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)

Comments

Fundamental solutions are also used in the study of Cauchy problems (cf. Cauchy problem) for hyperbolic and parabolic equations. The name "elementary solution of a linear partial differential equation" is also used.

See also Green function.

References

[a1]	A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964)
[a2]	O.A. Ladyzhenskaya, N.N. Ural'tseva, "Linear and quasilinear elliptic equations" , Acad. Press (1968) (Translated from Russian)
[a3]	O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural'tseva, "Linear and quasilinear parabolic equations" , Amer. Math. Soc. (1968) (Translated from Russian)
[a4]	L. Schwartz, "Théorie des distributions" , 1–2 , Hermann (1957–1959)
[a5]	I.M. Gel'fand, G.E. Shilov, "Generalized functions" , Acad. Press (1964) (Translated from Russian)