Simons’ cone and equivariant maximization of the first -Laplace eigenvalue

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• 作者：Sinan Ariturk ^{ariturk@mat.puc-rio.br}
• 关键词：Eigenvalue optimization ; pp-Laplacian ; Simons&rsquo ; cone
• 刊名：Nonlinear Analysis: Theory, Methods & Applications
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：150
• 期：Complete
• 页码：210-233
• 全文大小：727 K
• 卷排序：150

文摘

We consider an optimization problem for the first Dirichlet eigenvalue of the pp-Laplacian on a hypersurface in R2nR2n, with n≥2n≥2. If p≥2n−1p≥2n−1, then among hypersurfaces in R2nR2n which are O(n)×O(n)O(n)×O(n)-invariant and have one fixed boundary component, there is a surface which maximizes the first Dirichlet eigenvalue of the pp-Laplacian. This surface is either Simons’ cone or a C1C1 hypersurface, depending on pp and nn. If nn is fixed and pp is large, then the maximizing surface is not Simons’ cone. If p=2p=2 and n≤5n≤5, then Simons’ cone does not maximize the first eigenvalue.