@article{Yin2022, 
author = {Ke Yin and Kewei Zhang},
title = {Some computable quasiconvex multiwell models in linear subspaces without rank-one matrices},
year = {2022},
journal = {Electronic Research Archive},
volume = {30},
number = {5},
pages = {1632-1652},
keywords = {multiwell models, vectorial calculus of variations, quasiconvex functions, quasiconvex envelope, quasiconvex lower bounds, computational lower boundes, translation method, maximum function, compensated convex transforms, C1, 1-smooth approximation},
url = {https://www.sciopen.com/article/10.3934/era.2022082},
doi = {10.3934/era.2022082},
abstract = {In this paper we apply a smoothing technique for the maximum function, based on the compensated convex transforms, originally proposed by Zhang in [1] to construct some computable multiwell non-negative quasiconvex functions in the calculus of variations. Let    K  ⊆  E  ⊆      M          m      ×      n       with    K a finite set in a linear subspace    E without rank-one matrices of the space        M          m      ×      n       of real    m  ×  n matrices. Our main aim is to construct computable quasiconvex lower bounds for the following two multiwell models with possibly uneven wells: i) Let    f  :  K  ⊆  E  →      E    ⊥   be an    L-Lipschitz mapping with    0  ≤  L  ≤  1      /    α and        H    2    (  X  )  =  min  {      |        P    E    X  −      A    i              |        2    +  α      |        P                  E        ⊥              X  −  f  (      A    i    )            |        2    +      β    i    :    i  =  1  ,  2  ,  …  ,  k  }, where    α  >  0 is a control parameter, and ii)        H    1    (  X  )  =  α      |        P                  E        ⊥              X            |        2    +  min  {            |                      U            i        (          P      E        X    −          A      i        )                  |            2        +          γ      i        :  i  =  1  ,  2  ,  …  ,  k  }, where        A    i    ∈  E with        U    i    :  E  →  E invertible linear transforms for    i  =  1  ,  2  ,  …  ,  k. If the control paramenter    α  >  0 is sufficiently large, our quasiconvex lower bounds are 'tight' in the sense that near each 'well' the lower bound agrees with the original function, and our lower bound are of        C          1      ,      1      . We also consider generalisations of our constructions to other simple geometrical multiwell models and discuss the implications of our constructions to the corresponding variational problems.}
}