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Hilbert transform for the three-dimensional Vekua equation

Briceyda B. Delgado, R. Michael Porter

Abstract

The three-dimensional Hilbert transform takes scalar data on the boundary of a domain in R3 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 H in R3 given by T. Qian and Y. Yang (valid in Rn), we define the Hilbert transform Hf associated to the main Vekua equation DW = (Df/f)W in bounded Lipschitz domains in R3. This leads to an investigation of the three-dimensional analogue of the Dirichlet-to-Neumann map for the conductivity equation.

Hilbert transform for the three-dimensional Vekua equation

Abstract

The three-dimensional Hilbert transform takes scalar data on the boundary of a domain in R3 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 H in R3 given by T. Qian and Y. Yang (valid in Rn), we define the Hilbert transform Hf associated to the main Vekua equation DW = (Df/f)W in bounded Lipschitz domains in R3. This leads to an investigation of the three-dimensional analogue of the Dirichlet-to-Neumann map for the conductivity equation.

Paper Structure

This paper contains 15 sections, 30 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quaternionic analysis
  4. Components of the Cauchy and singular Cauchy integral operators
  5. Hilbert transform for monogenic functions
  6. Definition of $\operatorname{\mathcal{H}}$
  7. Properties of $\operatorname{\mathcal{H}}$ and its adjoint and inverse
  8. Dirichlet-to-Neumann map
  9. Hilbert transform associated to the main Vekua equation
  10. Construction of the Vekua-Hilbert transform
  11. Properties of $\operatorname{\mathcal{H}}_f$
  12. Dirichlet-to-Neumann map for the conductivity equation
  13. Quaternionic Dirichlet-to-Neumann map
  14. Norm properties of $\operatorname{\mathcal{H}}_f$
  15. Div-curl system and generalizations

Key Result

Theorem 2.1

Let $\Omega$ be a bounded domain and let $1<p<\infty$. The following operators are continuous:

Theorems & Definitions (54)

  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • ...and 44 more