Principal fundamental solution

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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.


[1] C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)
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Principal fundamental solution. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by Sh.A. Alimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article