Critical points and curvature in circular clamped plates

In this article we investigate some qualitative properties of the solutions of the classical linear model for clamped plates on circular domains, under constant sign external loads. In particular we prove that inside the circle there are at most a finite number of critical points, which in turn rules out the existence of critical curves. We also study the curvature of the level curves of the solutions, and we prove that the curvature function is continuous up to the border, even though the gradient of the solutions vanishes on the border circle.