Global Hölder regularity for discontinuous elliptic equations in the plane

$C^{1, \mu}$-regularity up to the boundary is proved for solutions of boundary value problems for elliptic equations with discontinuous coefficients in the plane. In particular we deal with Dirichlet boundary condition
\begin{equation*}
\begin{array}{ll}
u= g(x) &\ \rm on \:
\partial\Omega \end{array}
\end{equation*}
where $g(x) \in W^{2-\frac{1}{r}, r}(\partial \Omega)$, r>2, or with the following normal derivative boundary conditions
\begin{equation*}
\begin{array}{lclr}
\ds \frac{\partial u}{\partial n} = h( x) &\ \rm or &\ \ds
\frac{\partial u}{\partial n} + \sigma \;u = h( x) &\ \rm on \: \partial\Omega
\end{array}
\end{equation*}
where $h(x) \in W^{1-\frac{1}{r}, r}(\partial \Omega)$, r>2, $\sigma >0$ and $n$ is the unit outward normal to the boundary $\partial \Omega$.