Dirichlet problem with measurable data in rectifiable domains

Abstract

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