Kazdan–Warner equation on hypergraphs
UDC 519.17. 519.951 Let $H=(V, E)$ be a connected finite hypergraph, which is an extension of the graph theory in which the edges may connect more than two vertices and form hyperedges. We study the Kazdan-Warner equation\begin{gather*}\Delta \phi=c-he^{\phi}\end{gather*} on $H,$ where $c$ is a con...
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| Дата: | 2026 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
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Institute of Mathematics, NAS of Ukraine
2026
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/9163 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860513422937948160 |
|---|---|
| author | Zhang, Haigang Zhao, Juan Zhang, Haigang Zhao, Juan |
| author_facet | Zhang, Haigang Zhao, Juan Zhang, Haigang Zhao, Juan |
| author_sort | Zhang, Haigang |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-03-21T11:04:14Z |
| description | UDC 519.17. 519.951
Let $H=(V, E)$ be a connected finite hypergraph, which is an extension of the graph theory in which the edges may connect more than two vertices and form hyperedges. We study the Kazdan-Warner equation\begin{gather*}\Delta \phi=c-he^{\phi}\end{gather*} on $H,$ where $c$ is a constant and $h$ is a known function defined on $H$. Based on the work by Grigor'yan, Lin, and Yang [A. Grigor'yan, Y. Lin, Y. Yang, Kazdan–Warner equation on graph, Calc. Var. Partial Differential Equations, 55, № 4, Article 92 (2016)], we employ the variational calculus to extend the main results concerning the solutions to the Kazdan-Warner equation from finite graphs to hypergraphs. We obtain similar results for the cases where $c>0$ and $c<0$ provided that $h$ satisfies certain conditions on hypergraphs. However, for the case where $c=0,$ we cannot get the same results. |
| doi_str_mv | 10.3842/umzh.v78i1-2.9163 |
| first_indexed | 2026-03-24T03:44:26Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-9163 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:44:26Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-91632026-03-21T11:04:14Z Kazdan–Warner equation on hypergraphs Kazdan–Warner equation on hypergraphs Zhang, Haigang Zhao, Juan Zhang, Haigang Zhao, Juan Kazdan-Warner equation hypergraph calculus of variations Kazdan-Warner equation hypergraph calculus of variations UDC 519.17. 519.951 Let $H=(V, E)$ be a connected finite hypergraph, which is an extension of the graph theory in which the edges may connect more than two vertices and form hyperedges. We study the Kazdan-Warner equation\begin{gather*}\Delta \phi=c-he^{\phi}\end{gather*} on $H,$ where $c$ is a constant and $h$ is a known function defined on $H$. Based on the work by Grigor'yan, Lin, and Yang [A. Grigor'yan, Y. Lin, Y. Yang, Kazdan–Warner equation on graph, Calc. Var. Partial Differential Equations, 55, № 4, Article 92 (2016)], we employ the variational calculus to extend the main results concerning the solutions to the Kazdan-Warner equation from finite graphs to hypergraphs. We obtain similar results for the cases where $c>0$ and $c<0$ provided that $h$ satisfies certain conditions on hypergraphs. However, for the case where $c=0,$ we cannot get the same results. УДК 519.17. 519.951 Рівняння Каздана–Уорнера на гіперграфах Нехай $H=(V,E)$ є скінченним зв’язним гіперграфом – узагальненням графа, у якому ребра можуть з’єднувати більше ніж дві вершини, утворюючи гіперребра. Досліджено рівняння Каздана–Уорнера\[\Delta \phi=c-he^{\phi}\]на $H,$ де $c$ – стала, $h$ – відома функція, визначена на $H.$ На основі результатів роботи [A. Grigor'yan, Y. Lin, Y. Yang, Kazdan–Warner equation on graph, Calc. Var. Partial Differential Equations, 55, № 4, Article 92 (2016)] із застосуванням методів варіаційного числення узагальнено основні результати для рівняння Каздана-,Уорнера зі скінченних графів на гіперграфи. Одержано подібні результати для випадків $c>0$ і $c<0$ за умови, що $h$ задовольняє певні умови на гіперграфах. Проте для випадку $c=0$ аналогічні результати одержати не вдалося. Institute of Mathematics, NAS of Ukraine 2026-03-02 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/9163 10.3842/umzh.v78i1-2.9163 Ukrains’kyi Matematychnyi Zhurnal; Vol. 78 No. 1-2 (2026); 91 Український математичний журнал; Том 78 № 1-2 (2026); 91 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/9163/10628 Copyright (c) 2026 Haigang Zhang, Juan Zhao |
| spellingShingle | Zhang, Haigang Zhao, Juan Zhang, Haigang Zhao, Juan Kazdan–Warner equation on hypergraphs |
| title | Kazdan–Warner equation on hypergraphs |
| title_alt | Kazdan–Warner equation on hypergraphs |
| title_full | Kazdan–Warner equation on hypergraphs |
| title_fullStr | Kazdan–Warner equation on hypergraphs |
| title_full_unstemmed | Kazdan–Warner equation on hypergraphs |
| title_short | Kazdan–Warner equation on hypergraphs |
| title_sort | kazdan–warner equation on hypergraphs |
| topic_facet | Kazdan-Warner equation hypergraph calculus of variations Kazdan-Warner equation hypergraph calculus of variations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/9163 |
| work_keys_str_mv | AT zhanghaigang kazdanwarnerequationonhypergraphs AT zhaojuan kazdanwarnerequationonhypergraphs AT zhanghaigang kazdanwarnerequationonhypergraphs AT zhaojuan kazdanwarnerequationonhypergraphs |