On the continuity of solutions to degenerate elliptic equations
The very interesting paper under review deals with the local behaviour of weak solutions to the divergence form equation \[ \text{div\,} A(x)\nabla u=0\qquad \text{in}\;\Omega\subset\mathbb R^n \] with a real and symmetric matrix \(A(x)\) of measurable entries satisfying \[ w(x)|\xi|^2\leq \big\langle A(x)\xi,\xi\big\rangle\leq v(x)|\xi|^2 \] with suitable weights \(w(x)\geq0\) and \(v(x)\). In the special case of \(A_2\)-Muckenhoupt weights satisfying \(v(x)=\Lambda w(x)\) a.e. in \(\Omega\) with \(\Lambda\geq1,\) \textit{S. Chanillo} and \textit{R. L. Wheeden} [Commun. Partial Differ. Equations 11, 1111--1134 (1986; Zbl 0634.35035)] proved continuity of the weak solution under additional, very strong assumptions on the oscillations \(w(x)\) and \(v(x)\). In the paper under review, the authors consider the case \(w(x) = 1/K(x)\) and \(v(x) = K(x)\in A_2.\) Assuming \(K\) to satisfy the Gehring \(G_n\)-condition \[ G_n(K)=\sup_B { {\left({1\over{|B|}}\int_B K(x)^ndx\right)^{1/n}} \over{{1\over{|B|}}\int_B K(x)dx} }<\infty \] with the supremum taken over all balls \(B\subset \mathbb R^n,\) and setting \(S_v\) for the domain of the maximal function of \(v\), \(S_v =\{x\in\Omega:\;Mv(x)<\infty\}\), it is proved that the restriction to \(S_v\) of the precise representative \(\tilde u\) of any non-negative solution \(u\) is continuous.
scientific article; zbMATH DE number 5865665
scientific article