Partial symmetry and asymptotic behavior for some elliptic variational problems

Abstract

A short elementary proof based on polarizations yields a useful (new) rearrangement inequality for symmetrically weighted Dirichlet type functionals. It is then used to answer some symmetry related open questions in the literature. The non symmetry of the Hénon equation ground states (previously proved in [19]) as well as their asymptotic behavior are analyzed more in depth. A special attention is also paid to the minimizers of the Caffarelli-Kohn-Nirenberg [8] inequalities.

Symmetry of solutions to semilinear elliptic equations via Morse index

“…Afterwards a similar result for minimizers of some variational problems was obtained in [13] using a completely different approach based on symmetrization techniques.…”

“…This is, for example, the case for the simple nonlinearity given by f (s) = |s| p−1 s, p > 1. Moreover, also the results of [13] do not apply, because they essentially deal with cases where the minimizer is known to be a positive function. A first result, concerning variational problems where minimizers are sign changing functions, has been obtained recently in [3] where, using again symmetrization techniques, the authors extend the results of [13].…”

Some symmetric boundary value problems and non-symmetric solutions

“…Here f ′ is the derivative of a function f ∈ C 2 (R). Subsequent extensions include, among other things, classes of solutions that are not necessarily positive [6,8,9,10,11]. In particular, a results in [10] implies that, if Ω is a ball or annulus, w is radially symmetric, and f ′ is convex, then any solution u of (1.1) with Morse index n or less has an axial symmetry.…”

Approximation of symmetrizations by Markov processes

“…The spherical nonincreasing rearrangement has been approximated by other partial symmetrizations, such as cap symmetrizations [15] and polarizations [1,6,20,21,24]. Other symmetrizations such as the cap symmetrization [16] and discrete symmetrizations [14] have also been approximated in the deterministic case in order to prove isoperimetric theorems. Theorem 1.2 fails for the approximation of the spherical nonincreasing rearrangement by successive cap symmetrizations or polarizations.…”