Converse symmetry and intermediate energy values in rearrangement optimization problems

This paper discusses three rearrangement optimization problems where the energy functional is connected with the Dirichlet or Robin boundary value problems. First, we consider a simple model of Dirichlet type, derive a symmetry result, and prove an intermediate energy theorem. For this model, we show that if the optimal domain (or its complement) is a ball centered at the origin, then the original domain must be a ball. As for the intermediate energy theorem, we show that if α, β denote the optimal values of corresponding minimization and maximization problems, respectively, then every γ in (α, β) is achieved by solving a max-min problem. Second, we investigate a similar symmetry problem for the Dirichlet problems where the energy functional is nonlinear. Finally, we show the existence and uniqueness of rearrangement minimization problems associated with the Robin problems. In addition, we shall obtain a symmetry and a related asymptotic result.