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Equilibrium measures and partial balayage. (English) Zbl 1329.31001

Let \(\mu\) and \(\lambda\) be (signed) Radon measures with compact supports. Set \[ V^\mu_\lambda =\sup\bigg\{ V\in {\mathcal D}'({\mathbb C}): V \leq U^\mu \text{ in } {\mathbb C}, -\frac 1{2\pi } \Delta V \leq \lambda \text{ in } {\mathbb C} \bigg\}, \] where \(U^\mu(z)=-\int \log|z-\zeta|\, d\mu(\zeta)\) and \({\mathcal D}'({\mathbb C})\) denotes the set of distributions on \( {\mathbb C}\). If the set of all \(V\) under the supremum is not empty, the author defines the partial balayage of \(\mu\) to \(\lambda\) to be the signed Radon measure \[ \mathrm{Bal} (\mu, \lambda):=-\frac 1{2\pi} \Delta V^\mu_{\lambda}. \] In the case \(\lambda\equiv 0\) the following properties are established.
Theorem 5.1. Let \(\sigma=\sigma_+-\sigma_-\) be a signed Radon measure with compact support for which \(\sigma({\mathbb C})<0\) and such that \(U^{\sigma_-}\) is continuous on \({\mathbb C}\). Then \(\mathrm{Bal}(\sigma, 0)\) exists, has the same total mass as \(\sigma\), satisfies \(\mathrm{supp\,}\mathrm{Bal}(\sigma, 0)\subseteq \mathrm{supp\,} \sigma_-\), and has finite logarithmic energy. Moreover, with \(V^\sigma_0\) as in the definition, for \(\nu:=\mathrm{Bal}(\sigma,0)\), define the sets \(\omega:=\{z\in {\mathbb C}: V^\sigma(z)< U^\sigma(z) \}\) and \(\Omega:={\mathbb C} \setminus\mathrm{ supp\,} \nu\). Then \(\omega\) is an open set and \(\omega\subseteq \Omega\), so for every \(z\in \mathrm{supp\,} \nu\) we have \(V^\sigma_0=U^\sigma(z)\). Furthermore, there exists a constant \(c_0\) such that \(V^\sigma_0=U^\nu+c_0\).
A function \(Q: E\to (-\infty, \infty]\) is called a \(t\)-admissible background potential for \(t>0\) if the following holds: (i) \(Q\) is lower semicontinuous, (ii) \(\mathrm{cap}(\{ z\in E: Q(z)<\infty\})>0\), and (iii) \(Q(z)-t\log |z|\to \infty \) as \(|z|\to \infty\), \(z\in E\) (if \(E\) is unbounded).
Let \(\mu_{Q,t}\) denote the equilibrium measure of the background potential \(Q\), see, e.g., [T. Ransford, Potential theory in the complex plane. London Mathematical Society Student Texts. 28. Cambridge: Univ. Press (1995; Zbl 0828.31001)].
Let \(Q\) be a \(t\)-admissible background potential on \(E\subseteq {\mathbb C}\) for some \(t>0\). If \({ G}=(E', \sigma, c)\) is a triple with \(E'\subseteq E\) a compact set, \(\sigma\) a signed and compactly supported Radon measure with \(\sigma({\mathbb C})=-t\) and \(\mathrm{supp\,} \sigma_-\subseteq E'\), and \(c\in {\mathbb R}\) a constant such that the function \(\tilde Q:=c+U^\sigma(z)\) satisfies \[ \tilde Q(z)=Q(z) \text{ for q.e. } z\in E', \quad \tilde Q(z) \leq Q(z) \text{ for q.e. } z\in E, \] then we say that \( G\) defines a \(t\)-extension of \(Q\) (relative to \(E'\)).
{ Theorem 6.4.} Let \(Q\) be a \(t\)-admissible background potential on \(E\subseteq {\mathbb C}\), and assume \({ G}=(E', \sigma, c)\) defines a \(t\)-extension \(\tilde D=c+U^\sigma \) of \(Q\) relative to \(E'\). If \(U^{\sigma_-}\) is continuous on \({\mathbb C}\), then \[ \mu_{Q,t}+\mathrm{Bal}(\sigma, 0)=0. \]

MSC:

31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
30C85 Capacity and harmonic measure in the complex plane

Citations:

Zbl 0828.31001
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References:

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