Variation of the equilibrium energy and the \(S\)-property of stationary compact sets.

*(English. Russian original)*Zbl 1244.31001
Sb. Math. 202, No. 12, 1831-1852 (2011); translation from Mat. Sb. 202, No. 12, 113-136 (2011).

The authors study a variation of the equilibrium energy for a certain functional. The mentioned energy functional is a combination (with parameter) of the logarithmic and Green energy functionals related to the unit interval nested in a certain regular domain in \(\bar\mathbb C\). The extremal measure is the unique equilibrium measure for the mixed Green-logarithmic potential. After that the authors introduce the so-called stationary compact sets, which are the sets on which the supremum in a certain potential theoretic problem concerning the so-defined energy functional is actually attained. There is a short discussion from which it becomes clear that the existence of the stationary compact sets is a nontrivial problem and moreover these sets are generically different for different choices of the parameter, used to define the energy functional.

There are two main results in this paper. The first one says that a stationary compact set consisting of finite number of connected components obeys the following conditions which characterize it: The set has a connected complement in \(\mathbb C\), it has an empty interior and satisfies the \(S\)-property defined in the paper.

The \(S\)-property is roughly speaking the equity of the normal derivatives (on both sides of an analytic arc) of the Green potential corresponding to the given compact set.

The other result is a characterization of the equilibrium measure of a stationary compact set. It turns out that it’s Cauchy-transform is implicit in an identity (identity (9) in the paper) in which the right hand side is a ratio of certain polynomials. As a corollary of this, a stationary compact set consists of the critical trajectories of some quadratic differential.

The question of uniqueness of the stationary compact sets is also pursued in this paper.

The main results of this paper were announced earlier in [the authors, Russ. Math. Surv. 66, No. 1, 176–178 (2011; Zbl 1236.31001)].

There are two main results in this paper. The first one says that a stationary compact set consisting of finite number of connected components obeys the following conditions which characterize it: The set has a connected complement in \(\mathbb C\), it has an empty interior and satisfies the \(S\)-property defined in the paper.

The \(S\)-property is roughly speaking the equity of the normal derivatives (on both sides of an analytic arc) of the Green potential corresponding to the given compact set.

The other result is a characterization of the equilibrium measure of a stationary compact set. It turns out that it’s Cauchy-transform is implicit in an identity (identity (9) in the paper) in which the right hand side is a ratio of certain polynomials. As a corollary of this, a stationary compact set consists of the critical trajectories of some quadratic differential.

The question of uniqueness of the stationary compact sets is also pursued in this paper.

The main results of this paper were announced earlier in [the authors, Russ. Math. Surv. 66, No. 1, 176–178 (2011; Zbl 1236.31001)].

Reviewer: Zywomir Dinew (Kraków)

##### MSC:

31A15 | Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions |

41A20 | Approximation by rational functions |