On a class of weighted anisotropic Sobolev inequalities

In this article, motivated by a work of Caffarelli and Cordoba in phase transitions analysis, we prove new weighted anisotropic Sobolev type inequalities where different derivatives have different weight functions. These inequalities are also intimately connected to weighted Sobolev inequalities for Grushin type operators, the weights being not necessarily Muckenhoupt. For example we consider Sobolev inequalities on finite cylinders, the weight being a power of the distance function from the top or the bottom of the cylinder. We also prove similar inequalities in the more general case in which the weight is a power of the distance function from a higher codimension part of the boundary.