Difference between revisions of "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: | 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 | + | 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 | + | 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=18320
Non-linear potential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-linear_potential&oldid=18320
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article