Title: Compactness via symmetrization

Abstract: We determine a class of Carathéodory integrands G for which
the minimum of  non-compact variational problems is achieved at a Schwarz
symmetric function satisfying the constraint. Our hypotheses about G seem
natural and, as our examples show, they are optimal from some points of
view. We also discuss hypotheses on G under which all the minimizers are
Schwarz symmetric.