Infinite dimension of solutions of the Dirichlet problem
It is proved that the space of solutions of the Dirichlet problem for the harmonic functions in the unit disk with nontangential boundary limits 0 a.e. has the infinite dimension.
On the Riemann-Hilbert problem in multiply connected domains
We proved the existence of multivalent solutions with the infinite number of branches for the Riemann-Hilbert problem in the general settings of finitely connected domains bounded by mutually disjoint Jordan curves, measurable coefficients and measurable boundary data. The theorem is formulated in terms of harmonic measure and principal asymptotic values. It is also given the corresponding reinforced criterion for domains with rectifiable boundaries stated in terms of the natural parameter and nontangential...
On a theorem of Lindelöf
We give a quasiconformal version of the proof for the classical Lindelöf theorem: Let f map the unit disk D conformally onto the inner domain of a Jordan curve C. Then C is smooth if and only if arh f'(z) has a continuous extension to D. Our proof does not use the Poisson integral representation of harmonic functions in the unit disk.
Linear distortion of Hausdorff dimension and Cantor's function.
Let be a mapping from a metric space X to a metric space Y, and let α be a positive real number. Write dim (E) and H(E) for the Hausdorff dimension and the s-dimensional Hausdorff measure of a set E. We give sufficient conditions that the equality dim (f(E)) = αdim (E) holds for each E ⊆ X. The problem is studied also for the Cantor ternary function G. It is shown that there is a subset M of the Cantor ternary set such that H(M) = 1, with s = log2/log3 and dim(G(E)) = (log3/log2) dim (E), for every...
On a theorem of Lindelof
We give a quasiconformal version of the proof for the classical Lindelof theorem: Let map the unit disk conformally onto the inner domain of a Jordan curve : Then is smooth if and only if arg has a continuous extension to . Our proof does not use the Poisson integral representation of harmonic functions in the unit disk.
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