A further remark on the density estimate for degenerate Allen-Cahn equations: $\Delta_{p}$-type equations for $1<p<\frac{n}{n-1}$ with rough coefficients

TL;DR

This paper extends the analysis of degenerate Allen-Cahn equations with rough coefficients, establishing density estimates for minimizers under minimal regularity assumptions and specific parameter ranges.

Contribution

It removes all regularity assumptions on the Ginzburg-Landau energy and proves density estimates for minimizers with $1<p<\frac{n}{n-1}$ and monotone potential.

Findings

Density estimates for level sets of minimizers.

Extension to rough coefficients in Allen-Cahn equations.

Applicable for $1<p<\frac{n}{n-1}$ with monotone potential.

Abstract

In this short remark on a previous paper \cite{SZ25}, we continue the study of Allen-Cahn equations associated with Ginzburg-Landau energies \begin{equation*} J(v,\Omega)=\int_{\Omega}\Big\{F(\nabla v,v,x)+W(v,x)\Big\}dx, \end{equation*} involving a Dirichlet energy $F(\vec{\xi},\tau,x)\sim|\vec{\xi}|^{p}$ and a degenerate double-well potential $W(\tau,x)\sim(1-\tau^{2})^{m}$. In contrast to \cite{SZ25}, we remove all regularity assumptions on the Ginzburg-Landau energy. Then, with further assumptions that $1<p<\frac{n}{n-1}$ and that $W(\tau,x)$ is monotone in $\tau$ on both sides of $0$, we establish a density estimate for the level sets of nontrivial minimizers $|u|\leq1$.