A maximum principle for generalized analytic vectors

Abstract

In the theory of linear elliptic systems of differential equations of first order in the plane a lot of results are well known. The theory of two equations in two real functions was developed essentially by Rers [3] and Vekun [17] In the case of 2n equations,nt, in 2n real functions Douglis [5] constructed a normal form and developed a function theory in the special case of hyperanalytic functions Gilberi and Hile [7] succeeded in applying this theory to a more general class o linear elliptic systems of differential equations by using a hypercomplex algebra. In 1965 Bojarski [4] investigated the general case (generalized analytic vectors) with the help of singular integral equations. Especially he was able to prove a Cauchy integral formula and to solve boundary value problems. Further functions theoretic results, for example a Runge approximation theorem and the construction of global solutions and fundamental kernels in the case of only locally L_pintegrable coefficients are published in [9]. A summary of these results can be found in the monograph [19] of Wendland. Unfortunately in the formulation of the Cauchy integral formula Bojarski needed the values of the function in the whole domain and not only on the boundary. The aim of this paper is to show that the coefficients of the Cauchy integral formula in [4] are calculable by using only the values of the solution on the boundary of the domain. A maximum principle can be developed with the help of this explicit representation of the coefficients by the boundary values. Conclusions of the maximum principle are a uniqueness theorem and a convergence theorem. In particular it is shown that there exists no nontrivial solution with compact support. A few results from [9] can be proved with weaker assumptions.