A doubly nonlinear evolution for the optimal Poincaré inequality

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• 作者：Ryan Hynd ; Erik Lindgren
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2016
• 出版时间：August 2016
• 年：2016
• 卷：55
• 期：4
• 全文大小：580 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Systems Theory and Control
  Calculus of Variations and Optimal Control
  Mathematical and Computational Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0835
• 卷排序：55

文摘

We study the large time behavior of solutions of the PDE \(|v_t|^{p-2}v_t=\Delta _p v\). A special property of this equation is that the Rayleigh quotient \(\int _{\Omega }|Dv(x,t)|^pdx /\int _{\Omega }|v(x,t)|^pdx\) is nonincreasing in time along solutions. As t tends to infinity, this ratio converges to the optimal constant in Poincaré’s inequality. Moreover, appropriately scaled solutions converge to a function for which equality holds in this inequality. An interesting limiting equation also arises when p tends to infinity, which provides a new approach to approximating ground states of the infinity Laplacian.Mathematics Subject Classification35K1539B6235P3047J1035K55