Dirichlet problem with measurable data for quasilinear Poisson equations

Abstract

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk $\mathbb D$ goes to the famous dissertation of Luzin, see e.g. its reprint \cite{L}. His result was formulated in terms of angular limits (along nontangent paths) that are a traditional tool for the research of the boundary behavior in the geometric function theory. Following this way, we proved in \cite{GNR1} Theorem 7 on the solvability of the Dirichlet problem for the Poisson equations $\triangle\, U=G$ with sources in classes $G\in L^p,$ $ p>1$, in Jordan domains with arbitrary boundary data that are measurable with respect to the logarithmic capacity. There we assumed that the domains satisfy the quasihyperbolic boundary condition by Gehring--Martio, generally speaking, without the known $(A)-$condition by Ladyzhenskaya--Ural'tseva and, in particular, without the outer cone condition that were standard for boundary-value problems in the PDE theory. Note that such Jordan domains cannot be even locally rectifiable. With a view to further development of the theory of boundary value problems for semi-linear equations, the present paper is devoted to the Dirichlet problem with arbitrary measurable (over logarithmic capacity) boundary data for quasilinear Poisson equations in such Jordan domains. For this purpose, it is first constructed completely continuous operators generating nonclassical solutions of the Dirichlet boundary-value problem with arbitrary measurable data for the Poisson equations $\triangle\, U=G$ with the sources $G\in L^p,$ $ 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 $\triangle\, U(z)=H(z)\cdot Q(U(z))$ for multipliers $H\in L^p$ with $ p>1$ and continuous functions $Q: \mathbb R\to\mathbb R$ with $Q(t)/t\to 0$ as $t\to \infty$. As consequences, we give applications to some concrete quasilinear equations of mathematical physics, arising under modelling various phy\-si\-cal processes such as diffusion with absorption, plasma states, stationary burning etc. These results can be also applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.