Milton's conjecture on the regularity of solutions to isotropic equations (Q1406895)

The paper deals with the solutions of the isotropic equation \(\operatorname {div}p(z)\nabla u = 0 \) in a square \(Q\) of the plane. The main result is the following. Let \(K > 1\). There exist sequences of functions \(\{p_j \} \in L^\infty (Q,\{K,1/K\}) \) and \(\{u_j\} \in W^{1,2}(Q,R)\) with \(\|u_j\|_{W^{1,2} } \leq 1,\) such that \(\operatorname {div}((p_j(z) \nabla u_j(z))= 0 \) in \(Q,\) and for every compact set \(R\) of positive measure contained in \(Q\) \[ \lim_{j \to \infty }\int_R|\nabla u_j(z)|^{2K/(K-1)}dz = \infty. \]