Lane-Emden stars, selfgravitating disks and the Sobolev inequality

We estimate the minimal mass of selfgravitating polytropic disks using the famous Sobolev inequality. This bound resembles the well known mass formula for Lane-Emden stars. For ideal gas with the polytropic index n = 3 the minimal mass is not smaller than the Jeans mass. The accuracy of the estimate is verified in a number of numerical examples. The bound works well for heavy selfgravitating disks and is less useful for light disks.