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) |
Potential of a mass distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Potential_of_a_mass_distribution&oldid=49370