Eine einfache konstruktion der homotopien der gruppe

Abstract

The boundary condition Re(Pw) = ϕ of an elliptic system in a bounded domain of the complex plane can be transformed into the Dirichlet boundary condition Re ω = ϕ (by means of the reversible substitution ω= Pw) if it is possible to extend the matrix-valued function P continuously to the whole of [Gbar] without violating the Lopatinski condition det P ≠ 0. Due to W. L. Wendland [8] this problem was solved by W. Tutschke [7] under special assumptions. Also in connection with elliptic systems W. L. Wendland in [8] used a homotopy from P to the unit matrix, constructed by piecewise Gauss elimination, but this method can fail as was shown by a counter-example. A complete but rather deep-lying result is contained in Bott's periodicity theorem, stating that with the functions P a - valued functional Grad P can be associated and the above mentioned problems are solvable if one has Grad P = 0. It is then easy to prove Bojarski's theorem [1], that Grad P is the number of revolutions of the vector det P during one circulation around the boundary. Grad P is an important and well-known quantity in the theory of elliptic systems in the plane. In the present paper we give an elementary proof of the above mentioned special part of Bott's theorem by constructing a homotopy from P to the unit matrix in the case Grad P=0.