Non-uniqueness and failure of Calder\'on-Zygmund estimates below the critical exponent for non-monotone PDE with linear growth

We provide counterexamples to uniqueness of solutions as well as a priori Calder\'on-Zygmund estimates for solutions below $L^2$ using convex integration argument for equations of the type $$ \text{ div} (A (\nabla u)) = 0 \quad \text{ in } \mathbb{ B}^2, $$ where $A: \mathbb{ R}^{ 2} \to \mathbb{ R}^2$ is smooth, uniformly elliptic and has essentially linear growth, but fails to be monotone and asymptotically Uhlenbeck.

Categories: math.AP