The local behavior of solutions to a degenerate elliptic equation. divA(x)Δu=0inΩ⊂R^n where A(x)=t^A(x) and. w(x)|ξ|^2≤(A(x)ξ,ξ)≤v(x)|ξ|^2 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)=1/K(x) and v(x)=K(x). The assumption that v∈A_2, 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∈G_n, the Gehring class, and if S_v is the domain of the maximal function of v,. Sv={x∈Ω:Mv(x)<∞}, then the restriction to Sv of the precise representative ũ of any non-negative solution u is continuous.
On the continuity of solutions to degenerate elliptic equations
DI GIRONIMO, Patrizia;
2011
Abstract
The local behavior of solutions to a degenerate elliptic equation. divA(x)Δu=0inΩ⊂R^n where A(x)=t^A(x) and. w(x)|ξ|^2≤(A(x)ξ,ξ)≤v(x)|ξ|^2 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)=1/K(x) and v(x)=K(x). The assumption that v∈A_2, 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∈G_n, the Gehring class, and if S_v is the domain of the maximal function of v,. Sv={x∈Ω:Mv(x)<∞}, then the restriction to Sv of the precise representative ũ of any non-negative solution u is continuous.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
