About Smoothness of Solutions of the Heat Equations in Close

Hongjie Dong
University of Chicago

Abstract: We consider the probabilistic solutions of the heat equation $ u_{x^2}=u_{x^1x^1}+f$ in $ D$, where $ D$ is a bounded domain in $ {\mathbb{R}}^2=\{(x^1,x^2)\}$ of class $ C^{2k}$. We give sufficient conditions for $ u$ to have the $ k^{\text{th}}$ order continuous derivatives with respect to $ (x^1,x^2)$ in $ {\bar D}$, for integers $ k\geq 2$. The equation is supplemented with $ C^{2k}$ boundary data and we assume that $ f \in
C^{2(k-1)}$. We also prove that our conditions are sharp by given examples in the border cases.