Maximum-entropy regularization functional ($E(x)={\int }_{\Omega }(x\log x-x)$)

This convex function penalizes positive densities using negative entropy, with $0\log 0=0$ and infinite value for negative inputs. Its scalar derivative is $\log x$ on $x>0$. On a bounded domain, restricting to uniformly positive bounded functions gives a simple setting for a Frechet derivative. Its Bregman distance is the generalized Kullback–Leibler divergence. With fixed total mass, minimizing this negative-entropy expression maximizes the corresponding entropy.