Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2408.16141 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910581296988160 |
|---|---|
| author | Fang, Niufa Ye, Deping Zhang, Zengle |
| author_facet | Fang, Niufa Ye, Deping Zhang, Zengle |
| contents | We calculate the first order variation of the Riesz $α$-energy of a log-concave function $f$ with respect to the Asplund sum. Such a variational formula induces the Riesz $α$-energy measure of log-concave function $f$, which will be denoted by $\mathfrak{R}_α(f, \cdot)$. We pose the related Riesz $α$-energy Minkowski problem aiming to find necessary and/or sufficient conditions on a pregiven Borel measure $μ$ defined on $\Rn$ so that $μ=\mathfrak{R}_α(f,\cdot)$ for some log-concave function $f$. Assuming enough smoothness, the Riesz $α$-energy Minkowski problem reduces to a new Monge-Ampère type equation involving the Riesz $α$-potential. Moreover, this new Minkowski problem can be viewed as a functional counterpart of the recent Minkowski problem for the chord measures in integral geometry posed by Lutwak, Xi, Yang and Zhang (Comm.\ Pure\ Appl.\ Math.,\ 2024). The Riesz $α$-energy Minkowski problem will be solved under certain mild conditions on $μ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16141 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Riesz $α$-energy of log-concave functions and related Minkowski problem Fang, Niufa Ye, Deping Zhang, Zengle Functional Analysis Analysis of PDEs Metric Geometry 26B25, 52A40, 52A41, 35G20, 31B99 We calculate the first order variation of the Riesz $α$-energy of a log-concave function $f$ with respect to the Asplund sum. Such a variational formula induces the Riesz $α$-energy measure of log-concave function $f$, which will be denoted by $\mathfrak{R}_α(f, \cdot)$. We pose the related Riesz $α$-energy Minkowski problem aiming to find necessary and/or sufficient conditions on a pregiven Borel measure $μ$ defined on $\Rn$ so that $μ=\mathfrak{R}_α(f,\cdot)$ for some log-concave function $f$. Assuming enough smoothness, the Riesz $α$-energy Minkowski problem reduces to a new Monge-Ampère type equation involving the Riesz $α$-potential. Moreover, this new Minkowski problem can be viewed as a functional counterpart of the recent Minkowski problem for the chord measures in integral geometry posed by Lutwak, Xi, Yang and Zhang (Comm.\ Pure\ Appl.\ Math.,\ 2024). The Riesz $α$-energy Minkowski problem will be solved under certain mild conditions on $μ$. |
| title | The Riesz $α$-energy of log-concave functions and related Minkowski problem |
| topic | Functional Analysis Analysis of PDEs Metric Geometry 26B25, 52A40, 52A41, 35G20, 31B99 |
| url | https://arxiv.org/abs/2408.16141 |