@article{De Silva2023, 
author = {Daniela De Silva and Ovidiu Savin},
title = {Uniform density estimates and    Γ-convergence for the Alt-Phillips functional of negative powers},
year = {2023},
journal = {Mathematics in Engineering},
volume = {5},
number = {5},
pages = {1-27},
keywords = {free boundary problems, uniform estimates, minimal surfaces, Dirichlet-Perimeter functional},
url = {https://www.sciopen.com/article/10.3934/mine.2023086},
doi = {10.3934/mine.2023086},
abstract = {We obtain density estimates for the free boundaries of minimizers    u  ≥  0 of the Alt-Phillips functional involving negative power potentials         ∫    Ω        (                  |            ∇      u                        |                2            +              u                  −          γ                            χ                  {          u          >          0          }                      )      d  x  ,      γ  ∈  (  0  ,  2  )  .These estimates remain uniform as the parameter    γ  →  2. As a consequence we establish the uniform convergence of the corresponding free boundaries to a minimal surface as    γ  →  2. The results are based on the    Γ-convergence of these energies (properly rescaled) to the Dirichlet-perimeter functional         ∫          Ω            |    ∇  u            |        2    d  x  +  P  e      r          Ω        (  {  u  =  0  }  )  ,considered by Athanasopoulous, Caffarelli, Kenig, and Salsa.}
}