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On “balayage inwards” of charges in \(\mathbb{R}^ n\). (English. Russian original) Zbl 0836.31005
Math. Notes 54, No. 6, 1246-1260 (1993); translation from Mat. Zametki 54, No. 6, 90-112 (1993).
Let \(U^f\) stand for the potential of a given density \(f\) in a domain \(\Omega \subset \mathbb{R}^n\). The problem is to find a distribution \(w\) with compact support in \(\Omega\) such that \(U^f= U^w\) outside \(\Omega\) and such that the support of \(w\) is minimal in some sense. The problem is reduced to the complex analytic Cauchy problem, which is the central problem of the article. It is supposed that the boundary of \(\Omega\) is an algebraic hypersurface in \(\mathbb{R}^n\) and that \(f\) has an analytic continuation to the compactification \(\mathbb{C} P^n\) of \(\mathbb{C}^n\).
Reviewer: M.Dont (Praha)
MSC:
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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