View source for Non-linear potential

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.

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>

<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>

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" />.

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]]]).

====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>