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

A. A. Arkhipova

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 26, Tome 221 (1995), pp. 30-57

Résumé

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.