Explicit minimisers for anisotropic Riesz energies
-
1.
Ludwig-Maximilians-Universität München
-
2.
Munich Center for Quantum Science and Technology
-
3.
California Institute of Technology
-
4.
Autonomous University of Barcelona
-
5.
Centre de Recerca Matemàtica
-
6.
University of Pavia
-
7.
Heriot-Watt University
Abstract
In this paper we characterise the energy minimisers of a class of nonlocal interaction energies where the attraction is quadratic, and the repulsion is Riesz-like and anisotropic. In particular we show that, if the Fourier transform of the repulsive potential is positive, the minimiser is supported on a fully-dimensional ellipsoid, and its density is given by a Barenblatt-type profile. Our technique of proof is based on a Fourier representation of the potential of such measures, that extends a previous formula established by some of the authors in the Coulomb case.
Copyright and License
© The Author(s) 2025. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Acknowledgement
RLF was partially supported through US National Science Foundation grant DMS-1954995, as well as through the German Research Foundation through EXC-2111-390814868 and TRR 352-Project-ID 470903074. JM and JV have been partially supported by 2021SGR-00071 (Catalonia) and PID2024-155320NB-I00 (Mineco, Spain). MGM is a member of GNAMPA–INdAM. MGM acknowledges support from PRIN 2022 (Project no. 2022J4FYNJ), funded by MUR, Italy, and the European Union – Next Generation EU, Mission 4 Component 1 CUP F53D23002760006. LR is supported by the Italian MUR through the PRIN 2022 project n.2022B32J5C, under the National Recovery and Resilience Plan (PNRR), Italy, funded by the European Union - Next Generation EU, Mission 4 Component 1 CUP F53D23002710006, and by GNAMPA-INdAM through 2025 projects. LS acknowledges support by the EPSRC under the grants EP/V00204X/1 and EP/V008897/1. Part of this work was done during a visit of JM, MGM, LR, and JV to Heriot-Watt University, whose kind hospitality is gratefully acknowledged.
Funding
Open access funding provided by Università degli Studi di Pavia within the CRUI-CARE Agreement.
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Files
s00526-025-03185-1.pdf
Files
(489.5 kB)
| Name | Size | Download all |
|---|---|---|
|
md5:4feb78082aefbf895132420d42b1baeb
|
489.5 kB | Preview Download |
Additional details
Related works
- Describes
- Journal Article: https://rdcu.be/eVVTf (ReadCube)
- Is new version of
- Discussion Paper: arXiv:2504.11644 (arXiv)
Funding
- National Science Foundation
- DMS-1954995
- Deutsche Forschungsgemeinschaft
- EXC-2111-390814868
- Deutsche Forschungsgemeinschaft
- 470903074
- Generalitat de Catalunya
- 2021SGR-00071
- Ministerio de Ciencia, Innovación y Universidades
- PID2024-155320NB-I00
- Ministero dell'Università e della Ricerca
- PRIN 2022J4FYNJ
- Ministero dell'Università e della Ricerca
- PRIN 2022B32J5C
- Engineering and Physical Sciences Research Council
- EP/V00204X/1
- Engineering and Physical Sciences Research Council
- EP/V008897/1
- Università degli Studi di Pavia
Dates
- Submitted
-
2025-04-16
- Accepted
-
2025-10-21
- Available
-
2025-12-13Published