A doubly nonlinear evolution for the optimal Poincar\'e inequality

We study the large time behavior of solutions of the PDE $\mathrm{\mid }{v}_t{\mathrm{\mid }}^{p-2}{v}_t={\mathrm{\Delta }}_{p}v$. A special property of this equation is that the Rayleigh quotient ${\int }_{\mathrm{\Omega }}\mathrm{\mid }Dv(x,t){\mathrm{\mid }}^{p}dx\mathrm{/}{\int }_{\mathrm{\Omega }}\mathrm{\mid }v(x,t){\mathrm{\mid }}^{p}dx$ is nonincreasing in time along solutions. As $t$ tends to infinity, this ratio converges to the optimal constant in Poincar'{e}'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.