A doubly nonlinear evolution for the optimal Poincaré inequality

We study the large time behavior of solutions of the PDE |vt|p-2vt=Δpv\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|v_t|^{p-2}v_t=\Delta _p v$$\end{document}. A special property of this equation is that the Rayleigh quotient ∫Ω|Dv(x,t)|pdx/∫Ω|v(x,t)|pdx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _{\Omega }|Dv(x,t)|^pdx /\int _{\Omega }|v(x,t)|^pdx$$\end{document} 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.