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On the existence of a general solution of first order elliptic systems by fixed-point theorem. (English) Zbl 0890.35024

The goal of the authors is to investigate a real first order system for \(2n\) real-valued functions, transforming it into a complex normal form \[ {\partial w\over\partial\bar z}=F\Biggl( z,w,{\partial w\over\partial z}\Biggr)\tag{1} \] and using some methods of complex analysis. They essentially use the properties of \(T_D\) and \(\Pi_D\) operators. In such a way the differential equation (1) can be reduced to a system of operator equations (2) that can be solved by fixed-point theorems: \[ w(z)=\phi(z)+T_DF(\cdot,w,h)(z),\qquad h(z)=\phi'(z) +\Pi_DF(\cdot,w,h)(z).\tag{2} \] The final section regards \(W^{1,p}\)-solvability of the Dirichlet boundary value problem for the equation (1).

MSC:

35F20 Nonlinear first-order PDEs
30E25 Boundary value problems in the complex plane
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[1] Zygmund, A.; Calderon, A. P., On the Existence of Singular Integrals, Acta Math, 88, 85-139 (1952), Uppsall · Zbl 0047.10201
[2] Mshimba, A. S., Konstruktion Von Lo sungen nichtlinearer elliptischer Different ialeichungs Systeme erster ordnung in der Ebene durch Complexe Methoden in Sobolev Raum \(W_{1.p}(D)\), (Luther, A. Martin, Dissertation (1979), University Halle Wittenberg)
[3] Mshimba, A. S.; Tutschke, W., Functional Analytic Methods in Complex Analysis and Applications to Partial Differential Equations, 116-128 (1988)
[4] Lanckau, Eberhard; Tutschke, Wolfgang, Complex Analysis Methods, Trends and Applications, ((1985)), 51-58 · Zbl 0512.00014
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