Bitsadze equation

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 [1]. 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 [2].

References

[1]	A.V. Bitsadze, "On the uniqueness of the solution of the Dirichlet problem for elliptic partial differential operators" Uspekhi Mat. Nauk , 3 : 6 (1948) pp. 211–212 (In Russian)
[2]	A.V. Bitsadze, "Boundary value problems for second-order elliptic equations" , North-Holland (1968) (Translated from Russian)
[3]	C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)
[4]	L. Bers, F. John, M. Schechter, "Partial differential equations" , Interscience (1964)