# 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)