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)

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 equationelementary solution" 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)