A proof by foliation that lawson's cones are $ A_{\Phi} $-minimizing

Connor Mooney; Yang Yang

doi:10.3934/dcds.2021077

Article Contents

2021, Volume 41, Issue 11: 5291-5302. Doi: 10.3934/dcds.2021077

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A proof by foliation that lawson's cones are $ A_{\Phi} $-minimizing

Connor Mooney^, and
Yang Yang

Department of Mathematics, UC Irvine, 340 Rowland Hall, Irvine, CA 92697-3875, USA

^* Corresponding author: Connor Mooney
^* Corresponding author: Connor Mooney

Received: January 31, 2021

Revised: February 28, 2021

Early access: April 2021

Published: November 2021

The first author is supported by NSF grant DMS-1854788

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Abstract

We give a proof by foliation that the cones over $ \mathbb{S}^k \times \mathbb{S}^l $ minimize parametric elliptic functionals for each $ k, \, l \geq 1 $. We also analyze the behavior at infinity of the leaves in the foliations. This analysis motivates conjectures related to the existence and growth rates of nonlinear entire solutions to equations of minimal surface type that arise in the study of such functionals.

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Mathematics Subject Classification: Primary: 53A10, 49Q20, 35B08.

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Figure 1. The dilations of $ \Sigma_{kl} $ foliate one side of $ C_{kl} $

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Figure 2. The solution curve is contained in the region $ R $ bounded by $ \Gamma_1, \, \Gamma_2 $ and $ \Gamma_3 $

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