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On the symmetry and monotonicity of Morrey extremals

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This work was partially supported by NSF grant DMS-1554130
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  • 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.

    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

    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 $

    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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