Legendre-type integrands and convex integral functions

In this paper, we study the properties of integral functionals induced on $L^1_E (S,μ)$ 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.

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