Partial Regularity of Weak Solutions to Nonlinear Elliptic Systems Satisfying a Dini Condition

Abstract

This paper is concerned with systems of nonlinear partial differential equations

where the coefficients $a^{\alpha}_i$ are assumed to satisfy the condition

for all ${x, u}, {y, v} \in \Omega \times \mathbb R^N$ and all $\xi \in \mathbb R^{nN}$, and where $\int^1_0 \frac {\omega (t)}{t} dt < + \infty$ while the functions $\frac {\partial a_i^{\alpha}}{\partial \xi ^j_{\beta}}$ satisfy the standard boundedness and ellipticity conditions and the function $\xi \mapsto b_i (x, u, \xi)$ may have quadratic growth. With these assumptions we prove partial Hölder continuity of bounded weak solutions $u$ to the above system provided the usual smallness condition on $\|u \|_{L \infty ({\Omega})}$ is fulfilled.