On the symmetry and monotonicity of Morrey extremals

TL;DR

This paper proves that extremals of Morrey's inequality are axially symmetric, antisymmetric, and monotone in specific directions using Clarkson's inequality and symmetrization methods.

Contribution

It establishes new symmetry and monotonicity properties of Morrey's extremals, enhancing understanding of their structure.

Findings

Extremals are axially symmetric and antisymmetric.

Extremals are monotone relative to their axis and distance from it.

Uses Clarkson's inequality and symmetrization techniques.

Abstract

We employ Clarkson's inequality to deduce that each extremal of Morrey's inequality is axially symmetric and is antisymmetric with respect to reflection about a plane orthogonal to its axis of symmetry. We also use symmetrization methods to show that each extremal is monotone in the distance from its axis of symmetry and in the direction of its axis when restricted to spheres centered at the intersection of its axis and its antisymmetry plane.