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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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.
Bibliographic Details
Springer Science and Business Media LLC
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