On a class of weighted anisotropic Sobolev inequalities

TL;DR

This paper establishes new weighted anisotropic Sobolev inequalities with different weights for derivatives, extending to Grushin operators and boundary distance functions, motivated by phase transition analysis.

Contribution

It introduces novel weighted anisotropic Sobolev inequalities involving non-Muckenhoupt weights, applicable to cylinders and higher codimension boundary parts.

Findings

Derived Sobolev inequalities on finite cylinders with distinct weights

Extended inequalities to cases with weights as distance functions from boundary parts

Connected inequalities to weighted Sobolev inequalities for Grushin type operators

Abstract

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.