Open Access Research

H λ -regular vector functions and their boundary value problems

Piwen Yang* and Dan Li

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Department of Mathematics, Sichuan Normal University, Chengdu, 610066, P.R. China

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Citation and License

Boundary Value Problems 2012, 2012:75  doi:10.1186/1687-2770-2012-75

Published: 19 July 2012

Abstract

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M2">View MathML</a>, where λ is a positive real constant. In this paper, by using the methods from quaternion calculus, we investigate the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular vector functions, that is, the complex vector solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M4">View MathML</a> of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M5">View MathML</a>, and work out a systematic theory analogous to quaternionic regular functions. Differing from that, the component functions of quaternionic regular functions are harmonic, the component functions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular functions satisfy the modified Helmholtz equation, that is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M8">View MathML</a>. We give out a distribution solution of the inhomogeneous equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9">View MathML</a> and study some properties of the solution. Moreover, we discuss some boundary value problems for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular functions and solutions of equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9">View MathML</a>.

MSC: 30G35, 35J05.

Keywords:
quaternion calculus; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular vector function; modified Helmholtz equation; Riemann-Hilbert type boundary value problem