Rearrangement optimization problems with free boundary

This article is concerned with three optimization problems. In the first problem, a functional is maximized with respect to a set that is the weak closure of a rearrangement class; that is, a set comprising rearrangements of a prescribed function. Questions regarding existence, uniqueness, symmetry, and local minimizers are addressed. The second problem is of maximization type related to a Poisson boundary value problem. After defining a relevant function, we prove it is differentiable and derive an explicit formula for its derivative. Further, using the co-area formula, we establish a free boundary result. The third problem is the minimization version of the second problem.