Article

    Ufa Mathematical Journal
    Volume 4, Number 1, pp. 137-142

    Boundary problem for the generalized Cauchy--Riemann equation in the spaces, descibed by the modulus of continuity


    Timofeev A.Yu.

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    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 $\partial G,$ $\Im w = h$ in 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 in general not contained in $L_q(G),$ $q > 2.$ Thus I. Vekua’s theory is not applicable in this case. The article shows that similar to Dirichlet’s problem for holomorphic functions there appears a “logarithmic effect”. The solution outside the point $z = 0$ meets the demands of Lipsсhits with logarithmic factors. Nevertheless we are able to prove the existence of continuous solution of the problem in $\overline{G}.$