Optimizers in Sobolev-curl inequalities
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917056417366016 |
|---|---|
| author | Mederski, Jarosław Szulkin, Andrzej |
| author_facet | Mederski, Jarosław Szulkin, Andrzej |
| contents | We study a Sobolev-type inequality involving the $p$-curl operator in $\mathbb{R}^3$. We prove the existence of a minimizer which yields a solution to the $p$-curl-curl equation in the critical case. The problem is motivated both by nonlinear Maxwell equations and by the occurrence of zero modes in three-dimensional Dirac equations. Moreover, we introduce a new variational approach that allows to treat quasilinear strongly indefinite problems by direct minimization on a Nehari-type constraint. We also consider existence of minimizers under some symmetry assumptions. Finally, our approach offers a new proof of the compactness of minimizing sequences for the Sobolev inequalities in the critical case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_01432 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimizers in Sobolev-curl inequalities Mederski, Jarosław Szulkin, Andrzej Analysis of PDEs Primary: 35Q60, Secondary: 35Q61, 35J20 We study a Sobolev-type inequality involving the $p$-curl operator in $\mathbb{R}^3$. We prove the existence of a minimizer which yields a solution to the $p$-curl-curl equation in the critical case. The problem is motivated both by nonlinear Maxwell equations and by the occurrence of zero modes in three-dimensional Dirac equations. Moreover, we introduce a new variational approach that allows to treat quasilinear strongly indefinite problems by direct minimization on a Nehari-type constraint. We also consider existence of minimizers under some symmetry assumptions. Finally, our approach offers a new proof of the compactness of minimizing sequences for the Sobolev inequalities in the critical case. |
| title | Optimizers in Sobolev-curl inequalities |
| topic | Analysis of PDEs Primary: 35Q60, Secondary: 35Q61, 35J20 |
| url | https://arxiv.org/abs/2511.01432 |