Universal approximation of symmetrizations by polarizations

Abstract

Abstract. Any symmetrization (Schwarz, Steiner, cap or increasing rearrangement) can be approximated by a universal sequence of polarizations which converges in L p norm for any admissible function in L p for 1 ≤ p < +∞ and uniformly for admissible continuous functions. A new Pólya-Szegö inequality is proved for the increasing rearrangement.

On the symmetry and monotonicity of Morrey extremals

“…holds. This inequality and various other properties of cap and Steiner symmetrizations were verified by Van Schaftingen in[30].By definition, v ≥ 0, and we also have {u > c} ⊂ {x n > 0} for c > 0. As a result,{v > c} = {v > c} = {u > c} ⊂ {x n > 0} by (5.1).…”

“…These results were extended by Aubin for applications in Riemannian geometry [1,2]. These ideas lead mathematicians to employ rearrangement methods [9,27,29,30], to seek best constants [16,21,27], and to explore the role of symmetry in various functional inequalities [8,13,17,22]. In recent years, researchers have also been using new techniques such as optimal transport to pursue these types of results [3,12,23].…”

On the symmetry and monotonicity of Morrey extremals

“…holds. This inequality and various other properties of cap and Steiner symmetrizations were verified by Van Schaftingen in [32].…”

“…These results were extended by Aubin for applications in Riemannian geometry [1,2]. Moreover, these ideas led mathematicians to employ rearrangement methods [9,29,31,32], to seek best constants [16,21,29], and to explore the role of symmetry in various functional inequalities [8,13,17,22]. In recent years, researchers have also been using new techniques such as optimal transport to pursue these types of results [3,12,23].…”

Rearrangement and polarization

“…Then, in 2000, a landmark study by Brock and Solynin [8] gave further significance to polarization by showing that the Steiner or Schwarz rearrangements of a function (or symmetrals of a compact set) with respect to a subspace H can be approximated in L p (R n ) (or in the Hausdorff metric, respectively) via successive polarizations with respect to a sequence (H k ) of oriented subspaces. In [8], the sequence (H k ) may depend on the function or set, but this dependence was removed by Van Schaftingen [45,46]. Indeed, by [46, Theorem 1 and Section 4.3], the desired approximation of the Steiner or Schwarz rearrangement S H f of a suitable function f may be obtained by taking any sequence (H k ) dense in the set of oriented subspaces J such that J + contains H in its interior and defining f 1 = f and…”