Motivated by the sharp L1 Gagliardo-Nirenberg inequality, we prove by elementary arguments that given two increasing functions F and
G, solving the variational problem
inf E±(u) = ZRn
d|ru| ± ZRn
F(|u|) : ZRn
G(|u|) = 1
amounts to solve a one-dimensional optimization problem. Under appropriate conditions on the nonlinearities F and G, the infimum is attained and the minimizers are multiple of characteristic functions of balls. Several variants and applications are discussed, among which some sharp inequalities and nonexistence and existence results to some PDEs involving the 1-Laplacian.
(This is a joint work with G. Carlier).