@article{Carozza2023, 
author = {Menita Carozza and Luca Esposito and Raffaella Giova and Francesco Leonetti},
title = {Polyconvex functionals and maximum principle},
year = {2023},
journal = {Mathematics in Engineering},
volume = {5},
number = {4},
pages = {1-10},
keywords = {polyconvex functionals, minimizers, regularity},
url = {https://www.sciopen.com/article/10.3934/mine.2023077},
doi = {10.3934/mine.2023077},
abstract = {Let us consider continuous minimizers    u  :            Ω      ¯        ⊂            R        n    →            R        n   of         F    (  v  )  =      ∫          Ω        [      |    D  v            |        p      +        |        d    e    t      D  v            |        r    ]  d  x  ,with    p  >  1 and    r  >  0; then it is known that every component        u    α   of    u  =  (      u    1    ,  .  .  .  ,      u    n    ) enjoys maximum principle: the set of interior points    x, for which the value        u    α    (  x  ) is greater than the supremum on the boundary, has null measure, that is,              L        n    (  {  x  ∈  Ω  :      u    α    (  x  )  >      sup          ∂      Ω            u    α    }  )  =  0. If we change the structure of the functional, it might happen that the maximum principle fails, as in the case         F    (  v  )  =      ∫          Ω        [  max  {  (      |    D  v            |        p    −  1  )  ;  0  }    +        |        d    e    t      D  v            |        r    ]  d  x  ,with    p  >  1 and    r  >  0. Indeed, for a suitable boundary value, the set of the interior points    x, for which the value        u    α    (  x  ) is greater than the supremum on the boundary, has a positive measure, that is              L        n    (  {  x  ∈  Ω  :      u    α    (  x  )  >      sup          ∂      Ω            u    α    }  )  >  0. In this paper we show that the measure of the image of these bad points is zero, that is              L        n    (  u  (  {  x  ∈  Ω  :      u    α    (  x  )  >      sup          ∂      Ω            u    α    }  )  )  =  0, provided    p  >  n. This is a particular case of a more general theorem.}
}