# Potential of a mass distribution

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 ) 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 ).

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

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 ).

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