**V. Kokilashvili and V. Paatashvili**

## Neumann Problem in a Class of Harmonic Functions in Domains with a Piecewise-Lyapunov Boundary

**abstract:**

The Neumann boundary value problem is considered in a finite simply connected
domain $D$ with piecewise Lyapunov boundary free from zero interior angles.
The solution is sought in the class of harmonic functions $u$ satisfying $(1)$,
where $\Gm_r$ is the image of the circle $|z|=r$ under the conform mapping of
the unit disc onto $D$.

**Mathematics Subject Classification:**
35J25, 35J40, 35B65.

**Key words and phrases:**
Dirichlet and Neumann problem, Smirnov class,
conformal mapping, piecewise Lyapunov boundary, two-weight
inequality, Cauchy type integral.