On boundary-value problems for generalized analytic and harmonic functions - Gutlyanskiĭ, V.Ya.; Nesmelova, O.V.; Ryazanov, V.I.; Yefimushkin, A.S. (Reports of the National Academy of Sciences of Ukraine, 2020)
The present paper is a natural continuation of our last articles on the Riemann, Hilbert, Dirichlet, Poincaré, and, in particular, Neumann boundary-value problems for quasiconformal, analytic, harmonic functions and the so-called A-harmonic functions with arbitrary boundary data that are measurable with respect to the logarithmic capacity. Here, we extend the corresponding results to generalized analytic functions :h D → С with sources :g : ∂z-h = g ∈ Lp , p > 2 , and to generalized harmonic functions U with sources G : ΔU =G ∈Lp , p > 2 . Our approach is based on the geometric (functional-theoretic) interpretation of boundary values in comparison with the classical operator ap- proach in PDE. Here, we will establish the corresponding existence theorems for the Poincaré problem on direc- tional derivatives and, in particular, for the Neumann problem to the Poisson equations U GΔ = with arbitrary boundary data that are measurable with respect to the logarithmic capacity. A few mixed boundary-value problems are considered as well. These results can be also applied to semilinear equations of mathematical physics in aniso- tropic and inhomogeneous media.