On the continuity of solutions to degenerate elliptic equations
Abstract
The local behavior of solutions to a degenerate elliptic equation divA(x)∇u=0 in Ω⊂R where A(x)=At(x) and w(x)⩽<A(x)ξ,ξ>⩽v(x) for weights w(x)⩾0 and v(x), has been studied by Chanillo and Wheeden. In Chanillo and Wheeden (1986) [7], they generalize the results of Fabes, Kenig, and Serapioni (1961) [8] relative to the case v(x)=Λw(x).
We consider the case where w(x)=<mml:mfrac>1K(x)</mml:mfrac> and v(x)=K(x). The assumption that v∈A2, the Muckenhoupt class, is not sufficient as it was in the case v(x)=Λw(x) to obtain the continuity of local solutions. However, if v∈Gn, the Gehring class, and if Sv is the domain of the maximal function of v, Sv={x∈Ω:Mv(x)<∞}, then the restriction to Sv of the precise̲ representative u∼ of any nonnegative solution u is continuous.
 Publication:

Journal of Differential Equations
 Pub Date:
 March 2011
 DOI:
 10.1016/j.jde.2010.12.014
 Bibcode:
 2011JDE...250.2671C