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A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067180/n0671801.png" /> generated by a [[Radon measure|Radon measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067180/n0671802.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067180/n0671803.png" /> being a point of the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067180/n0671804.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067180/n0671805.png" />, that depends non-linearly on the generating measure.
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A function  $  U _  \mu  ( x) $
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generated by a [[Radon measure|Radon measure]] $  \mu $,  
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$  x $
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being a point of the Euclidean space $  \mathbf R  ^ {N} $,  
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$  N \geq  2 $,  
 +
that depends non-linearly on the generating measure.
  
 
For example, in the study of properties of solutions of partial differential equations and of boundary properties of analytic functions, non-linear potentials of the following form turn out to be useful:
 
For example, in the study of properties of solutions of partial differential equations and of boundary properties of analytic functions, non-linear potentials of the following form turn out to be useful:
  
<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/n/n067/n067180/n0671806.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$ \tag{* }
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U _  \mu  ( x)  = U _  \mu  ( x ; p , l ) =
 +
$$
  
<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/n/n067/n067180/n0671807.png" /></td> </tr></table>
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$$
 +
= \
 +
\int\limits \left [ \int\limits
 +
\frac{d \mu ( z) }{| y - z |  ^ {N-} l }
 +
\right ] ^ {1 / ( p - 1 ) }
 +
\frac{dy}{| x - y |  ^ {N-} l }
 +
,\  x \in \mathbf R  ^ {N} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067180/n0671808.png" /> is the distance between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067180/n0671809.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067180/n06718010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067180/n06718011.png" /> is a Radon measure with compact support, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067180/n06718012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067180/n06718013.png" /> are real numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067180/n06718014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067180/n06718015.png" />.
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where $  | x - y | $
 +
is the distance between $  x $
 +
and $  y $,  
 +
$  \mu $
 +
is a Radon measure with compact support, and $  p $
 +
and $  l $
 +
are real numbers, $  1 < p < \infty $,
 +
0 < l < \infty $.
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067180/n06718016.png" /> the non-linear potentials (*) turn into the linear Riesz potentials (cf. [[Riesz potential|Riesz potential]]), and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067180/n06718017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067180/n06718018.png" /> into the classical [[Newton potential|Newton potential]]. The concepts of capacity and energy have been constructed, and analogues of certain basic theorems of [[Potential theory|potential theory]] have been proved for the non-linear potential (*) (see [[#References|[1]]]).
+
For $  p = 2 $
 +
the non-linear potentials (*) turn into the linear Riesz potentials (cf. [[Riesz potential|Riesz potential]]), and for $  p = 2 $
 +
and $  l = 1 $
 +
into the classical [[Newton potential|Newton potential]]. The concepts of capacity and energy have been constructed, and analogues of certain basic theorems of [[Potential theory|potential theory]] have been proved for the non-linear potential (*) (see [[#References|[1]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.G. Maz'ya,  V.P. Khavin,  "Nonlinear potential theory"  ''Russian Math. Surveys'' , '''27''' :  6  (1972)  pp. 71–148  ''Uspekhi Mat. Nauk'' , '''27''' :  6  (1972)  pp. 67–138</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.G. Maz'ya,  V.P. Khavin,  "Nonlinear potential theory"  ''Russian Math. Surveys'' , '''27''' :  6  (1972)  pp. 71–148  ''Uspekhi Mat. Nauk'' , '''27''' :  6  (1972)  pp. 67–138</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:03, 6 June 2020


A function $ U _ \mu ( x) $ generated by a Radon measure $ \mu $, $ x $ being a point of the Euclidean space $ \mathbf R ^ {N} $, $ N \geq 2 $, that depends non-linearly on the generating measure.

For example, in the study of properties of solutions of partial differential equations and of boundary properties of analytic functions, non-linear potentials of the following form turn out to be useful:

$$ \tag{* } U _ \mu ( x) = U _ \mu ( x ; p , l ) = $$

$$ = \ \int\limits \left [ \int\limits \frac{d \mu ( z) }{| y - z | ^ {N-} l } \right ] ^ {1 / ( p - 1 ) } \frac{dy}{| x - y | ^ {N-} l } ,\ x \in \mathbf R ^ {N} , $$

where $ | x - y | $ is the distance between $ x $ and $ y $, $ \mu $ is a Radon measure with compact support, and $ p $ and $ l $ are real numbers, $ 1 < p < \infty $, $ 0 < l < \infty $.

For $ p = 2 $ the non-linear potentials (*) turn into the linear Riesz potentials (cf. Riesz potential), and for $ p = 2 $ and $ l = 1 $ into the classical Newton potential. The concepts of capacity and energy have been constructed, and analogues of certain basic theorems of potential theory have been proved for the non-linear potential (*) (see [1]).

References

[1] V.G. Maz'ya, V.P. Khavin, "Nonlinear potential theory" Russian Math. Surveys , 27 : 6 (1972) pp. 71–148 Uspekhi Mat. Nauk , 27 : 6 (1972) pp. 67–138

Comments

In recent years, non-linear versions of different branches of potential theory, concrete or axiomatic, have been constructed. A sample of these developments is given by [a1][a6].

References

[a1] D.R. Adams, "Weighted nonlinear potential theory" Trans. Amer. Math. Soc. , 297 (1986) pp. 73–94
[a2] E.M.J. Bertin, "Fonctions convexes et théorie du potentiel" Indag. Math. , 41 (1979) pp. 385–409
[a3] S. Grandlund, P. Lindqvist, O. Martio, "Note on the PWB-method in the non-linear case" Pacific J. Math. , 125 (1986) pp. 381–395
[a4] L.I. Hedberg, Th.H. Wolff, "Thin sets in nonlinear potential theory" Ann. Inst. Fourier (Grenoble) , 33 : 4 (1983) pp. 161–187
[a5] I. Laine, "Axiomatic non-linear potential theories" J. Král (ed.) J. Lukeš (ed.) J. Veselý (ed.) , Potential theory. Survey and problems (Prague, 1987) , Lect. notes in math. , 1344 , Springer (1988) pp. 118–132
[a6] Y. Mizuta, T. Nakai, "Potential theoretic properties of the subdifferential of a convex function" Hiroshima Math. J. , 7 (1977) pp. 177–182
How to Cite This Entry:
Non-linear potential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-linear_potential&oldid=47999
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article