Caltech/UCLA Joint Analysis Seminar
"Between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain." P. Painleve, 1900.
- Newton noticed that the gravitational potential of a spherical mass with constant density equals, outside the ball, the potential of the point-mass at the center. Rephrasing, the gravitational potential of the ball with constant mass density continues as a harmonic function inside the ball except for the center. Fairly recently, it was noted that the latter statement holds for any polynomial, or even for entire densities.
- If a harmonic in a spherical shell function vanishes on one piece of a line through the center piercing the shell, then it must vanish on the second piece of that line. Yet, the similar statement fails for tori.
- If we solve the Dirichlet problem in an ellipse with entire data, the solution will always be an entire harmonic function. Yet, if we do that in a domain bounded by the curve x4+y4=1, with the data as simple as x2+y2, the solution will have infinitely many singularities outside the curve.
- Where and why do eigenfunctions of the Laplacian in domains bounded by algebraic curves start having singularities?
We shall discuss these and some other questions under the unified umbrella of analytic continuation of solutions to analytic pde in Cn.