@article{Zhang2025, 
author = {Chilin Zhang},
title = {A further remark on the density estimate for degenerate Allen-Cahn equations:        Δ          p      -type equations for    1  <  p  <      n          n      −      1       with rough coefficients},
year = {2025},
journal = {Mathematics in Engineering},
volume = {7},
number = {5},
pages = {668-677},
keywords = {Ginzburg-Landau theory, phase transitions, degenerate Allen-Cahn equations, density estimate, weak Harnack principle},
url = {https://www.sciopen.com/article/10.3934/mine.2025027},
doi = {10.3934/mine.2025027},
abstract = {In this short remark on a previous paper [1], we continue the study of Allen-Cahn equations associated with Ginzburg-Landau energies     J  (  v  ,  Ω  )  =      ∫          Ω            {    F  (  ∇  v  ,  v  ,  x  )  +  W  (  v  ,  x  )      }    d  x  ,involving a Dirichlet energy    F  (            ξ      →        ,  τ  ,  x  )  ∼      |              ξ      →                  |              p       and a degenerate double-well potential    W  (  τ  ,  x  )  ∼  (  1  −      τ          2            )          m      . In contrast to [1], we remove all regularity assumptions on the Ginzburg-Landau energy. Then, with further assumptions that    1  <  p  <      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      |    ≤  1.}
}