##### Actions

The partial differential equation that can be written in complex form as follows:

$$4w _ {\overline{z}\; \overline{z}\; } \equiv \ w _ {xx} +2 iw _ {xy} - w _ {yy} = 0,$$

where $w(z) = u + iv, z = x + iy$, and that can be reduced to the elliptic system

$$u _ {xx} - u _ {yy} -2v _ {xy} = 0,$$

$$v _ {xx} - v _ {yy} + 2u _ {xy} = 0,$$

in the real independent variables $x$ and $y$. The homogeneous Dirichlet problem in a disc $C$: $| z - z _ {0} | < \epsilon$, where the radius $\epsilon$ is as small as one pleases, for the Bitsadze equation has an infinite number of linearly independent solutions . The Dirichlet problem for the inhomogeneous equation $w _ {\overline{z}\; \overline{z}\; } = f$ in the disc $C$ is normally solvable according to Hausdorff, since it is neither a Fredholm problem nor Noetherian; in a bounded domain containing a segment of the straight line $y = 0$, this problem is not even a Hausdorff problem, even though the homogeneous problem has only one zero solution .

How to Cite This Entry: