G. Manjavidze, W. Tutschke, H. L. Vasudeva
abstract:
Very general boundary data for which boundary value problems are solvable,
belong to fractional order spaces. Suppose, e.g., that
the boundary data $g$ belongs to the fractional order space ${\mathcal W}_p^s
(\partial \Omega)$, where $p > 2$, $s=1 - {1 \over
p}$ and $\partial \Omega$ is sufficiently smooth. Then the solution of the
boundary value problem
$$ \Delta u = 0 \quad \mbox{in} \quad \Omega, \quad u = g \quad \mbox{on} \quad
\partial \Omega $$
belongs to ${\mathcal W}_p^1 (\Omega)$. On the other hand, piecewise H\"older
continuous functions do not belong to a
fractional order space, in general. However, sufficiently general boundary data
$g$ can be split up into the sum $g=\tilde g + \hat g$, where $\tilde g$ belongs
to a fractional order space and $\hat g$ is piecewise H\"older continuous.
The paper constructs the solution of the Dirichlet boundary value problem with
such boundary data for non-linear partial complex differential equations of the
type ${\partial w \over \partial \overline{z}} =F \left( z, w, {\partial w \over
\partial z} \right)$ provided the right hand side satisfies the global Lipschitz
condition with respect to $w$ and ${\partial w \over \partial {z}}$, and the
Lipschitz constants are small enough.
The method which will be applied to the Dirichlet boundary value problem can
also be used in order to solve the modified Dirichlet boundary value problem and
the Riemann-Hilbert boundary value problem with H\"older continuous
coefficients.
Mathematics Subject Classification: 30E25, 26A33
Key words and phrases: Dirichlet problem, fractional order space, integral operator, weighted Sobolev spaces