Generalized Moisil-Théodoresco Systems and Cauchy Integral Decompositions. (17th March 2008)

• Record Type: Journal Article
• Authors: Abreu Blaya, Ricardo; Bory Reyes, Juan; Delanghe, Richard; Sommen, Frank
• Other Names: Begehr Heinrich Academic Editor.
• Is Part Of: International journal of mathematics and mathematical sciences. Volume 2008(2008)
• Publication Date: 2008-03-17
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1155/2008/746946
• Languages: English
• ISSNs: 0161-1712

Abstract

Abstract : Letℝ 0, m + 1 ( s ) be the space ofs -vectors( 0 ≤ s ≤ m + 1 ) in the Clifford algebraℝ 0, m + 1 constructed over the quadratic vector spaceℝ 0, m + 1, letr, p, q ∈ ℕ with0 ≤ r ≤ m + 1, 0 ≤ p ≤ q, andr + 2 q ≤ m + 1, and letℝ 0, m + 1 ( r, p, q ) = ∑ j = p q ⨁ ℝ 0, m + 1 ( r + 2 j ) . Then, anℝ 0, m + 1 ( r, p, q ) -valued smooth functionW defined in an open subsetΩ ⊂ ℝ m + 1 is said to satisfy the generalized Moisil-Théodoresco system of type( r, p, q ) if∂ x W = 0 inΩ, where∂ x is the Dirac operator inℝ m + 1 . A structure theorem is proved for such functions, based on the construction of conjugate harmonic pairs. Furthermore, ifΩ is bounded with boundaryΓ, whereΓ is an Ahlfors-David regular surface, and ifW is aℝ 0, m + 1 ( r, p, q ) -valued Hölder continuous function onΓ, then necessary and sufficient conditions are given under whichW admits onΓ a Cauchy integral decompositionW = W + + W − .