@article{Hu2023, 
author = {Yuwei Hu and Jun Zheng},
title = {Local porosity of the free boundary in a minimum problem},
year = {2023},
journal = {Electronic Research Archive},
volume = {31},
number = {9},
pages = {5457-5465},
keywords = {free boundary, minimizer, porosity, non-degeneracy, p-Laplacian},
url = {https://www.sciopen.com/article/10.3934/era.2023277},
doi = {10.3934/era.2023277},
abstract = {Given certain set        K   and functions    q and    h, we study geometric properties of the set    ∂  {  x  ∈  Ω  :  u  (  x  )  >  0  } for non-negative minimizers of the functional        J    (  u  )  =      ∫          Ω                      (                  1        p                    |            ∇      u                        |                p            +      q      (              u        +                    )        γ            +      h      u        )    d  x over        K  , where        Ω    ⊂              R        n    (  n  ≥  2  ) is an open bounded domain,    p  ∈  (  1  ,  +  ∞  ) and    γ  ∈  (  0  ,  1  ] are constants,        u    +   is the positive part of    u and    ∂  {  x  ∈  Ω  :  u  (  x  )  >  0  } is the so-called free boundary. Such a minimum problem arises in physics and chemistry for    γ  =  1 and    γ  ∈  (  0  ,  1  ), respectively. Using the comparison principle of    p-Laplacian equations, we establish first the non-degeneracy of non-negative minimizers near the free boundary, then prove the local porosity of the free boundary.}
}