View source for Potential of a mass distribution

An expression of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p0741201.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p0741202.png" /> is a bounded domain in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p0741203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p0741204.png" />, bounded by a closed Lyapunov surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p0741205.png" /> (a curve for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p0741206.png" />, cf. [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p0741207.png" /> is the fundamental solution of the Laplace operator:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p0741208.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p0741209.png" /> is the area of the unit sphere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412011.png" /> is the distance between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412013.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412014.png" /> is the volume element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412015.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412016.png" />, then the potential is defined for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412018.png" />. In the complementary domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412019.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412020.png" /> then has derivatives of all orders and satisfies the [[Laplace equation|Laplace equation]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412021.png" />, that is, is a [[Harmonic function|harmonic function]]; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412022.png" /> this function is regular at infinity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412023.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412024.png" /> the potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412025.png" /> belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412026.png" /> and satisfies the [[Poisson equation|Poisson equation]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412027.png" />.

These properties can be generalized in various ways. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412028.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412031.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412033.png" /> has generalized second derivatives in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412034.png" />, and the Poisson equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412035.png" /> is satisfied almost-everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412036.png" />. Properties of potentials of an arbitrary [[Radon measure|Radon measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412037.png" /> concentrated on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412038.png" />-dimensional domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412039.png" /> have also been studied:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412040.png" /></td> </tr></table>

Here again <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412042.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412044.png" /> almost-everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412045.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412046.png" /> is the derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412047.png" /> with respect to Lebesgue measure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412048.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412049.png" /> with variable coefficients of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412050.png" />; then the properties listed above still hold with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412051.png" /> replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412052.png" /> (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]]]).

For the solution of boundary value problems for parabolic partial differential equations the concept of a heat potential of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412053.png" /></td> </tr></table>

is used, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412054.png" /> is a fundamental solution of the heat equation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412055.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412056.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412057.png" /> is the density. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412058.png" /> and its generalizations to the case of an arbitrary second-order parabolic partial differential equation have properties similar to those given above for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412059.png" /> (see [[#References|[3]]]–[[#References|[6]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.M. Günter,   "Potential theory and its applications to basic problems of mathematical physics" , F. Ungar  (1967)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Miranda,   "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tichonoff,   A.A. Samarskii,   "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.I. Smirnov,   "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Friedman,   "Partial differential equations of parabolic type" , Prentice-Hall  (1964)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.V. Bitsadze,   "Boundary value problems for second-order elliptic equations" , North-Holland  (1968)  (Translated from Russian)</TD></TR></table>



====Comments====
A Levi function of a linear partial differential equation is also called a [[Fundamental solution|fundamental solution]] of this equation, or a parametrix of this equation. This function is named after E.E. Levi, who anticipated [[#References|[a1]]], [[#References|[a2]]] what is known today as the [[Parametrix method|parametrix method]].

See also [[Potential theory|Potential theory]]; [[Logarithmic potential|Logarithmic potential]]; [[Newton potential|Newton potential]]; [[Non-linear potential|Non-linear potential]]; [[Riesz potential|Riesz potential]]; [[Bessel potential|Bessel potential]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.E. Levi,   "Sulle equazioni lineari alle derivate parziali totalmente ellittiche"  ''Rend. R. Acc. Lincei, Classe Sci. (V)'' , '''16'''  (1907)  pp. 932–938</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.E. Levi,   "Sulle equazioni lineari totalmente ellittiche alle derivate parziali"  ''Rend. Circ. Mat. Palermo'' , '''24'''  (1907)  pp. 275–317</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  O.D. Kellogg,   "Foundations of potential theory" , F. Ungar  (1929)  (Re-issue: Springer, 1967)</TD></TR></table>