# Principal fundamental solution

From Encyclopedia of Mathematics

A fundamental solution , defined throughout the space , of a second-order elliptic equation

(*) |

that satisfies the conditions

for certain positive constants and if .

If the coefficients , and satisfy a Hölder condition on and if the inequality is satisfied for some , then a principal fundamental solution exists. If the coefficients of the operator are defined in a certain bounded domain with smooth boundary, then they can be extended to the entire space so that a principal fundamental solution will exist for the extended operator.

#### References

[1] | C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian) |

**How to Cite This Entry:**

Principal fundamental solution.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Principal_fundamental_solution&oldid=14964

This article was adapted from an original article by Sh.A. Alimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article