# Bitsadze equation

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

where , and that can be reduced to the elliptic system

in the real independent variables and . The homogeneous Dirichlet problem in a disc : , where the radius 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 in the disc 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 , 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 equation. A.M. Nakhushev (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Bitsadze_equation&oldid=17152