Extension of Talenti's Inequality and Maximum Values Relative to Rearrangement Classes

The article starts by revisiting and extending the Talenti's inequality where the sharpness of the extended inequality is also addressed. The process leading to the extension comprises two steps. First, an observation that the Talenti's inequality indeed can be formulated in terms of a rearrangement class. Second, proving that the inequality holds even when the rearrangement class is replaced by a much bigger (modulo trivial cases) set namely an appropriate closure of the class. The article then continues to introduce and explore a related maximization problem, associated to the classical Poisson equation, where the admissible set is the class of rearrangements of a given function. The article briefly explains the physical interest in this optimization problem. The existence of optimal solutions is proved and the optimality conditions they satisfy are explicitly derived. The particular case where the rearrangement class is built out of a characteristic function is also discussed.