faleiros 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(11):3478-3486
Using the classical Lie method we obtain the full Lie point symmetry group of the Aronsson equation in two independent variables. Some group invariant solutions of this equation are found and a conjecture on the Lie point symmetry group of the Aronsson equation in Rn is presented. 相似文献
15.
Ricardo Abreu-Blaya Juan Bory-Reyes Richard Delanghe Frank Sommen 《Advances in Applied Clifford Algebras》2007,17(4):589-610
The space HF
k
(Ω) of harmonic multi-vector fields in a domain
as introduced in [1] is closely connected to the space of harmonic forms. The main aim of this paper is to characterize the
dual space of HF
k
(E) being
a compact set. It is proved that HF
k
(E)* is isomorphic to a certain quotient space of so-called harmonic pairs outside E vanishing at infinity.
Research of the third author was supported by the FWO Research Network WO. 003. 01N, research of the fourth author was supported
by the FWO “Krediet aan Navorsers: 1.5.106.02” 相似文献
16.
We study mixed boundary value problems for an elliptic operator on a manifold with boundary , i.e., in on , where is subdivided into subsets with an interface and boundary conditions on that are Shapiro–Lopatinskij elliptic up to from the respective sides. We assume that is a manifold with conical singularity . As an example we consider the Zaremba problem, where is the Laplacian and Dirichlet, Neumann conditions. The problem is treated as a corner boundary value problem near which is the new point and the main difficulty in this paper. Outside the problem belongs to the edge calculus as is shown in Bull. Sci. Math. (to appear).With a mixed problem we associate Fredholm operators in weighted corner Sobolev spaces with double weights, under suitable edge conditions along of trace and potential type. We construct parametrices within the calculus and establish the regularity of solutions. 相似文献
17.
Given asymptotics types P, Q, pseudodifferential operators
are constructed in such a way that if u(t) possesses conormal asymptotics
of type P as t +0, then Au(t) possesses conormal asymptotics of
type Q as t +0. This is achieved by choosing the operators A in Schulzes
cone algebra on the half-line
, controlling their complete Mellin symbols
{
},
and prescribing the mapping properties of the residual
Green operators. The constructions lead to a coordinate invariant calculus,
including trace and potential operators at t = 0, in which a parametrix construction
for the elliptic elements is possible. Boutet de Monvels calculus for
pseudodifferential boundary problems occurs as a special case when P = Q is
the type resulting from Taylor expansion at t = 0. 相似文献
18.
V. Bangert 《Aequationes Mathematicae》1987,34(2-3):153-166
19.
20.
Mixed elliptic boundary value problems are characterised by conditions which have a jump along an interface of codimension 1 on the boundary. We study such problems in weighted edge spaces and show the Fredholm property and the existence of parametrices under additional conditions of trace and potential type on the interface. We develop a new method for computing the interface conditions in terms of the index of boundary value problems in weighted spaces on infinite cones, combined with structures from the calculus of boundary value problems on a manifold with edges. This will be illustrated by the Zaremba problem and other mixed problems for the Laplace operator. The approach itself is completely general. 相似文献