On behaviour of solution of boundary value problem for generalized Cauchy--Riemann equation

A. S. Il'chukov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 2 (2013), pp. 27-34

Résumé

The following boundary value problem for generalized Cauchy–Riemann equation in the unit disk $G = \{z \in \mathbb{C}: |z| 1 \}$ is considered in the paper: $\partial_{\overline{z}} w + b(z) \overline{w} = 0,$ $\Re w = g$ on $\partial G,$ $\Im w = h$ at the point $z_0 = 1.$ The coefficient $b(z)$ is chosen from some set $S_P,$ constructed by scales, with $S_P \subsetneq L_2,$ $S_P \not\subset L_q,$ $q > 2.$ The boundary value $g$ is chosen from the space, constructed by a modulus of continuity $\mu$ with some special properties. It is shown that the problem has unique solution $w = w(z)$ in the unit disk $G$ with $w \in C(\overline{G}).$ Moreover, outside the point $z = 0$ the behaviour of the solution $w(z)$ is defined by the same modulus of continuity $\mu;$ it means there is no “logarithmic effect” for the solution.

Détail

Keywords: generalized Cauchy–Riemann equation, Dirichlet problem, modulus of continuity.