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} $:

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

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.

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)