Sobolev Inequality

PDEsFunctionalAnalysis It turns out that Sobolev Spaces are nested in a dimension dependent way. This can be shown by bounding the norm by the Sobolev norm for some appropriately chosen . Subsequently, such has a differing form depending on the following conditions:

1. : Gagliardo-Niremberg-Sobolev (GNS) Inequality: where is the Sobolev Conjugate of p for and for where is open and bounded with Examples:
   • , :
   • , :
2. : Morrey’s Inequality for for and for , with and where is open and bounded with
3. : (Consequence of) Poincaré’s Inequality

More generally Poincaré’s Inequality does not depend on the dimension of the space and only depends on the gradient of .