The study of the Dirichlet problem in the unit disk $\mathbb D$ with
arbitrary measurable data for harmonic functions is due to the
famous dissertation of Luzin [31]. Later on, the known
monograph of Vekua \cite{Ve} has been devoted to boundary-value
problems (only with H\"older continuous data) for the generalized
analytic functions, i.e., continuous complex valued functions $h(z)$
of the complex variable $z=x+iy$ with generalized first partial
derivatives by Sobolev satisfying equations of the form
$\partial_{\bar z}h\, +\, ah\, +\ b{\overline h}\, =\, c\, ,$ where
it was assumed that the complex valued functions $a,b$ and $c$
belong to the class $L^{p}$ with some $p>2$ in smooth enough domains
$D$ in $\mathbb C$.
The present paper is a natural continuation of our previous articles
on the Riemann, Hilbert, Dirichlet, Poincar\'{e} and, in particular,
Neumann boundary-value problems for quasiconformal, analytic,
harmonic, and the so-called $A-$harmonic functions with boundary data
that are measurable with respect to logarithmic capacity. Here, we
extend the corresponding results to the generalized analytic
functions $h:D\to\mathbb C$ with the sources $g$ : $\partial_{\bar
z}h\ =\ g\in L^p$, $p>2\,$, and to generalized harmonic functions
$U$ with sources $G$ : $\triangle\, U=G\in L^p$, $p>2\,$.
This paper contains various theorems on the existence of
nonclassical solutions of the Riemann and Hilbert boundary-value
problems with arbitrary measurable (with respect to logarithmic
capacity) data for generalized analytic functions with
sources. Our approach is based on the geometric
(theoretic-functional) interpretation of boundary-values in
comparison with the classical operator approach in PDE. On this
basis, it is established the corresponding existence theorems for
the Poincar\'{e} problem on directional derivatives and, in particular,
for the Neumann problem to the Poisson equations $\triangle\, U=G$
with arbitrary boundary data that are measurable with respect to
logarithmic capacity. These results can be also applied to
semi-linear equations of mathematical physics in anisotropic and
inhomogeneous media.
On the Mountain Pass Solutions to Boundary Value Problems on the Sierpinski Gasket
