# Fundamental solution

of a linear partial differential equation

A solution of a partial differential equation $Lu ( x) = 0$, $x \in \mathbf R ^ {n}$, with coefficients of class $C ^ \infty$, in the form of a function $E ( x, y)$ that satisfies, for fixed $y \in \mathbf R ^ {n}$, the equation

$$L E ( x, y) = \delta ( x - y),\ \ x \neq y,$$

which is interpreted in the sense of the theory of generalized functions, where $\delta$ 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

$$\sum _ {i, j = 1 } ^ { n } a _ {ij} \frac{\partial ^ {2} u }{\partial x _ {i} \partial x _ {j} } = 0$$

with constant coefficients $a _ {ij}$ forming a positive-definite matrix $a$, a fundamental solution is provided by the function

$$E ( x, y) = \begin{cases} \displaystyle \left [ \sum _ {i, j = 1 } ^ { n } A _ {ij} ( x _ {i} - y _ {i} ) ( x _ {j} - y _ {j} ) \right ] ^ {( 2 - n)/2 } , & n > 2 , \\ \displaystyle \log \left [ \sum _ {i, j = 1 } ^ { n } A _ {ij} ( x _ {i} - y _ {i} ) ( x _ {j} - y _ {j} ) \right ] , & n = 2 , \end{cases}$$

where $A _ {ij}$ is the cofactor of $a _ {ij}$ in the matrix $a$.

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)