Compactness via symmetrization

摘要

Consider two types of translation-invariant functionals I and J on R-m, and a sequence of functions f(n), whose corresponding symmetric rearrangements f(n)* are convergent. We show that f(n) themselves converge up to translations if either lim(n-->proportional to) I (f(n))= lim(n-->infinity)(f(n)*) or lim(n-->infinity) J (f(n)) = lim(n-->infinity)J(f(n)*). These compactness results lead to applications in variational problems and stability problems in stellar dynamics. (C) 2004 Elsevier Inc. All rights reserved.