Condensers with infinitely many touching Borel plates and minimum energy problems

• Autorzy: Zorii Natalia

Identyfikatory

DOI

10.4064/ba190414-29-11

• Języki publikacji: EN

Abstrakty

EN

Defining a condenser in a locally compact space as a locally finite, countable collection of Borel sets A_i, i ϵ I, with the sign s_i = ±1 prescribed such that A_i ∩ A_j = Ø; whenever s_is_j = −1, we consider a minimum energy problem with an external field over infinite-dimensional vector measures (µⁱ) i ϵ I, where µⁱ is a suitably normalized positive Radon measure carried by A_i and such that µⁱ ≤ ξⁱ for all i ϵ I₀, I₀ ⊂ I and constraints ξⁱ, i ϵ I₀, being given. If I₀ = Ø, the problem reduces to the (unconstrained) Gauss variational problem, which is in general unsolvable even for a condenser of two closed, oppositely signed plates in R³ and the Coulomb kernel. Nevertheless, we provide sufficient conditions for the existence of solutions to the stated problem in its full generality, establish the vague compactness of the solutions, analyze their uniqueness, describe their weighted potentials, and single out their characteristic properties. The strong and the vague convergence of minimizing nets to the minimizers is studied. The phenomena of non-existence and non-uniqueness of solutions to the problem are illustrated by examples. The results obtained are new even for the classical kernels on Rⁿ, n ≥ 2, and closed A_i, i ϵ I, which is import ant for applications.

Słowa kluczowe

EN

infinite-dimensional vector measures; minimum energy problems; condensers of infinitely many touching Borel plates; external fields; constraints

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2019
• Tom: Vol. 67, no. 2
• Strony: 125--163
• Opis fizyczny: Bibliogr. 45 poz., rys.

Twórcy

autor

Zorii Natalia

• Institute of Mathematics, Academy of Sciences of Ukraine, Tereshchenkivska 3, 01601 Kyiv, Ukraine

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Uwagi

Dedicated to the memory of Professor Bogdan Bojarski

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).