Hilbert transform for the three-dimensional Vekua equation

The three-dimensional Hilbert transform takes scalar data on the boundary of a domain and produces the boundary value of the vector part of a quaternionic monogenic (hyperholomorphic) function of three real variables, for which the scalar part coincides with the original data. This is analogous to t...

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Bibliographic Details
Published inComplex variables and elliptic equations Vol. 64; no. 11; pp. 1797 - 1824
Main Authors Delgado, Briceyda B., Porter, R. Michael
Format Journal Article
LanguageEnglish
Published Colchester Taylor & Francis 02.11.2019
Taylor & Francis Ltd
Subjects
conductivity equation
Dirichlet problem
Dirichlet-to-Neumann map
div-curl system
Domains
Formulas (mathematics)
Hilbert transform
Hilbert transformation
main Vekua equation
Mathematical analysis
Real variables
Vekua-Hilbert transform
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Summary:The three-dimensional Hilbert transform takes scalar data on the boundary of a domain and produces the boundary value of the vector part of a quaternionic monogenic (hyperholomorphic) function of three real variables, for which the scalar part coincides with the original data. This is analogous to the question of the boundary correspondence of harmonic conjugates. Generalizing a representation of the Hilbert transform in given by T. Qian and Y. Yang (valid in ), we define the Hilbert transform associated to the main Vekua equation in bounded Lipschitz domains in . This leads to an investigation of the three-dimensional analogue of the Dirichlet-to-Neumann map for the conductivity equation.
ISSN:1747-6933
1747-6941
DOI:10.1080/17476933.2018.1555246