# Potential of a mass distribution

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An expression of the form

$$\tag{* } u ( x) = \int\limits _ { D } h ( | x - y | ) f ( y) d v ( y) ,$$

where $D$ is a bounded domain in a Euclidean space $\mathbf R ^ {N}$, $N \geq 2$, bounded by a closed Lyapunov surface $S$( a curve for $N = 2$, cf. Lyapunov surfaces and curves), $h ( | x - y | )$ is the fundamental solution of the Laplace operator:

$$h ( | x - y | ) = \ \left \{ \begin{array}{ll} \frac{1}{( N - 2 ) \omega _ {N} | x - y | ^ {N-} 2 } , & N \geq 3 ; \\ \frac{1}{2 \pi } \mathop{\rm ln} \frac{1}{| x - y | } , & N = 2 ; \\ \end{array} \right .$$

where $\omega _ {N} = 2 \pi ^ {N/2} / \Gamma ( N / 2 )$ is the area of the unit sphere in $\mathbf R ^ {N}$, $| x - y |$ is the distance between the points $x$ and $y$, and $d v ( y)$ is the volume element in $D$.

If $f \in C ^ {(} 1) ( \overline{D}\; )$, then the potential is defined for all $x \in \mathbf R ^ {N}$ and $u \in C ^ {(} 1) ( \mathbf R ^ {N} )$. In the complementary domain $\overline{D}\; {} ^ {c}$, the function $u$ then has derivatives of all orders and satisfies the Laplace equation: $\Delta u = 0$, that is, is a harmonic function; for $N \geq 3$ this function is regular at infinity, $u ( \infty ) = 0$. In $D$ the potential $u$ belongs to the class $C ^ {(} 2) ( D)$ and satisfies the Poisson equation: $\Delta u = - f$.

These properties can be generalized in various ways. For example, if $f \in L _ \infty ( D)$, then $u \in C ( \mathbf R ^ {N} )$, $u \in C ^ \infty ( \overline{D}\; {} ^ {c} )$, $\Delta u = 0$ in $\overline{D}\; {} ^ {c}$, $u$ has generalized second derivatives in $D$, and the Poisson equation $\Delta u = - f$ is satisfied almost-everywhere in $D$. Properties of potentials of an arbitrary Radon measure $\mu$ concentrated on an $N$- dimensional domain $D$ have also been studied:

$$u ( x) = \int\limits h ( | x - y | ) d \mu ( y) .$$

Here again $u \in C ^ \infty ( \overline{D}\; {} ^ {c} )$ and $\Delta u = 0$ in $\overline{D}\; {} ^ {c}$, $\Delta u = - \mu ^ \prime$ almost-everywhere in $D$, where $\mu ^ \prime$ is the derivative of $\mu$ with respect to Lebesgue measure in $\mathbf R ^ {n}$. 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 $L$ with variable coefficients of class $C ^ {( 0 , \lambda ) } ( \overline{D}\; )$; then the properties listed above still hold with $\Delta u$ replaced by $L u$( 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

$$v ( x , t ) = \ \int\limits _ { 0 } ^ { t } d \tau \int\limits _ { D } G ( x , t ; y, \tau ) f ( y , \tau ) d v ( y)$$

is used, where $G ( x , t ; y , \tau )$ is a fundamental solution of the heat equation in $\mathbf R ^ {N}$:

$$G ( x , t ; y , \tau ) = \ \frac{1}{( 2 \sqrt \pi ) ^ {N} ( t - \tau ) ^ {N/2} } \mathop{\rm exp} ^ {- | x - y | ^ {2} / 4 ( t - \tau ) } ,$$

and $f ( y , \tau )$ is the density. The function $v ( x , t )$ and its generalizations to the case of an arbitrary second-order parabolic partial differential equation have properties similar to those given above for $u$( 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.

#### 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)
How to Cite This Entry:
Potential of a mass distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Potential_of_a_mass_distribution&oldid=49529
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article