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 ).
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Non-linear potential. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Non-linear_potential&oldid=18320