A further remark on the density estimate for degenerate Allen-Cahn equations: $Δ_{p}$-type equations for $1<p<\frac{n}{n-1}$ with rough coefficients
Chilin Zhang
TL;DR
This work extends the density-estimate theory for degenerate Allen-Cahn equations to Ginzburg-Landau energies with rough coefficients, proving that minimizers $u:\mathbb{R}^{n}\to[-1,1]$ satisfy a uniform density bound $|B_R\cap\{u\ge0\}|\ge\delta R^{n}$ and $|B_R\cap\{u\le0\}|\ge\delta R^{n}$ for all large $R$, under $1<p<\frac{n}{n-1}$, $m>p$, and monotonicity of $W$ in $\tau$. The approach avoids regularity assumptions on $F$ and $W$ by leveraging energy estimates, translation to a near-one point, and a weak-Harnack–type iterative scheme via carefully constructed comparison functions. This solidifies the link between degenerate phase-field energies and minimal-surface-type limits, generalizing prior results (e.g., SZ25, DFV18) to rough energies and broad coefficient ranges. The results have implications for understanding phase transitions and the geometry of level sets in degenerate variational problems.
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,Ω)=\int_Ω\Big\{F(\nabla v,v,x)+W(v,x)\Big\}dx, \end{equation*} involving a Dirichlet energy $F(\vecξ,τ,x)\sim|\vecξ|^{p}$ and a degenerate double-well potential $W(τ,x)\sim(1-τ^{2})^{m}$. 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 $W(τ,x)$ is monotone in $τ$ on both sides of $0$, we establish a density estimate for the level sets of nontrivial minimizers $|u|\leq1$.