On starshaped rearrangement and applications

Abstract

A radial symmetrization technique is investigated and new properties are proven. The method transforms functions u u into new functions u ∗ {u^\ast } with starshaped level sets and is therefore called starshaped rearrangement. This rearrangement is in general not equimeasurable, a circumstance with some surprising consequences. The method is then applied to certain variational and free boundary problems and yields new results on the geometrical properties of solutions to these problems. In particular, the Lipschitz continuity of free boundaries can now be easily obtained in a new fashion.