Summary: Using fundamental solutions, boundary value problems for elliptic equations and elliptic systems can be reduced to fixed-point problems for a suitably defined operator. In order to solve the related fixed-point problems, we apply both the contraction-mapping principle and the second version of the Schauder Fixed-point Theorem as well. Since the right-hand sides are supposed to be only locally bounded or only locally Lipschitz-continuous, the fixed-point theorems are applicable only in balls of the underlying function space, not in the whole function space. We show also how one can determine the best radius of the ball which leads to solvability conditions which are as weak as possible. Elliptic first order systems in higher dimensions can be reduced to operators whose definition contains a monogenic function. The real-valued components of a monogenic function are solutions of the Laplace equation. The paper constructs also so-called distinguishing parts of the boundary from which the real-valued components can completely be recovered.
For the entire collection see [Zbl 1298.00309].