Symmetrizations are a tool used to prove that solutions of variational problems are symmetrical. To every function \(u\), a more symmetrical fonction \(u^*\) is associated. This nonlinear transformation preserves the measure of sublevel sets, so that many integral functionals are preserved or decrease when the function they contain is symmetrized.
I have worked on
- the relationships between the properties of the symmetrization of sets and the symmetrization of functions,
- the approximation of symmetrizations by simpler symmetrizations, by simpler symmetrizations, in particular by polarizations, and random approximation,
- the symmetry of critical points obtained by minimax methods (Mountain Pass Theorem, Linking Theorem, Krasnsoselskii genus),
- anisotropic symmetrizations, i.e. symmetrization with respect to a noneuclidean norm.
Justin Dekeyser and Jean Van Schaftingen, Approximation of symmetrizations by Markov processes, Indiana Univ. Math. J. 66 (2017), no. 4, 1145–1172.
doi:10.1512/iumj.2017.66.6118 DIAL:191989 arXiv:1508.00464
Jean Van Schaftingen, Equivalence between Pólya–Szegő and relative capacity inequalities under rearrangement, Arch. Math. (Basel) 103 (2014), no. 4, 367–379.
doi:10.1007/s00013-014-0695-4 SharedIt DIAL:152136 arXiv:1401.2780
Jean Van Schaftingen, Explicit approximation of the symmetric rearrangement by polarizations, Arch. Math. (Basel) 93 (2009), no. 2, 181–190.
doi:10.1007/s00013-009-0018-3 SharedIt DIAL:35391 arXiv:0902.0637
Jean Van Schaftingen, Approximation of symmetrizations and symmetry of critical points, Topol. Methods Nonlinear Anal. 28 (2006), no. 1, 61–85.
MR:2262256 DIAL:38258 preprint
Jean Van Schaftingen, Anisotropic symmetrization, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), no. 4, 539–565.
doi:10.1016/j.anihpc.2005.06.001 MR:2245755 DIAL:38319 preprint
Jean van Schaftingen, Universal approximation of symmetrizations by polarizations, Proc. Amer. Math. Soc. 134 (2006), no. 1, 177–186.
doi:10.1090/S0002-9939-05-08325-5 MR:2170557 DIAL:39089 preprint
Jean Van Schaftingen, Symmetrization and minimax principles, Commun. Contemp. Math. 7 (2005), no. 4, 463–481.
doi:10.1142/S0219199705001817 MR:2166661 DIAL:39111 preprint
Jean Van Schaftingen and Michel Willem, Set transformations, symmetrizations and isoperimetric inequalities, in V. Benci and A. Masiello (eds.), Nonlinear analysis and applications to physical sciences, Springer Italia, Milan, 2004, 135–152.
