2018/02/08 by Hagerty, Adrian · 1 citation
#49J45 #49Q20 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1802.02994
In this paper it is shown that if Ω⊂ ℝN is an open, bounded Lipschitz set, and if f: Ω× ℝd × N × N → [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] = ∫Ω f(x, ∇2u(x)) dx for bounded sequences in W2,1 is given by \cal F[u] = ∫Ω\cal Q2f(x, ∇2u) dx + ∫Ω(\cal Q2f)∞(x, (d Ds(∇ 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 and Fonseca and Müller. 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.