Boundary problem for the generalized Cauchy–Riemann equation in spaces, described by the modulus of continuity

Abstract

The article is devoted to the Dirichlet problem in the unit disk $G$ for $\partial_{\bar z}w+b(z)\overline w=0$, $\Re w=g$ on $\partial G$, $\Im w=h$ at the point $z_0=1$, where $g$ is a given Lipsсhitz continuous function. The coefficient $b$ belongs to a subspace of $L_2(G)$ which is not contained in $L_q(G)$, $q>2$ in the general case. Thus, I. Vekua's theory is not applicable in this case. The article shows that, as well as in the case of Dirichlet's problem for holomorphic functions, there appears a “logarithmic effect”. The solution outside the point $z=0$ satisfies the Lipsсhits conditions with logarithmic factors. The existence of a continuous solution of the problem in $\overline G$ is proved.