Polynomial solutions to boundary value problems for the heat equation: It is well-known that Dirichlet's problem for the Laplace equation with polynomial data on an ellipsoid always has a polynomial solution. D. Khavinson and H. S. Shapiro also showed that entire data gives entire solution. Matt Cecil, Ronald A. Walker, and I are investigating parallel questions about the solutions of boundary value problems for the heat equation. A more natural data set for the heat equation is the boundary of a cylinder. If the base of the cylinder is an ellipsoid, then the heat equation with polynomial data has a polynomial solution. We extend this to entire data under an additional assumption. We also consider the heat equation posed on a space-time parabola. If the parabola opens in the direction of positive time, then the heat equation with polynomial data has a polynomial solution. If the parabola opens in the negative direction of time, then the

Open Problem: Suppose U is a cylinder in n+1 dimesions whose base D is a quadrature domain. It is well-known that the solution to the Dirichlet problem in D for Laplace's equation with polynomial data on the boundary has a solution that can be analytically continued by reflection to a neighborhood of the closure of D.

This implies continuation (to a neighborhood of U) of the stationary solutions of the heat equation with polynomial data. Can the non-stationary solutions with polynomial data be analytically continued to the same neighborhood?

Remark about boundary control:

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