# 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 [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 $\mathbf R ^ {N}$:

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

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)