Boundary Value Problems for Non-linear First Order Systems of Partial Differential Equations in Higher Dimensions, Especially in Three Dimensions

Starting from distinguishing boundaries for monogenic functions (see Tutschke in Adv. Appl. Clifford Algebr 25:441-451, 2015), one can solve boundary value problems for monogenic functions (see Dao in Boundary value problems for monogenic functions in higher dimensions. Ph.D Thesis, Hanoi University of Science and Technology, Vietnam, 2019). A boundary value problem for monogenic function in R-3 is the following: prescribe the vector components u(1) and u(2) on the whole boundary, the bi-vector component u(12) on a (one-dimensional) curve on the boundary and the real part u(0) at one point. The goal of the present paper is to reduce the same boundary value problems for more general (linear or fully non-linear) first order systems to fixed-point problems for an operator containing the monogenic solution of the boundary value problem under consideration (provided the given system is written in its Clifford-analytic normal form). In case of R-3 the existence of a uniquely determined solution of the boundary value problem is proved as fixed-point of the corresponding fixed-point problem in case the right hand side is Lipschitz-continuous with sufficiently small Lipschitz constants.