On the continuity of solutions to degenerate elliptic equations

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 non-negative solution u is continuous.