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 = ∆ p v. A special property of this equation is that the Rayleigh quotient Ω |Dv(x, t)| p dx/ Ω |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.

Hölder estimates and large time behavior for a nonlocal doubly nonlinear evolution

“…Since the proof is exactly the same as the proof of the corresponding result in [HL14], Corollary 2.5, we have chosen to omit it.…”

“…One of the novelties of the present paper in comparison with [HL14], is that we obtain uniform convergence to a ground state and a uniform Hölder estimate for the doubly nonlinear, nonlocal equation (1.1). No such results are known for equation (1.6).…”

“…In addition, v and −v are upper semicontinuous. We recall the following result, which is Lemma 4.4 in [HL14]. The statement of the result in [HL14] is for smooth φ, but the proof holds also for φ as below.…”

“…The main results are that the Rayleigh quotient is monotone along the flow (Proposition 3.6) and that "bounded" sequences of weak solutions are compact (Theorem 3.8). The interested reader could also consult [HL14] where a similar theory is built for equation (1.6).…”

“…In our previous work [HL14], we studied the large time behavior of the doubly nonlinear, local equation |v t | p−2 v t = ∆ p v.…”