Hilbert transform for the three-dimensional Vekua equation

Abstract

The three-dimensional Hilbert transform takes scalar data on the boundary of a domain Ω ⊆ 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 H in R 3 given by T. Qian and Y. Yang (valid in R n ), we define the Hilbert transform H f associated to the main Vekua equation DW = (Df /f )W in bounded Lipschitz domains in R 3 . This leads to an investigation of the three-dimensional analogue of the Dirichlet-to-Neumann map for the conductivity equation.

An Exterior Neumann Boundary-Value Problem for the Div-Curl System and Applications

“…Before closing this section, we must point out that Theorem 2 and Corollary 1 generalize to n dimensions those results recorded as [8] [Th. A.1] and [8] [Cor. A.3], respectively, which were valid for bounded Lipschitz domains in R 3 .…”

“…Additionally, recall that Sc(ab) = Sc(ab) = a • b, for all a, b ∈ Cl 0,n . On the other hand, if n = 3 and w = w 0 + w is a quaternion-valued function, then the integrand of T 2,Ω reduces to the cross product between E n and w [7,8]. A similar decomposition of Equation ( 24) was also used in [9] for the perturbed Teodorescu transform, whose analysis allowed to give the explicit form of a right inverse of curl + λ, with λ ∈ C.…”

“…∂Ω y − x 4π|y − x| 3 • η(y) α 0 (y) ds y = tr ψ 0 (x), x ∈ ∂Ω. (8) where ψ 0 defined in Equation (2). Moreover, div v * = 0.…”

“…It is worth pointing out that each of the operators appearing in the expression of the general solution Equation ( 1) is important an operator from quaternionic analysis. In the present work, we will frequently employ an important decomposition of the Teodorescu operator used in [7][8][9] for bounded domains of R 3 . In turn, the radial operator appearing in the last term of Equation ( 1) was defined in [10,11] as a generalization to exterior domains of an important family of radial operators.…”

Note on Calderón's inverse problem for measurable conductivities

“…Note that quaternionic analytic techniques have been used in connection with the inverse conductivity problem also in the works [7,6,5,13,14].…”