Solution of Initial Value Problems of Cauchy-Kovalevsky Type in the Space of Generalized Monogenic Functions

Abstract

This paper deals with the initial value problem of the type partial derivative(t)u(t, x) = Lu(t, x), u(0, x) = u(0)(x) where t is an element of R(0)(+) is the time, x is an element of R(n+1), u(0)(x) is a generalized monogenic function and the operator L, acting on a Clifford-algebra-valued function u(t, x) = Sigma(B) u(B)(t, x)e(B) with real-valued components u(B)(t, x), is defined by Lu(t, x) := Sigma(A,B,i) c(B,i)((A)) (t, x)partial derivative(xi) u(B)(t, x)e(A) + Sigma(A,B) d(B)((A)) (t, x)u(B)(t, x)e(A) + Sigma(A)gA(t,x)e(A). We formulate sufficient conditions on the coefficients of the operator L under which L transforms generalized monogenic functions again into generalized monogenic functions. For such an operator the initial value problem (0.1) is solvable for an arbitrary generalized monogenic initial function u(0) and the solution is also generalized monogenic for each t.