FIRST-ORDER DIFFERENTIAL OPERATORS ASSOCIATED TO THE CAUCHY-RIEMANN OPERATOR OF CLIFFORD ANALYSIS

Abstract:

Consider initial value problems of type

where is a linear first-order differential operator and the desired solution is sought in a function space defined as the kernel of a linear differential operator . Mainly two assumptions are required for such initial value problems to be solvable. Firstly, the operators have to be associated, i.e., must ransform solutions of again into solutions of this equation. Secondly, an interior estimate must be true, where Ω′ is a subdomain of Ω having the positive distance δ from the boundary of Ω, and ∥·∥ denotes the norm of a suitably chosen function space (with respect to Ω′ and Ω).

This paper formulates sufficient conditions on the real-valued coefficients of the operator having the form

where is a Clifford-algebra-valued function with real-valued components u_B under which turns out be associated to the Cauchy-Riemann operator of Clifford analysis, i.e., transforms monogenic functions into monogenic functions again. The operator is constructed in the special case of n = 2 using this criterion. See [2] for the construction of for n = 1. In [2] the initial value problem (1) is also solved in Banach spaces with L_p-norm.