# Bitsadze equation

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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)
How to Cite This Entry:
Bitsadze-Samarskii problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bitsadze-Samarskii_problem&oldid=43269