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arXiv · 2506.17000

Density estimates for Ginzburg-Landau energies with degenerate double-well potentials

Ovidiu Savin·Chilin Zhang
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Abstract

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,\Omega)=\int_{\Omega}\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-C\'ordoba.

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Ovidiu Savin, Chilin Zhang. 2025-06-20. Density estimates for Ginzburg-Landau energies with degenerate double-well potentials. https://doi.org/10.1137/25m1778468

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