Partial regularity of solutions of nonlinear quasimonotone systems via a lower \(L^{p}\) estimate

We prove partial regularity of solutions u of the nonlinear quasimonotone system \({\text {div}}A\left( x,u,Du\right) +B\left( x,u,Du\right) =0\) $\text{div}A\left(x,,,u,,,D,u\right)+B\left(x,,,u,,,D,u\right)=0$ under natural polynomial growth of its coefficient functions A and B. We propose a new direct method based on an \(L^{p}\) $L^p$ estimate with low exponent \(p>1\) $p>1$ for a linear elliptic system with constant coefficient.