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The discontinuous oblique derivative problem for nonlinear elliptic systems of first order. (English) Zbl 0682.35033

One of the well known boundary value problems for analytic functions is the PoincarĂ© problem also called the skew derivative or the oblique derivative boundary value problem. In this note this problem is studied for nonlinear uniformly elliptic complex equation of first order, which include the generalized linear Beltrami equation and thus the theory of generalized analytic functions. Moreover, the coefficients of the boundary conditions are allowed to have finitely many points of discontinuity of the first kind i.e. finite jumps. In this case the boundary value problem is addressed as discontinuous. For analytic functions the theory of discontinuous Hilbert boundary value problems was found by Muskhelishvili. The results of this theory are used here. The relation of the complex equation under consideration to elliptic systems of first or real equations can be found e.g. in the paper by A. N. Fang [Acta Math. Sin. 23, 280-292 (1980; Zbl 0457.35018)]. The problem is studied in the unit disc \({\mathbb{D}}\) of the complex plane. Using conformal mappings the results can be seen to hold for any (bounded) simply connected domain with a smooth boundary.
Reviewer: M.Biroli

MSC:

35J60 Nonlinear elliptic equations
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35R05 PDEs with low regular coefficients and/or low regular data

Citations:

Zbl 0457.35018
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