Effective energy integral functionals for thin films on curl-free vector fields in the Orlicz–Sobolev space setting

Volume 119 / 2019

Włodzimierz Laskowski, Hong Thai Nguyen Banach Center Publications 119 (2019), 259-277 MSC: 49J45, 74B20, 74K35, 74K15, 46E30, 46E35, 47H30. DOI: 10.4064/bc119-15


We consider an elastic thin film $\omega\subset \mathbb{R}^2$ with three dimensional bending moment. The effective energy functional defined on the Orlicz–Sobolev space over $\omega$ is obtained by $\Gamma$-convergence and $3D$-$2D$ dimension reduction techniques in the case when the energy density function is cross-quasiconvex. In the case when the energy density function is not cross-quasiconvex we obtained both upper and lower bounds for the $\Gamma$-limit. These results are proved in the case when the energy density function has the growth prescribed by an Orlicz convex function $M$. Here $M, M^*$ are assumed to be non-power-growth-type and to satisfy the condition $\Delta_{2}^{\text{glob}}$ (that imply the reflexivity of Orlicz and Orlicz–Sobolev spaces generated by $M$), and $M^*$ denotes the complementary (conjugate) Orlicz $N$-function of $M$.


  • Włodzimierz LaskowskiSchool of Mathematics
    West Pomeranian University of Technology
    Al. Piastów 48
    70-311 Szczecin, Poland
  • Hong Thai NguyenInstitute of Mathematics
    Center for N-Physics and N-quantized Mathematics
    Szczecin University
    ul. Wielkopolska 15
    70-451 Szczecin, Poland

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