The sets of critical points of solutions to elliptic partial differential equations have been well-studied but many questions remain open. With somewhat weak hypotheses on the regularity of the coefficients, one knows that the Hausdorff codimension of the set of critical points is at least one. Eigenfunctions of the Laplacian on a rectangle can have critical sets whose codimension is exactly one. In joint work with Sugata Mondal, we show that if a polygon has a second Neumann eigenfunction with infinitely many critical points, then the polygon is a rectangle. One conjectures that one can replace ‘polygon’ with ‘planar domain with smooth boundary’.
