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arXiv · 1404.5077

A doubly nonlinear evolution for the optimal Poincaré inequality

Ryan Hynd·Erik Lindgren
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Abstract

We study the large time behavior of solutions of the PDE $|v_t|^{p-2}v_t=Δ_p v$. A special property of this equation is that the Rayleigh quotient $\int_Ω|Dv(x,t)|^pdx /\int_Ω|v(x,t)|^pdx$ 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.

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Ryan Hynd, Erik Lindgren. 2016-06-30. A doubly nonlinear evolution for the optimal Poincaré inequality. https://arxiv.org/abs/1404.5077

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