On the regularity of solutions of model nonlinear elliptic systems with the oblique derivative type boundary condition

Properties of generalized solutions of model nonlinear elliptic systems of second order are studied in the semiball $B^+_1=B_1(0)\cap\{x_n>0\}\subset\mathbb R^n$, with the oblique derivative type boundary condition on $\Gamma_1=B_1(0)\cap\{x_n=0\}$. For solutions $u\in H^1(B_1^+)$ of systems of the form $\frac d{dx_\alpha}a^k_\alpha(u_x)=0$, $k\le N$, it is proved that the derivatives $u_x$ are Hölder in $(B^+_1\cup\Gamma_1)\setminus\Sigma$, where $\mathcal H_{n-p}(\Sigma)=0$, $p>2$. It is shown for continuous solutions $u$ from $H^1(B_1^+)$ of systems $\frac d{dx_\alpha}a^k_\alpha(u,u_x)=0$ that the derivatives $u_x$ are Hölder on the set $(B^+_1\cup\Gamma_1)\setminus\Sigma$, $\dim_\mathcal H\Sigma\le n-2$. Bibliography: 13 titles.