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Relaxation of functionals in the space of vector-valued functions of bounded Hessian

Calculus of Variations and Partial Differential Equations, ISSN: 0944-2669, Vol: 58, Issue: 1
2019
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  • Citations
    2
    • Citation Indexes
      2
  • Captures
    3
  • Social Media
    82
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      82
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        82

Article Description

In this paper it is shown that if Ω⊂ R is an open, bounded Lipschitz set, and if f: Ω× R→ [0 , ∞) is a continuous function with f(x, ·) of linear growth for all x∈ Ω, then the relaxed functional in the space of functions of Bounded Hessian of the energy F[u]=∫Ωf(x,∇2u(x))dxfor bounded sequences in W is given by F[u]=∫ΩQ2f(x,∇2u)dx+∫Ω(Q2f)∞(x,dDs(∇u)d|Ds(∇u)|)d|Ds(∇u)|.This result is obtained using blow-up techniques and establishes a second order version of the BV relaxation theorems of Ambrosio and Dal Maso (J Funct Anal 109:76–97, 1992) and Fonseca and Müller (Arch Ration Mech Anal 123:1–49, 1993). The use of the blow-up method is intended to facilitate future study of integrands which include lower order terms and applications in the field of second order structured deformations.

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