Non-uniqueness and failure of Calderón-Zygmund estimates below the critical exponent for non-monotone PDE with linear growth

Abstract

We provide counterexamples to uniqueness of solutions as well as a priori Calderón-Zygmund estimates for solutions below $L^2$ using convex integration argument for equations of the type $$\text{div}(A(\nabla u))=0\text{in }{\mathbb{B}}^2,$$ 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.

This version includes an additional theorem on Calderon-Zygmund estimates for exponent above 2 with non-zero boundary datum