Dirichlet problem with measurable data in rectifiable domains

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in unit disk D is due to dissertation Luzin. paper [11] was devoted continuous boundary quasilinear Poisson equations smooth (C1) domains. present (over natural parameter) any Jordan domains rectifiable boundaries. For this purpose, it constructed completely operators generating nonclassical solutions boundary-value ?U = G sources ? Lp, p > 1. latter makes possible apply Leray-Schauder approach proof theorems on existence regular form ?U(z) H(z)?Q(U(z)) multipliers H Lp 1 and Q : R Q(t)/t 0 as t ?. Here values are interpreted sense angular (along nontangential paths) limits that a traditional tool geometric function theory comparison variational interpretations PDE. As consequences, we give applications some concrete semi-linear mathematical physics, arising under modelling various physical processes such diffusion absorption, plasma states, stationary burning etc.