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 $a_i^{\alpha }$ 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 ${\int }_0^1\frac{\omega (t)}{t}dt<+\infty$ while the functions $\frac{\partial a_i^{\alpha }}{\partial {\xi }_{\beta }^j}$ 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 $\Vert u{\Vert }_{L^{\infty }(\Omega )}$ is fulfilled.