Some Riemann boundary value problems in Clifford analysis

In this paper, we mainly study the R-m (m > 0) Riemann boundary value problems for functions with values in a Clifford algebra C(sic)(V-3,V-3). We prove a generalized Liouville-type theorem for harmonic functions and biharmonic functions by combining the growth behaviour estimates with the series expansions for k-monogenic functions. We obtain the result under only one growth condition at infinity by using the integral representation formulas for harmonic functions and biharmonic functions. By using the Plemelj formula and the integral representation formulas, a more generalized Liouville theorem for harmonic functions and biharmonic functions are presented. Combining the Plemelj formula and the integral representation formulas with the above generalized Liouville theorem, we prove that the R-m (m > 0) Riemann boundary value problems for monogenic functions, harmonic functions and biharmonic functions are solvable. Explicit representation formulas of the solutions are given. Copyright (c) 2009 John Wiley & Sons, Ltd.