On the upper semicontinuity of a quasiconcave functional

In the recent paper [21], the second author proved a divergence-quasiconcavity inequality for the following functional D(A)=∫Tndet⁡(A(x))1n−1dx defined on the space of positive definite matrices in Lp(Tn,Sym+(n)) with zero divergence. We consider the space Xp of tensor-fields in Lp(Tn,Sym+(n)) whose divergence is a Radon measure. We endow Xp with the weak topology given by the weak convergence in Lp and the weak-⁎ convergence of the measures representing the divergence of the tensor-fields. Our main result proves the weak upper semicontinuity of the functional D(⋅) on Xp if and only if p>nn−1. We also consider the case p≤nn−1 and show that D(⋅) is upper semicontinuous along sequences satisfying additional conditions. We use the positive result to show some properties of multi-dimensional Burgers equation.