We investigate the Hilbert boundary-value
problem for Beltrami equations $\overline\partial f=\mu\partial f$
with singularities in generalized quasidisks $D$ whose Jordan
boundary $\partial D$ consists of a countable collection of open
quasiconformal arcs and, maybe, a countable collection
of points. Such generalized quasicircles can be nowhere even locally
rectifiable but include, for instance, all piecewise smooth curves, as well
as all piecewise Lipschitz Jordan curves.
Generally speaking, generalized quasidisks do not satisfy the
standard $(A)-$condition in PDE by Ladyzhenskaya-Ural'tseva, in
particular, the outer cone touching condition, as well as the
quasihyperbolic boundary condition by Gehring-Martio that we
assumed in our last paper for the uniformly elliptic Beltrami
equations.
In essence, here, we admit any countable collection of singularities
of the Beltrami equations on the boundary and arbitrary
singularities inside the domain $D$ of a general nature. As
usual, a point in $\overline D$ is called a singularity of the
Beltrami equation, if the dilatation quotient
$K_{\mu}:=(1+|\mu|)/(1-|\mu|)$ is not essentially bounded in all its
neighborhoods.
Presupposing that the coefficients of the problem are arbitrary
functions of countable bounded variation and the boundary data
are arbitrary measurable with respect to the logarithmic
capacity, we prove the existence of regular solutions of the
Hilbert boundary-value problem. As a consequence, we derive the
existence of nonclassical solutions of the Dirichlet, Neumann, and
Poincar\'{e} boundary-value problems for equations of mathematical
physics with singularities in anisotropic and inhomogeneous media.
A Note on Rational $$L^p$$ L p Approximation on Jordan Curves