SBH-ellipticity of the relaxed interfacial energy density in the context of second-order structured deformations

Starting from an energy comprised of both a bulk term and a surface term, set in the space of special functions of bounded hessian, $SBH$, a relaxation problem in the context of second-order structured deformations was studied in Fonseca-Hagerty-Paroni. It was shown, via the global method for relaxation, that the relaxed functional admits an integral representation and the relaxed energy densities were identified. In this paper we show that, under certain hypotheses on the original densities, the corresponding relaxed energy densities verify the same type of growth conditions and the surface energy density satisfies a specific ``convexity-type'' property, i.e. it is $SBH$-elliptic.