About Smoothness of Solutions of the Heat Equations in Close

Hongjie Dong
University of Chicago
hjdong@math.uchicago.edu

Abstract: We consider the probabilistic solutions of the heat equation in , where is a bounded domain in of class . We give sufficient conditions for to have the order continuous derivatives with respect to in , for integers . The equation is supplemented with boundary data and we assume that . We also prove that our conditions are sharp by given examples in the border cases.