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.
Categories: math.AP
