On the Hilbert problem for semi-linear Beltrami equations

Abstract

The presented paper is devoted to the study of the well-known Hilbert boundary-value problem for semi-linear Beltrami equations with arbitrary boundary data that are measurable with respect to logarithmic capacity. Namely, we prove here the corresponding results on the existence, regularity, and representation of its nonclassical solutions with a geometric interpretation of boundary values as the angular (along the nontangential paths) limits in comparison with the classical approach in PDE. For this purpose, we apply completely continuous operators by Ahlfors-Bers, first of all to obtain solutions of semi-linear Beltrami equations, generally speaking with no boundary conditions, and then to derive their representation through the solutions of the Vekua-type equations and the so-called generalized analytic functions with sources. Besides, we obtain similar results for nonclassical solutions of the Poincaré boundary-value problem on directional derivatives and, in particular, of the Neumann problem with arbitrary measurable data to semi-linear equations of the Poisson type. The obtained results are applied to some problems of mathematical physics describing such phenomena as diffusion with physical and chemical absorption, plasma states, and stationary burning in anisotropic and inhomogeneous media.

On the Dirichlet problem for beltrami equations with sources in simply connected domains

“…see Corollary 4.1.8 and Th eorem 4.2.1 in [12], the existence of a holomorphic function A u iv D C      (unique up to an additive pure imaginary constant) in an arbitrary bounded simple connected domain D and the factorization h f    of solutions of (2) in terms of suitable generalized analytic functions with sources, see Lemma 1 and Remark 2 in [13] for the uniformly elliptic case. Th e existence of the given   conformal mapping f follows from Proposition 1; see further details of the proof in [3].…”