Boundary regularity for elliptic systems under a natural growth condition

Abstract

We consider weak solutions $${u \in u_0 + W^{1,2}_0(\Omega,\mathbb{R}^N) \cap L^{\infty}(\Omega,\mathbb{R}^N)}$$ of second-order nonlinear elliptic systems of the type $$- {\rm div} \,a (\, \cdot \,, u, Du ) = b(\, \cdot \,,u,Du)\qquad \text{ in }\Omega$$ with an inhomogeneity satisfying a natural growth condition. In dimensions $${n \in \{2,3,4\}}$$ , we show that $${\mathcal{H}^{n-1}}$$ -almost every boundary point is a regular point for Du, provided that the boundary data and the coefficients are sufficiently smooth.