# Fundamental solution

of a linear partial differential equation

A solution of a partial differential equation , , with coefficients of class , in the form of a function that satisfies, for fixed , the equation

which is interpreted in the sense of the theory of generalized functions, where is the delta-function. There is a fundamental solution for every partial differential equation with constant coefficients, and also for arbitrary elliptic equations. For example, for the elliptic equation

with constant coefficients forming a positive-definite matrix , a fundamental solution is provided by the function

where is the cofactor of in the matrix .

Fundamental solutions are widely used in the study of boundary value problems for elliptic equations.

#### References

 [1] V.S. Vladimirov, "Generalized functions in mathematical physics" , MIR (1979) (Translated from Russian) [2] L. Bers, F. John, M. Schechter, "Partial differential equations" , Interscience (1964) [3] F. John, "Plane waves and spherical means: applied to partial differential equations" , Interscience (1955)