Partial regularity of solutions of quasilinear elliptic systems with nonsmooth condition on the conormal derivative

Partial regularity of a generalized solution $u\colon\Omega\subset\mathbb R^n\to\mathbb R^N$, $n>2$, $N>1$, of a quasilinear elliptic system is proved under a nonsmooth condition on the conormal derivative. The singular set $\Sigma\subset\overline\Omega$ is described; it is proved that for some $p>2$ the Hausdorff dimension of $\Sigma$ is equal to $n-p$. In the proof essential use is made of a theorem proved earlier by the author on reverse inequalities with surface integrals.