A fourth-order elliptic Riemann type problem in R3

• Published in: Boundary value problems Vol. 2016; no. 1
• Main Author: Gu, Longfei
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 05.08.2016
• Subjects: Analysis; Approximations and Expansions; Difference and Functional Equations; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; micro-electro-mechanical systems; fourth-order elliptic equations; Clifford analysis; Riemann type problem; 30G35; 45J05
• ISSN: 1687-2770
• DOI: 10.1186/s13661-016-0656-x

This article is concerned with a fourth-order elliptic equation i.e. , ( Δ 2 − κ 2 Δ ) [ u ] = 0 ( κ > 0 ) coupled by Riemann boundary value conditions in Clifford analysis. In the framework of a Clifford algebra Cl ( V 3 , 3 ) , we obtain factorizations of the fourth-order elliptic equation and construct the explicit expressions of higher-order kernel functions. Some integral representation formulas and properties of the null solution of the fourth-order elliptic equations in Clifford analysis are presented. Based on these integral representation formulas, the boundary behavior of some singular integral operators, and the Clifford analytic approach, we prove that the fourth-order elliptic Riemann type problem in R 3 is solvable. The explicit representation formula of the solution is also established.