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