About smoothness of solutions of the heat equations in closed, smooth space‐time domains
Hongjie Dong · Communications on Pure and Applied Mathematics · 2005
Abstract We consider the probabilistic solutions of the heat equation u = u + f in D, where D is a bounded domain in ℝ2 = {(x1, x2)} of class C2k. We give sufficient conditions for u to have kth‐order continuous derivatives with respect to (x1, x2) in D̄ for integers k ≥ 2. The equation is supplemented with C2k boundary data, and we assume that f ϵ C2(k−1). We also prove that our conditions are sharp by examples in the border cases. © 2005 Wiley Periodicals, Inc.