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

TL;DR

This paper extends density estimates for minimizers of Ginzburg-Landau energies with degenerate double-well potentials, covering a broader range of degeneracy and generalizing classical results.

Contribution

It generalizes density estimates for minimizers of Ginzburg-Landau energies with degenerate potentials to a wider range of parameters, extending prior bounded results.

Findings

Established density estimates for nontrivial minimizers with degenerate potentials.

Extended previous results to a broader range of degeneracy parameter m.

Generalized classical estimates to more complex energy functionals.

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.