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On extremal problems related to inverse balayage. (English) Zbl 1109.31001

Summary: Suppose \(G\) is a body in \(\mathbb R^d\), \(D\subset G\) is compact, and \(\rho\) a unit measure on \(\partial G\). Inverse balayage refers to the question of whether there exists a measure \(\nu\) supported inside \(D\) such that \(\rho\) and \(\nu\) produce the same electrostatic field outside \(G\). Establishing a duality principle between two extremal problems, it is shown that such an inverse balayage exists if and only if
\[ \sup_\mu \Bigl\{ \inf_{y\in D}U^\mu(y)- \int U^\rho\,d\mu\Bigr\}=0, \]
where the supremum is taken over all unit measures \(\mu\) on \(\partial G\) and \(U^\mu\) denotes the electrostatic potential of \(\mu\). A consequence is that pairs \((\rho,D)\) admitting such an inverse balayage can be characterized by a \(\rho\)-mean-value principle, namely,
\[ \sup_{z\in D} h(z)\geq\int h\,d\rho\geq \inf_{z\in D}h(z) \]
for all \(h\) harmonic in \(G\) and continuous up to the boundary. In addition, two approaches for the construction of an inverse balayage related to extremal point methods are presented, and the results are applied to problems concerning the determination of restricted Chebyshev constants in the theory of polynomial approximation.

MSC:

31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
30C85 Capacity and harmonic measure in the complex plane
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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