Non-linear potential

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