The equilibrium measure for an anisotropic nonlocal energy
In this paper we characterise the minimisers of a one-parameter family of nonlocal and anisotropic energies 𝐼𝛼 defined on probability measures in ℝ𝑛, with 𝑛≥3. The energy 𝐼𝛼 consists of a purely nonlocal term of convolution type, whose interaction kernel reduces to the Coulomb potential for 𝛼=0 and...
| Main Authors: | , , , , , |
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| Format: | Journal article |
| Language: | English |
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Springer
2021
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| _version_ | 1797105967466283008 |
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| author | Carrillo, JA Mateu, J Mora, MG Rondi, L Scardia, L Verdera, J |
| author_facet | Carrillo, JA Mateu, J Mora, MG Rondi, L Scardia, L Verdera, J |
| author_sort | Carrillo, JA |
| collection | OXFORD |
| description | In this paper we characterise the minimisers of a one-parameter family of nonlocal and anisotropic energies 𝐼𝛼 defined on probability measures in ℝ𝑛, with 𝑛≥3. The energy 𝐼𝛼 consists of a purely nonlocal term of convolution type, whose interaction kernel reduces to the Coulomb potential for 𝛼=0 and is anisotropic otherwise, and a quadratic confinement. The two-dimensional case arises in the study of defects in metals and has been solved by the authors by means of complex-analysis techniques. We prove that for 𝛼∈(−1,𝑛−2], the minimiser of 𝐼𝛼 is unique and is the (normalised) characteristic function of a spheroid. This result is a paradigmatic example of the role of the anisotropy of the kernel on the shape of minimisers. In particular, the phenomenon of loss of dimensionality, observed in dimension 𝑛=2, does not occur in higher dimension at the value 𝛼=𝑛−2 corresponding to the sign change of the Fourier transform of the interaction potential. |
| first_indexed | 2024-03-07T06:54:57Z |
| format | Journal article |
| id | oxford-uuid:fdd1dcf0-42de-49b2-9af6-8bb394f6d0e5 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T06:54:57Z |
| publishDate | 2021 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:fdd1dcf0-42de-49b2-9af6-8bb394f6d0e52022-03-27T13:31:43ZThe equilibrium measure for an anisotropic nonlocal energyJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:fdd1dcf0-42de-49b2-9af6-8bb394f6d0e5EnglishSymplectic ElementsSpringer2021Carrillo, JAMateu, JMora, MGRondi, LScardia, LVerdera, JIn this paper we characterise the minimisers of a one-parameter family of nonlocal and anisotropic energies 𝐼𝛼 defined on probability measures in ℝ𝑛, with 𝑛≥3. The energy 𝐼𝛼 consists of a purely nonlocal term of convolution type, whose interaction kernel reduces to the Coulomb potential for 𝛼=0 and is anisotropic otherwise, and a quadratic confinement. The two-dimensional case arises in the study of defects in metals and has been solved by the authors by means of complex-analysis techniques. We prove that for 𝛼∈(−1,𝑛−2], the minimiser of 𝐼𝛼 is unique and is the (normalised) characteristic function of a spheroid. This result is a paradigmatic example of the role of the anisotropy of the kernel on the shape of minimisers. In particular, the phenomenon of loss of dimensionality, observed in dimension 𝑛=2, does not occur in higher dimension at the value 𝛼=𝑛−2 corresponding to the sign change of the Fourier transform of the interaction potential. |
| spellingShingle | Carrillo, JA Mateu, J Mora, MG Rondi, L Scardia, L Verdera, J The equilibrium measure for an anisotropic nonlocal energy |
| title | The equilibrium measure for an anisotropic nonlocal energy |
| title_full | The equilibrium measure for an anisotropic nonlocal energy |
| title_fullStr | The equilibrium measure for an anisotropic nonlocal energy |
| title_full_unstemmed | The equilibrium measure for an anisotropic nonlocal energy |
| title_short | The equilibrium measure for an anisotropic nonlocal energy |
| title_sort | equilibrium measure for an anisotropic nonlocal energy |
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