# Non-linear potential

A function generated by a Radon measure , being a point of the Euclidean space , , 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:

(*) |

where is the distance between and , is a Radon measure with compact support, and and are real numbers, , .

For the non-linear potentials (*) turn into the linear Riesz potentials (cf. Riesz potential), and for and 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. E.D. Solomentsev (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Non-linear_potential&oldid=18320