Convex solutions to the power - of - mean curvature flow , conformally invariant inequalities and regularity results in some applications of optimal transportation

Convex solutions to the power-of-mean curvature flow, conformally invariant inequalities and regularity results in some applications of optimal transportation Shibing Chen Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2012 In this thesis we study three different problems: convex ancient solutions to the power-ofmean curvature flow; Sharp inequalities; regularity results in some applications of optimal transportation. The second chapter is devoted to the power-of-mean curvature flow; We prove some estimates for convex ancient solutions (the existence time for the solution starts from −∞) to the power-of-mean curvature flow, when the power is strictly greater than 1 2 . As an application, we prove that in two dimension, the blow-down of an entire convex translating solution, namely uh = 1 h u(h 1 1+αx), locally uniformly converges to 1 1+α |x| as h→∞. The second application is that for generalized curve shortening flow (convex curve evolving in its normal direction with speed equal to a power of its curvature), if the convex compact ancient solution sweeps the whole space R, it must be a shrinking circle. Otherwise the solution must be defined in a strip region. In the first section of the third chapter, we prove a one-parameter family of sharp conformally invariant integral inequalities for functions on the n-dimensional unit ball. As a limiting case, we obtain an inequality that generalizes Carleman’s inequality for harmonic functions in the plane to poly-harmonic functions in higher dimensions. The second section represents joint work with Tobias Weth and Rupert Frank; the main result is that, one can always put a sharp remainder term on the righthand side of the sharp fractional sobolev inequality.