Potential of a mass distribution

An expression of the form

(*)

where is a bounded domain in a Euclidean space , , bounded by a closed Lyapunov surface (a curve for , cf. Lyapunov surfaces and curves), is the fundamental solution of the Laplace operator:

where is the area of the unit sphere in , is the distance between the points and , and is the volume element in .

If , then the potential is defined for all and . In the complementary domain , the function then has derivatives of all orders and satisfies the Laplace equation: , that is, is a harmonic function; for this function is regular at infinity, . In the potential belongs to the class and satisfies the Poisson equation: .

These properties can be generalized in various ways. For example, if , then , , in , has generalized second derivatives in , and the Poisson equation is satisfied almost-everywhere in . Properties of potentials of an arbitrary Radon measure concentrated on an -dimensional domain have also been studied:

Here again and in , almost-everywhere in , where is the derivative of with respect to Lebesgue measure in . In definition (*) the fundamental solution of the Laplace operator may be replaced by an arbitrary Levi function (see [2]) for a general second-order elliptic operator with variable coefficients of class ; then the properties listed above still hold with replaced by (see [2]–[4]).

Potentials of mass distributions are applied in the solution of boundary value problems for elliptic partial differential equations (see [2]–[5]).

For the solution of boundary value problems for parabolic partial differential equations the concept of a heat potential of the form

is used, where is a fundamental solution of the heat equation in :

and is the density. The function and its generalizations to the case of an arbitrary second-order parabolic partial differential equation have properties similar to those given above for (see [3]–[6]).

References

[1]	N.M. Günter, "Potential theory and its applications to basic problems of mathematical physics" , F. Ungar (1967) (Translated from French)
[2]	C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)
[3]	A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[4]	V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian)
[5]	A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964)
[6]	A.V. Bitsadze, "Boundary value problems for second-order elliptic equations" , North-Holland (1968) (Translated from Russian)

Comments

A Levi function of a linear partial differential equation is also called a fundamental solution of this equation, or a parametrix of this equation. This function is named after E.E. Levi, who anticipated [a1], [a2] what is known today as the parametrix method.

See also Potential theory; Logarithmic potential; Newton potential; Non-linear potential; Riesz potential; Bessel potential.

References

[a1]	E.E. Levi, "Sulle equazioni lineari alle derivate parziali totalmente ellittiche" Rend. R. Acc. Lincei, Classe Sci. (V) , 16 (1907) pp. 932–938
[a2]	E.E. Levi, "Sulle equazioni lineari totalmente ellittiche alle derivate parziali" Rend. Circ. Mat. Palermo , 24 (1907) pp. 275–317
[a3]	O.D. Kellogg, "Foundations of potential theory" , F. Ungar (1929) (Re-issue: Springer, 1967)