Difference between revisions of "Potential of a mass distribution"
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is a bounded domain in a Euclidean space $ \mathbf R ^ {N} $, | is a bounded domain in a Euclidean space $ \mathbf R ^ {N} $, | ||
$ N \geq 2 $, | $ N \geq 2 $, | ||
| − | bounded by a closed Lyapunov surface $ S $( | + | bounded by a closed Lyapunov surface $ S $ (a curve for $ N = 2 $, |
| − | a curve for $ N = 2 $, | ||
cf. [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]), $ h ( | x - y | ) $ | cf. [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]), $ h ( | x - y | ) $ | ||
is the fundamental solution of the Laplace operator: | is the fundamental solution of the Laplace operator: | ||
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$$ | $$ | ||
h ( | x - y | ) = \ | h ( | x - y | ) = \ | ||
| − | \left \{ | + | \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 ) $ | where $ \omega _ {N} = 2 \pi ^ {N/2} / \Gamma ( N / 2 ) $ | ||
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is the volume element in $ D $. | is the volume element in $ D $. | ||
| − | If $ f \in C ^ {( | + | If $ f \in C ^ {( 1)} ( \overline{D} ) $, |
then the potential is defined for all $ x \in \mathbf R ^ {N} $ | then the potential is defined for all $ x \in \mathbf R ^ {N} $ | ||
| − | and $ u \in C ^ {( | + | and $ u \in C ^ {( 1)} ( \mathbf R ^ {N} ) $. |
| − | In the complementary domain $ \overline{D | + | In the complementary domain $ \overline{D} ^ {c} $, |
the function $ u $ | the function $ u $ | ||
then has derivatives of all orders and satisfies the [[Laplace equation|Laplace equation]]: $ \Delta u = 0 $, | then has derivatives of all orders and satisfies the [[Laplace equation|Laplace equation]]: $ \Delta u = 0 $, | ||
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In $ D $ | In $ D $ | ||
the potential $ u $ | the potential $ u $ | ||
| − | belongs to the class $ C ^ {( | + | belongs to the class $ C ^ {( 2)} ( D) $ |
and satisfies the [[Poisson equation|Poisson equation]]: $ \Delta u = - f $. | and satisfies the [[Poisson equation|Poisson equation]]: $ \Delta u = - f $. | ||
These properties can be generalized in various ways. For example, if $ f \in L _ \infty ( D) $, | These properties can be generalized in various ways. For example, if $ f \in L _ \infty ( D) $, | ||
then $ u \in C ( \mathbf R ^ {N} ) $, | then $ u \in C ( \mathbf R ^ {N} ) $, | ||
| − | $ u \in C ^ \infty ( \overline{D} | + | $ u \in C ^ \infty ( \overline{D} ^ {c} ) $, |
$ \Delta u = 0 $ | $ \Delta u = 0 $ | ||
| − | in $ \overline{D | + | in $ \overline{D} ^ {c} $, |
$ u $ | $ u $ | ||
has generalized second derivatives in $ D $, | has generalized second derivatives in $ D $, | ||
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is satisfied almost-everywhere in $ D $. | is satisfied almost-everywhere in $ D $. | ||
Properties of potentials of an arbitrary [[Radon measure|Radon measure]] $ \mu $ | Properties of potentials of an arbitrary [[Radon measure|Radon measure]] $ \mu $ | ||
| − | concentrated on an $ N $- | + | concentrated on an $ N $-dimensional domain $ D $ |
| − | dimensional domain $ D $ | ||
have also been studied: | have also been studied: | ||
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$$ | $$ | ||
| − | Here again $ u \in C ^ \infty ( \overline{D} | + | Here again $ u \in C ^ \infty ( \overline{D} ^ {c} ) $ |
and $ \Delta u = 0 $ | and $ \Delta u = 0 $ | ||
| − | in $ \overline{D} | + | in $ \overline{D} ^ {c} $, |
$ \Delta u = - \mu ^ \prime $ | $ \Delta u = - \mu ^ \prime $ | ||
almost-everywhere in $ D $, | almost-everywhere in $ D $, | ||
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with respect to Lebesgue measure in $ \mathbf R ^ {n} $. | 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 [[#References|[2]]]) for a general second-order elliptic operator $ L $ | In definition (*) the fundamental solution of the Laplace operator may be replaced by an arbitrary Levi function (see [[#References|[2]]]) for a general second-order elliptic operator $ L $ | ||
| − | with variable coefficients of class $ C ^ {( 0 , \lambda ) } ( \overline{D} | + | with variable coefficients of class $ C ^ {( 0 , \lambda ) } ( \overline{D} ) $; |
then the properties listed above still hold with $ \Delta u $ | then the properties listed above still hold with $ \Delta u $ | ||
| − | replaced by $ L u $( | + | replaced by $ L u $ (see [[#References|[2]]]–[[#References|[4]]]). |
| − | see [[#References|[2]]]–[[#References|[4]]]). | ||
Potentials of mass distributions are applied in the solution of boundary value problems for elliptic partial differential equations (see [[#References|[2]]]–[[#References|[5]]]). | Potentials of mass distributions are applied in the solution of boundary value problems for elliptic partial differential equations (see [[#References|[2]]]–[[#References|[5]]]). | ||
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G ( x , t ; y , \tau ) = \ | G ( x , t ; y , \tau ) = \ | ||
| − | \frac{1}{( 2 \sqrt \pi ) ^ {N} ( t - \tau ) ^ {N/2} } | + | \frac{1}{( 2 \sqrt \pi ) ^ {N} ( t - \tau ) ^ {N/2} } \exp ^ {- | x - y | ^ {2} / 4 ( t - \tau ) } , |
| − | |||
| − | |||
$$ | $$ | ||
and $ f ( y , \tau ) $ | and $ f ( y , \tau ) $ | ||
is the density. The function $ v ( x , t ) $ | is the density. The function $ v ( x , t ) $ | ||
| − | and its generalizations to the case of an arbitrary second-order parabolic partial differential equation have properties similar to those given above for $ u $( | + | and its generalizations to the case of an arbitrary second-order parabolic partial differential equation have properties similar to those given above for $ u $ (see [[#References|[3]]]–[[#References|[6]]]). |
| − | see [[#References|[3]]]–[[#References|[6]]]). | ||
====References==== | ====References==== | ||
Latest revision as of 05:56, 13 June 2022
An expression of the form
$$ \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) |
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=48264