View source for Non-linear potential

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A function  $  U _  \mu  ( x) $
generated by a [[Radon measure|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|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====
<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====
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 [[#References|[a1]]]–[[#References|[a6]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.R. Adams,   "Weighted nonlinear potential theory"  ''Trans. Amer. Math. Soc.'' , '''297'''  (1986)  pp. 73–94</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.M.J. Bertin,   "Fonctions convexes et théorie du potentiel"  ''Indag. Math.'' , '''41'''  (1979)  pp. 385–409</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Grandlund,   P. Lindqvist,   O. Martio,   "Note on the PWB-method in the non-linear case"  ''Pacific J. Math.'' , '''125'''  (1986)  pp. 381–395</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L.I. Hedberg,   Th.H. Wolff,   "Thin sets in nonlinear potential theory"  ''Ann. Inst. Fourier (Grenoble)'' , '''33''' :  4  (1983)  pp. 161–187</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  Y. Mizuta,   T. Nakai,   "Potential theoretic properties of the subdifferential of a convex function"  ''Hiroshima Math. J.'' , '''7'''  (1977)  pp. 177–182</TD></TR></table>