Hilbert transform for the three-dimensional Vekua equation

ABSTRACT

The three-dimensional Hilbert transform takes scalar data on the boundary of a domain $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ 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 $\mathcal{H}$ in ${\mathbb{R}}^3$ given by T. Qian and Y. Yang (valid in ${\mathbb{R}}^n$), we define the Hilbert transform ${\mathcal{H}}_f$ associated to the main Vekua equation $DW=\left(Df/f\right)\overline{W}$ in bounded Lipschitz domains in ${\mathbb{R}}^3$. This leads to an investigation of the three-dimensional analogue of the Dirichlet-to-Neumann map for the conductivity equation.

Keywords:

• Hilbert transform
• Vekua–Hilbert transform
• main Vekua equation
• conductivity equation
• Dirichlet-to-Neumann map
• div-curl system

AMS SUBJECT CLASSIFICATIONS:

• 44A15
• 30E20
• 30G20
• 35J25
• 35Q60

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Partially supported by Consejo Nacional de Ciencia y Tecnología (CONACyT) #166183