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 $\mathbb D$ is due to the
dissertation of Luzin.

The paper \cite{GNR2} was devoted to the Dirichlet problem with
continuous boundary data for quasilinear Poisson equations in smooth
($C^1$) domains.

The present paper is devoted to the Dirichlet problem with arbitrary
measurable (over natural parameter) boundary data for the
quasilinear Poisson equations in 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
$\triangle\, U=G$ 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 $t\to \infty$.

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
phy\-si\-cal processes such as diffusion with absorption, plasma
states, stationary burning etc.