Self-gravitating discs and the Sobolev inequality

We estimate the minimal mass of self-gravitating polytropic discs using the famous Sobolev inequality. This 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 self-gravitating discs and is less useful for light discs.