A doubly nonlinear evolution for the optimal Poincaré inequality

Abstract

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 ${\int }_{\Omega }|Dv(x,t){|}^{p}dx/{\int }_{\Omega }|v(x,t){|}^{p}dx$ 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.