Legendre-Type Integrands and Convex Integral Functions

We study the properties of integral functionals induced on $L^1_E(S,\mu)$ by closed convex functions on a Euclidean space E. We give sufficient conditions for such integral functions to be strongly rotund (well-posed). We show that in this generality functions such as the Boltzmann-Shannon entropy and the Fermi-Dirac entropy are strongly rotund. We also study convergence in measure and give various limiting counterexample.