$p$-Energy forms on fractals: recent progress
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866929677015187456 |
|---|---|
| author | Kajino, Naotaka Shimizu, Ryosuke |
| author_facet | Kajino, Naotaka Shimizu, Ryosuke |
| contents | In this article, we survey recent progress on self-similar $p$-energy forms on self-similar fractals, where $p\in(1,\infty)$. While for $p=2$ the notion of such forms coincides with that of self-similar Dirichlet forms and there have been plenty of studies on them since the late 1980s, studies on the case of $p\in(1,\infty)\setminus\{2\}$ was initiated much later in 2004 by Herman, Peirone and Strichartz [Potential Anal. 20 (2004), 125--148] and Strichartz and Wong [Nonlinearity 17 (2004), 595--616] and no essential progress on this case had been made since then until a few years ago. The recent progress by Kigami, Shimizu, Cao--Gu--Qiu and Murugan--Shimizu has established the existence of such $p$-energy forms on general post-critically finite (p.-c.f.) self-similar sets and on large classes of low-dimensional infinitely ramified self-similar sets, and the authors have proved further detailed properties of these forms and associated $p$-harmonic functions, mainly for p.-c.f. self-similar sets. This article is devoted to a review of these results, focusing on the most recent developments by the authors and illustrating them in the simplest non-trivial setting of the two-dimensional standard Sierpiński gasket. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_09002 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $p$-Energy forms on fractals: recent progress Kajino, Naotaka Shimizu, Ryosuke Functional Analysis Analysis of PDEs Probability 28A80, 31C45, 31E05, 60G30 (Primary) 31C25, 46E36 (Secondary) In this article, we survey recent progress on self-similar $p$-energy forms on self-similar fractals, where $p\in(1,\infty)$. While for $p=2$ the notion of such forms coincides with that of self-similar Dirichlet forms and there have been plenty of studies on them since the late 1980s, studies on the case of $p\in(1,\infty)\setminus\{2\}$ was initiated much later in 2004 by Herman, Peirone and Strichartz [Potential Anal. 20 (2004), 125--148] and Strichartz and Wong [Nonlinearity 17 (2004), 595--616] and no essential progress on this case had been made since then until a few years ago. The recent progress by Kigami, Shimizu, Cao--Gu--Qiu and Murugan--Shimizu has established the existence of such $p$-energy forms on general post-critically finite (p.-c.f.) self-similar sets and on large classes of low-dimensional infinitely ramified self-similar sets, and the authors have proved further detailed properties of these forms and associated $p$-harmonic functions, mainly for p.-c.f. self-similar sets. This article is devoted to a review of these results, focusing on the most recent developments by the authors and illustrating them in the simplest non-trivial setting of the two-dimensional standard Sierpiński gasket. |
| title | $p$-Energy forms on fractals: recent progress |
| topic | Functional Analysis Analysis of PDEs Probability 28A80, 31C45, 31E05, 60G30 (Primary) 31C25, 46E36 (Secondary) |
| url | https://arxiv.org/abs/2501.09002 |