Operator Calculus for Elliptic Boundary Value Problems in Unbounded Domains

In the last years operator principles for the solution of elliptic boundary value problems have got an increasing importance. K. GUrlebeck and W. Sprossig have found in [2] an operator calculus for several boundary value problems in bounded domains which is based on quaternionic analysis. The aim of the present paper is to obtain similar results for unbounded domains. Let H be the quaternionic algebra with the basis e0 , e1 , e2 , e3 . Every element a E H has the unique representation a a,e1 with the real coefficients a. The conjugate quay 3 ternion a is defined by a = a0 e0 a.e1 = Rea Ima and the norm by lal = (aa)I'2. Let G C R 3 be a domain which lies outside a closed compact, sufficiently smooth surface àG = F. An H-valued function fwill be written by f(x) 0fj(x)e1 with I',: G W. Each point x € R' may be identified with a quaternion of the form x = xi e1 . The following operator will play an essential role in our considerations: Let u C C 1(G) n then we have