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:
Back to my home page.