Free boundary regularity for the porous medium equation

In this paper, we resolve the local cone conjecture proposed by Kienzler, Koch, and Vázquez concerning the free-boundary regularity of the porous medium equation. We prove that the local cone condition guarantees the smoothness of the free boundary lying outside the initial support. The proof relies on new gradient estimates obtained through maximum principle arguments and comparison principles. As an application, we establish the global-in-time smoothness of the free boundary for solutions with convex initial support under standard assumptions on the initial data.