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

In this paper it is shown that if Ω ⊂ R N is an open, bounded Lipschitz set, and if f : Ω × R d × 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 , ∇ 2 u ( x ) ) d x for bounded sequences in W 2 , 1 is given by F [ u ] = ∫ Ω Q 2 f ( x , ∇ 2 u ) d x + ∫ Ω ( Q 2 f ) ∞ ( x , d D s ( ∇ u ) d | D s ( ∇ u ) | ) d | D s ( ∇ 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.