A priori estimates for elliptic systems
A priori estimates for the solution of the Riemann-Hilbert boundary value problem for the Beltrami equation \[ w_ z-\mu_ 1w_ z-\mu_ 2\bar w_ z=aw+b\bar w+c \] are obtained. Here \(| \mu_ 1(z)| +| \mu_ 2(z)| \leq q<1\) is assumed for ellipticity. The estimates follow from a representation formula [see \textit{W. Haack} and \textit{W. Wendland}, ``Vorlesungen über partielle und Pfaffsche Differentialgleichungen'' (1969; Zbl 0196.114)] in terms of Green and Neumann functions of a \(C^{1+\alpha}\) domain for a canonical problem to which the original problem is reduced by means of a complete homeomorphism [see \textit{I. N. Vekua}, ``Generalized analytic functions'' (1962; Zbl 0100.076)]. These estimates are useful for existence and uniqueness results for related nonlinear problems.
scientific article; zbMATH DE number 4056193