On the left linear Riemann problem in Clifford analysis

Abstract

We consider a left-linear analogue to the classical Riemann problem: Dau =0 inR n n u + = H(x)u +h(x )o n ju(x)j = O(jxj n 2 1 )a sjxj!1: For this purpose, we state a Borel-Pompeiu formula for the disturbed Dirac operatorDa = D +a with a paravectora and some functiontheoretical results. We reformulate the Riemann problem as an integral equation: Pau +HQau = h on ; where Pa = 1 (I +Sa )a ndQa = I Pa: We demonstrate that the essential part of the singular integral operator Sa which is constructed by the aid of a fundamental solution of D +a is just the singular integral operator S associated toD: In case Sa is simplyS and = R n 1 , then under the assumptions 1.H= P He and allH are real-valued, measurable and essentially bounded; 2. (1 +H(x))(1 +H(x)) and H(x) H(x) are real numbers for all x2R n 1 ; 3. the scalar part H0 of H fulls H0(x) >> 0 for all x2R n 1 ; the Riemann problem is uniquely solvable in L2;C(R n 1 ) and the successive approximation