**Title:** Regularity of solutions to the heat equations near singular boundary points

**Speaker:** Professor Hongjie Dong

**Speaker Info:** University of Chicago

**Brief Description:**

**Special Note**:

**Abstract:**

This talk is devoted to the smoothness of the solutions of the heat equation u_t=u_{xx}+f in bounded space-time domains of class C^{2k}. I will give a sharp condition for the solution to have kth order continuous derivatives with respect to both x and t near singular boundary points, for integers k\geq 2. The equation is supplemented with C^{2k} boundary data and we assume that f is C^{2(k-1)}.The approach here is probabilistic. We don't know at present whether it is possible to give a proof only in terms of the theory of partial differential equations. The main difficulty in terms of the theory of partial differential equations is the absence of the analogues for the Burkholder-Davis-Gundy inequality, which plays an important role in the proofs.

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