Theory of Potential with Respect to Consistent Kernels; Theorem on Completeness and Sequences of Potentials
The concept of consistent kernels introduced by Fuglede in 1960 is widely used in extremal problems of the theory of potential on classes of positive measures. In the present paper, we show that this concept is also efficient for the investigation of extremal problems on fairly broad classes of sign...
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| Datum: | 2004 |
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| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2004
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509997016809472 |
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| author | Zorii, N. V. Зорий, Н. В. Зорий, Н. В. |
| author_facet | Zorii, N. V. Зорий, Н. В. Зорий, Н. В. |
| author_sort | Zorii, N. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:12:48Z |
| description | The concept of consistent kernels introduced by Fuglede in 1960 is widely used in extremal problems of the theory of potential on classes of positive measures. In the present paper, we show that this concept is also efficient for the investigation of extremal problems on fairly broad classes of signed measures. In particular, for an arbitrary consistent kernel in a locally compact space, we prove a theorem on the strong completeness of fairly general subspaces E of all measures with finite energy. (Note that, according to the well-known Cartan counterexample, the entire space E is strongly incomplete even in the classical case of the Newton kernel in ℝn Using this theorem, we obtain new results for the Gauss variational problem, namely, in the non-compact case, we give a description of vague and (or) strong limiting measures of minimizing sequences and obtain sufficient solvability conditions. |
| first_indexed | 2026-03-24T02:49:59Z |
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| id | umjimathkievua-article-3862 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | rus English |
| last_indexed | 2026-03-24T02:49:59Z |
| publishDate | 2004 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/15/0383d74bca10f713170429ce5fa19e15.pdf |
| spelling | umjimathkievua-article-38622020-03-18T20:12:48Z Theory of Potential with Respect to Consistent Kernels; Theorem on Completeness and Sequences of Potentials Теория потенциала относительно согласованных ядер: теорема о полноте, последовательности потенциалов Zorii, N. V. Зорий, Н. В. Зорий, Н. В. The concept of consistent kernels introduced by Fuglede in 1960 is widely used in extremal problems of the theory of potential on classes of positive measures. In the present paper, we show that this concept is also efficient for the investigation of extremal problems on fairly broad classes of signed measures. In particular, for an arbitrary consistent kernel in a locally compact space, we prove a theorem on the strong completeness of fairly general subspaces E of all measures with finite energy. (Note that, according to the well-known Cartan counterexample, the entire space E is strongly incomplete even in the classical case of the Newton kernel in ℝn Using this theorem, we obtain new results for the Gauss variational problem, namely, in the non-compact case, we give a description of vague and (or) strong limiting measures of minimizing sequences and obtain sufficient solvability conditions. Концепція узгоджених ядер, уведена Фугледе у I960 p., отримала широке застосування в екстремальних задачах теорії потенціалу на класах додатних мір. У роботі показано ефективність цієї концепції у дослідженні екстремальних задач на досить загальних класах знакозмігших мір. Так, для довільного узгодженого ядра у локально компактному просторі доведено теорему про сильну повноту вельми загальних підпросторів простору $E$ всіх мір зі скінченною енергією. (Зазначимо, що відповідно до відомого коїпрприкладу Картана весь простір $E$ є сильно неповним навіть у класичному випадку ядра Ньютона в $ℝ^n$). З допомогою згаданої теореми отримано нові результат у дослідженні варіаційної задачі Гаусса: у некомпактному випадку наведено опис широких та (або) сильних граничних мір мінімізуючих послідовностей, знайдено достатні умови розв'язності. Institute of Mathematics, NAS of Ukraine 2004-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3862 Ukrains’kyi Matematychnyi Zhurnal; Vol. 56 No. 11 (2004); 1513-1526 Український математичний журнал; Том 56 № 11 (2004); 1513-1526 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3862/4443 https://umj.imath.kiev.ua/index.php/umj/article/view/3862/4444 Copyright (c) 2004 Zorii N. V. |
| spellingShingle | Zorii, N. V. Зорий, Н. В. Зорий, Н. В. Theory of Potential with Respect to Consistent Kernels; Theorem on Completeness and Sequences of Potentials |
| title | Theory of Potential with Respect to Consistent Kernels; Theorem on Completeness and Sequences of Potentials |
| title_alt | Теория потенциала относительно согласованных ядер: теорема о полноте, последовательности потенциалов |
| title_full | Theory of Potential with Respect to Consistent Kernels; Theorem on Completeness and Sequences of Potentials |
| title_fullStr | Theory of Potential with Respect to Consistent Kernels; Theorem on Completeness and Sequences of Potentials |
| title_full_unstemmed | Theory of Potential with Respect to Consistent Kernels; Theorem on Completeness and Sequences of Potentials |
| title_short | Theory of Potential with Respect to Consistent Kernels; Theorem on Completeness and Sequences of Potentials |
| title_sort | theory of potential with respect to consistent kernels; theorem on completeness and sequences of potentials |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3862 |
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