A class of total variation minimization problems on the whole space

Motivated by the sharp L1L1 Gagliardo–Nirenberg inequality, we prove by elementary arguments that given two increasing functions FF and GG, solving the variational problem inf{E±(u)=∫Rnd|∇u|±∫RnF(|u|):∫RnG(|u|)=1} amounts to solving a one-dimensional optimization problem. Under appropriate conditions on the nonlinearities FF and GG, the infimum is attained and the minimizers are multiples of characteristic functions of balls. Several variants and applications are discussed, among which are some sharp inequalities and nonexistence and existence results to some PDEs involving the 1-Laplacian.