A fourth-order elliptic Riemann type problem in R 3 $\mathbb{R}^{3}$

Longfei Gu

doi:10.1186/s13661-016-0656-x

Abstract

This article is concerned with a fourth-order elliptic equation i.e., $(\Delta^{2}-\kappa^{2}\Delta)[u]=0$ ( $\kappa>0$ ) coupled by Riemann boundary value conditions in Clifford analysis. In the framework of a Clifford algebra $\operatorname{Cl}(V_{3,3})$ , 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 $\mathbb{R}^{3}$ is solvable. The explicit representation formula of the solution is also established.

Highlights

1 Introduction Fourth-order elliptic equations have become a very important and useful area of mathematics over the last few decades, which is caused both by the intensive development of the theory of partial differential equations and their applications in various fields of physics and engineering such as theory of elasticity, micro-electro-mechanical systems, bi-harmonic systems, and so on
It is natural and important to study fourth-order elliptic equations coupled by the Riemann boundary conditions in Rn (n ≥ )
In Section, in the framework of the Clifford algebra Cl(V, ), we construct the explicit expressions of the kernel functions and obtain some integral representation formulas, we study some properties of null solutions for the fourthorder elliptic equations ( –κ )u =, for instance, the mean value formula, the Painlevé principle, and so on

Summary

Introduction

Fourth-order elliptic equations have become a very important and useful area of mathematics over the last few decades, which is caused both by the intensive development of the theory of partial differential equations and their applications in various fields of physics and engineering such as theory of elasticity, micro-electro-mechanical systems, bi-harmonic systems, and so on. It is natural and important to study fourth-order elliptic equations coupled by the Riemann boundary conditions in Rn (n ≥ ). The first method fails to solve a system of the fourth-order elliptic equation i.e., ( – κ )u = , coupled by the Riemann boundary conditions.

Results

Conclusion