On the symmetry and monotonicity of Morrey extremals

Ryan Hynd; Francis Seuffert

doi:10.3934/cpaa.2020238

Article Contents

2020, Volume 19, Issue 11: 5285-5303. Doi: 10.3934/cpaa.2020238

This issue Previous Article Monotonicity of solutions for a class of nonlocal Monge-Ampère problem Next Article

On the symmetry and monotonicity of Morrey extremals

Ryan Hynd and
Francis Seuffert^,

Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104-6395

^* Corresponding author
^* Corresponding author

Received: March 31, 2020

Revised: April 30, 2020

Early access: September 2020

Published: November 2020

This work was partially supported by NSF grant DMS-1554130

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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 and in the direction of its axis when restricted to spheres centered at the intersection of its axis and its antisymmetry plane.

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Mathematics Subject Classification: Primary: 46E35, 58E15; Secondary: 35J60, 58J70.

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Figure 1. These diagrams illustrate the monotonicity properties of the Morrey extremal satisfying (1.3) and (1.4) for $ n = 2 $. Theorems 1.2 and 1.3 respectively assert that $ u(x^1)\le u(x^2) $ for $ x^1,x^2\in \mathbb{R}^2 $ which are ordered as on the horizontal lines in the top diagram and as on each circle in the bottom diagram

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Figure 2. The spherical cap $ C_{t,\theta} $ of radius $ t $ and opening angle $ \theta $. The cap contains all points $ x \in \mathbb{R}^2 $ such that $ |x| = t $ and $ x_2 > t \cos \theta $

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Figure 3. The shaded region on top represents a (closed) subset $ A \subset \mathbb{R}^2 $. The shaded region on bottom is $ A^\star\subset \mathbb{R}^2 $, the cap symmetrization of $ A $ in the direction of the positive $ x_2 $ axis

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