Unlike Green's functions for elliptic equations in divergence form, Green's function for elliptic operators in nondivergence form do not possess nice pointwise bounds even in the case when the coefficients are uniformly continuous. In this talk, I will describe how to construct and get pointwise estimates for elliptic PDEs in non-divergence form with coefficients satisfying the so called Dini mean oscillation condition. I will also mention the parallel result for parabolic equations in non-divergence form.

A family of surfaces is called mean curvature flow (MCF) if the velocity of surface is equal to the mean curvature of the surface at that point. Even starting from smooth surface, the MCF typically encounters some singularities and various generalized notions of MCF have been proposed to extend the existence past singularities. They are level set flow, Brakke flow and BV flow, just to name a few. In my talk I explain a recent global-in-time existence result of a particular generalized solution which has some desirable properties. I describe a basic outline of how to construct the solution.

A family of surfaces is called mean curvature flow (MCF) if the velocity of surface is equal to the mean curvature of the surface at that point. Even starting from smooth surface, the MCF typically encounters some singularities and various generalized notions of MCF have been proposed to extend the existence past singularities. They are level set flow, Brakke flow and BV flow, just to name a few. In my talk I explain a recent global-in-time existence result of a particular generalized solution which has some desirable properties. I describe a basic outline of how to construct the solution.