Density estimates for Ginzburg-Landau energies with degenerate double-well potentials
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911015581515776 |
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| author | Savin, Ovidiu Zhang, Chilin |
| author_facet | Savin, Ovidiu Zhang, Chilin |
| contents | We consider a class of Allen-Cahn equations associated with Ginzburg-Landau energies involving degenerate double-well potentials that vanish of order $m$ at the minima \begin{equation}
J(v,Ω)=\int_Ω\Big\{|\nabla v|^{p}+(1-v^{2})^{m}\Big\}dx,\quad 1<p<m, \end{equation} and establish density estimates for the level sets of nontrivial minimizers $|v| \leq 1$. This extends a result of Dipierro-Farina-Valdinoci where the density estimates for such degenerate potentials were obtained for a bounded range of $m$'s. The original estimates for the classical case $p=m=2$ were established by Caffarelli-Córdoba. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_17000 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Density estimates for Ginzburg-Landau energies with degenerate double-well potentials Savin, Ovidiu Zhang, Chilin Analysis of PDEs We consider a class of Allen-Cahn equations associated with Ginzburg-Landau energies involving degenerate double-well potentials that vanish of order $m$ at the minima \begin{equation} J(v,Ω)=\int_Ω\Big\{|\nabla v|^{p}+(1-v^{2})^{m}\Big\}dx,\quad 1<p<m, \end{equation} and establish density estimates for the level sets of nontrivial minimizers $|v| \leq 1$. This extends a result of Dipierro-Farina-Valdinoci where the density estimates for such degenerate potentials were obtained for a bounded range of $m$'s. The original estimates for the classical case $p=m=2$ were established by Caffarelli-Córdoba. |
| title | Density estimates for Ginzburg-Landau energies with degenerate double-well potentials |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.17000 |