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 unique polynomial solution. If the parabola opens in the negative direction of time, then the answer depends on the ratio of the focal length of the parameter and the heat coefficient. In fact it has a "chaotic" dependence on this parameter.

Remark about boundary control: It is interesting to interpret polynomial solvability from a control-theoretic point of view. Namely, a polynomial solution to a boundary value on an elliptic cylinder allows immediate analytic continuation to all of space-time (it is a polynomial!). Thus, if we view the restriction of the polynomial to the cylinder boundary as an "objective function", which we wish to achieve through assignment of boundary values from further away (say the boundary of a much larger cylinder), then this boundary control problem is exactly controllable from any distance. Usually, one only gets L^2-approximate controllability when dealing with the heat equation.

Open Problem: Suppose U is a cylinder in 2+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?

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