# 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