An inequality for the normal derivative of the Lane–Emden ground state
An inequality for the normal derivative of the Lane–Emden ground state
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DOI：
10.1515/acv-2022-0005
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发表时间：
2022-01
影响因子：
1.7
通讯作者：
R. Frank;S. Larson
中科院分区：
文献类型：
--
作者：
R. Frank;S. Larson
摘要考虑多向性指数0≤q-1≤10 {\leq q-1 \leq 1的Lane-Emden基态，}即L q L^{q{归一化函数间Dirichlet积分的极小值}}。我们的主要结果{是以能量表示的法向导数的{l2l ^}}2范数的一个明显的下界，这意味着一个相应的等周不等式。我们的界适用于任意有界开李普希兹集Ω∧∈d{\Omega\subset\mathbb{R} ^{d}}，不假设凹凸性。
Abstract We consider Lane–Emden ground states with polytropic index 0 ≤ q - 1 ≤ 1 {0\leq q-1\leq 1} , that is, minimizers of the Dirichlet integral among L q {L^{q}} -normalized functions. Our main result is a sharp lower bound on the L 2 {L^{2}} -norm of the normal derivative in terms of the energy, which implies a corresponding isoperimetric inequality. Our bound holds for arbitrary bounded open Lipschitz sets Ω ⊂ ℝ d {\Omega\subset\mathbb{R}^{d}} , without assuming convexity.