Boundary value problems for a second-order elliptic partial differential equation system in Euclidean space

Let (Figure presented.) be a bounded regular domain, let (Figure presented.) be the standard Dirac operator in (Figure presented.), and let (Figure presented.) be the Clifford algebra constructed over the quadratic space (Figure presented.). For (Figure presented.) fixed, (Figure presented.) denotes the space of (Figure presented.) -vectors in (Figure presented.). In the framework of Clifford analysis, we consider two boundary value problems for a second-order elliptic system of partial differential equations of the form (Figure presented.) in (Figure presented.), where (Figure presented.) is a smooth (Figure presented.) -vector valued function. The boundary conditions of the problems contain the inner and outer products of the (Figure presented.) -vector solution (Figure presented.) with both the Dirac operator and the normal vector to (Figure presented.), ensuring the well-posedness for the problems. Investigation of the spectral properties of the sandwich operator (Figure presented.) is considered by using the Fredholm theory. Finally, it is shown that satisfactory problem-solving properties, in general, fail when we replace the standard Dirac operator by those, obtained via unusual orthogonal bases of (Figure presented.).