Compactness via symmetrization

Consider two types of translation-invariant functionals I and J on Rm, and a sequence of functions fn whose corresponding symmetric rearrangements f∗n are convergent. We show that fn themselves converge up to translations if either limn→∞I(fn)=limn→∞I(fn∗) or limn→∞J(fn)=limn→∞J(fn∗). These compactness results lead to applications in variational problems and stability problems in stellar dynamics.