Comparison and regularity results for degenerate elliptic equations related to Gauss measure

Abstract

This article deals with a Dirichlet problem relative to a linear elliptic equation with zero-order terms, whose ellipticity condition is given in terms of the function ν(x)ϕ(x), where is the density of Gauss measure, ν(x) is a nonnegative integrable function. Using the notion of rearrangement with respect to Gauss measure, we prove a comparison result with a problem of the same type defined in a half space, with data depending only on the first variable. Also, some regularity results for the solutions in Lorentz–Zygmund spaces are obtained by us.

Keywords:

• comparison results
• regularity
• rearrangements
• Gauss measure
• degenerate elliptic equation

AMS Subject Classifications::

• 26D20
• 35B65
• 35J70

Acknowledgements

This project was supported by NSFC(No:10401009) and NCET of China (No:060275).