A doubly nonlinear evolution for the optimal Poincaré inequality
نویسندگان
چکیده
We study the large time behavior of solutions of the PDE |vt |p−2vt = pv. A special property of this equation is that the Rayleigh quotient ∫ |Dv(x, t)|pdx/ ∫ |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 Classification 35K15 · 39B62 · 35P30 · 47J10 · 35K55
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