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\(H_\lambda\)-regular vector functions and their boundary value problems. (English) Zbl 1276.30061
Summary: Let \[ D=\left(\begin{matrix} \lambda +\frac {\partial}{\partial t}&2\frac{\partial}{\partial\overline z}\\ 2\frac{\partial}{\partial z}&\lambda -\frac{\partial} {\partial t}\end{matrix}\right), \]

where \(\lambda\) is a positive real constant. In this paper, by using the methods from quaternion calculus, we investigate the \(H_\lambda\)-regular vector functions, that is, the complex vector solutions
\[ \mathbf{\Psi}(t, z) = \left(\begin{matrix} \psi_1(t,z)\\ \psi_2(t,z)\end{matrix}\right) \]
of the equation \(D\mathbf{\Psi}=0\), 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 \(H_\lambda\)-regular functions satisfy the modified Helmholtz equation, that is, \((\lambda^2 -\Delta)\psi_i=0\), \(i = 1, 2\). We give out a distribution solution of the inhomogeneous equation \(Du = f\) and study some properties of the solution. Moreover, we discuss some boundary value problems for \(H_\lambda\)-regular functions and solutions of equation \(Du = f\).

MSC:
30G35 Functions of hypercomplex variables and generalized variables
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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References:
[1] doi:10.1080/02781070310001617529 · Zbl 1065.30050 · doi:10.1080/02781070310001617529
[2] doi:10.1007/BF01082451 · Zbl 0790.30039 · doi:10.1007/BF01082451
[3] doi:10.1080/17476938208814464 · Zbl 0762.47023 · doi:10.1080/17476938208814464
[4] doi:10.1016/j.jde.2010.09.007 · Zbl 1205.35303 · doi:10.1016/j.jde.2010.09.007
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