Solution of Initial Value Problems with Monogenic Initial Functions in Banach Spaces with <i>L_p</i>-Norm

Abstract

This paper deals with the initial value problem of the type partial derivative u(t,x)/partial derivative t = Lu(t,x), u(0,x) = u(0)(x) (0.1) in Banach spaces with L-p-norm, where t is the time, u(0) is a monogenic function and the operator L is of the form Lu(t,x) := Sigma(A,B,i) C-B,i((A))(t,x)partial derivative u(B)(t,x)/partial derivative x(i)e(A). (0.2) The desired function u(t,x) = Sigma(B) u(B)(t,x)e(B) defined in [0, T] x Omega subset of R-0(+) x Rn+1 is a Clifford-algebra-valued function with real-valued components u(B)(t, x). We give sufficient conditions on the coefficients of the operator L under which L is associated to the Cauchy-Riemann operator D of CLIFFORD analysis. For such an operator L the initial value problem (0.1) is solvable for an arbitrary monogenic initial function u(0) and the solution is also monogenic for each t.