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

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,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 $|v|\le 1$. 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.