The Lp-Minkowski problem introduced by Lutwak is solved for p > n + 1 in the smooth category. The relevant Monge-Ampère equation (1) is solved for all p > 1. The same equation for p < 1 is also… (More)

By studying a negative gradient flow of certain Hessian functionals we establish the existence of critical points of the functionals and consequently the existence of ground states to a class of… (More)

Integrable equations satisfied by the curvature of plane curves or curves on the real line under inextensible motions in some Klein geometries are identified. These include the Euclidean, similarity,… (More)

whereV andk are, respectively, the normal velocity and curvature of the interface. Equation (1) is sometimes called the “curve shortening problem” because it is the negativeL2-gradient flow of the… (More)

Let X be a compact, strictly convex C-hypersurface in the (n+1)-dimensional Euclidean space R. The Gauss map of X maps the hypersurface one-to-one and onto the unit n-sphere S. One may parametrize X… (More)

The connection between motion of space or plane curves and integrable equations has drawn wide interest in the past and many results have been obtained. The pioneering work is due to Hasimoto where… (More)

Upper bounds on the Hausdorff dimensions of the rupture set of a weak solution of the thin film equation in space-time and in space slices are derived. Finite time rupture is shown to occur for a… (More)

The thin film equation, which is derived from the Navier-Stokes equations, is a degenerate fourth order parabolic equation describing the motion of thin films on a plate. In this talk we shall give… (More)