A new proof of a Harnack inequality for the curve shortening flow

Abstract

We offer an algorithmic approach for determining Harnack quantities for the curve shortening flow and we show how, following this procedure, one can obtain Hamilton's Harnack inequality for this flow κ t + 1 2t κ ≥ κ 2 s κ , where κ is the curvature of the curve being deformed by the flow.

A Harnack inequality for the parabolic Allen–Cahn equation

“…The method both determines the Harnack quantity and proves that it's non-negative, using the maximum principle. The method has proven to be able to recover classical results for other non-linear parabolic equation (see [6,7]) or for the curve shortening flow ( [4]). The procedure has its roots in the study of Ricci flow, but recently its efficiency has been observed for other heat-type equations also (see, for example, [7] for a non-linear heat equation, or [6] for the Endangered Species Equation).…”