### 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.

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