Continuous symmetrizations and uniqueness of solutions to nonlocal equations
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866918194540707840 |
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| author | Delgadino, Matias G. Vaughan, M. |
| author_facet | Delgadino, Matias G. Vaughan, M. |
| contents | We show that nonlocal seminorms are strictly decreasing under the continuous Steiner rearrangement. This implies that all solutions to nonlocal equations which arise as critical points of nonlocal energies are radially symmetric and decreasing. Moreover, we show uniqueness of solutions by exploiting the convexity of the energies under a tailored interpolation in the space of radially symmetric and decreasing functions. As an application, we consider the long time dynamics of a higher order nonlocal equation which models the growth of symmetric cracks in an elastic medium. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_13221 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Continuous symmetrizations and uniqueness of solutions to nonlocal equations Delgadino, Matias G. Vaughan, M. Analysis of PDEs 35R11, 35G20, 35C06 We show that nonlocal seminorms are strictly decreasing under the continuous Steiner rearrangement. This implies that all solutions to nonlocal equations which arise as critical points of nonlocal energies are radially symmetric and decreasing. Moreover, we show uniqueness of solutions by exploiting the convexity of the energies under a tailored interpolation in the space of radially symmetric and decreasing functions. As an application, we consider the long time dynamics of a higher order nonlocal equation which models the growth of symmetric cracks in an elastic medium. |
| title | Continuous symmetrizations and uniqueness of solutions to nonlocal equations |
| topic | Analysis of PDEs 35R11, 35G20, 35C06 |
| url | https://arxiv.org/abs/2310.13221 |