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

Chilin Zhang; Chilin Zhang

doi:10.3934/mine.2025027

Mathematics in Engineering

2025, Volume 7, Issue 5: 668-677. doi: 10.3934/mine.2025027

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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

Chilin Zhang ^,

School of Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai 200433, China

Received: 25 July 2025 Revised: 08 October 2025 Accepted: 15 October 2025 Published: 20 October 2025

In this short remark on a previous paper ^[1], 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 ^[1], 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 $.
- Ginzburg-Landau theory,
- phase transitions,
- degenerate Allen-Cahn equations,
- density estimate,
- weak Harnack principle
Citation: Chilin Zhang. 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[J]. Mathematics in Engineering, 2025, 7(5): 668-677. doi: 10.3934/mine.2025027

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Abstract

In this short remark on a previous paper ^[1], 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 ^[1], 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 $.

References

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