Some particular Riemann-Hilbert boundary value problems for linear complex second order equations with \(\rho:= \partial^2w/ \partial\overline z^2\) as leading term are investigated. Representing the general solution as \(w=\varphi +\overline z\psi+ T^2\rho\) with two arbitrary analytic functions \(\varphi\) and \(\psi\) and the Pompeiu operator \(T\), the boundary conditions are used to determine \(\varphi\) and \(\psi\) in terms of \(\rho\). The differential equation then leads to a singular integral equation which is shown to obey the Fredholm alternative. Some complications occur for the case of negative index of the boundary value problem. Also quasilinear and nonlinear equations of second order are studied.
For higher order equations a similar treatment is used. The respective integral operator is taken from a hierarchy of operators which turn out from proper iterations of the Pompeiu operator and its complex conjugate, see the reviewer and G. N. Hile [A hierarchy of integral operators. Rocky Mountains J. Math. (to appear)]. If the leading term is \(\rho= \partial^nw/ \partial\overline z^n\) then \[ w(z)= \sum^{n-1}_{k=0} \varphi_k(z) \overline z^k+ {(-1)^n \over(n-1)!\pi} \int_D \rho (\zeta) {(\overline {\zeta-z})^{n-1} \over \zeta-z} d\xi d\eta, \] where \(\varphi_k\) are arbitrary analytic functions. Particular equations of higher order were investigated by A. Dzhuraev [Sov. Math. Dokl. 33, 763-768 (1986); transl. from Dokl. Akad. Nauk SSSR 288, 802-808 (1986; Zbl 0628.35029)]. While Dzhuraev assumed \(n\) to be even, \(n\) here is odd. In particular, the case \(n=3\) is treated in detail.