About the Young measures associated with Y. Brenier's ABI model

The Born–Infeld system is a nonlinear model of electromagnetism, intended to serve in high-energy physics. It has been embeded by Y. Brenier into a quasilinear system of conservation laws endowed with a convex entropy [3]. The latter system, denoted as ABI, governs the evolution of the electromagnetic field (B,D)(B,D), together with the energy density h and the Poynting vector P.The recent preprint [6] stresses that the convex hull in the (h,B,D,P)(h,B,D,P)-space R10R10 of the equilibrium manifold defined byP=D×B,h=1+|B|2+|D|2+|P|2 plays a key role in the approach by convex integration. In this article, S. Müller and M. Palombaro point out a mistake in the description of this hull given in our previous paper [8].We give below the correct description of the convex hull and show that it is not positively invariant under the evolution of ABI. Its points are mass centers of the Young measures permitted by the ABI system. We give additional restrictions on these measures.