Upper Semicontinuity for Divergence-Free Positive Tensors

Recently, D. Serre proposed the framework of divergence-free positive definite symmetric tensors (DPTs) to obtain better a priori estimates for various compressible fluid models. He proved, in particular, a Jensen-type inequality for the determinant to the power 1/(d-1), which suggests that a weak semicontinuity result could be obtained by virtue of by now standard methods of Fonseca-Müller. However, the positivity constraint excludes these methods and requires a novel approach. We elaborate such an approach within a very general framework. Joint work with Jack Skipper.