chapter  19
16 Pages

Symmetrization - or How to Prove Symmetry of Solutions to a PDE

ByB. Kawohl

Many partial differential equations are Euler equations of variational problems, and this circumstance can be used to derive symmetry or monotonicity properties of their solutions. I survey some recent results on symmetrization methods, i.e. on procedures in which a supposedly nonsymmetric minimizer is deformed into a symmetric one with lower energy. I also compare this variational approach with arguments involving the maximum principle. There are specific advantages and disadvantages of both methods. Another symmetry approach uses the unique continuation principle. Finally I address the recently emerging subject of isoparametric surfaces.