Boundary regularity for elliptic systems under a natural growth condition

TL;DR

This paper proves boundary regularity for weak solutions of certain nonlinear elliptic systems in low dimensions, under natural growth conditions and smooth boundary data, ensuring almost every boundary point is regular for the gradient.

Contribution

It establishes boundary regularity results for elliptic systems with natural growth conditions, extending understanding of solution behavior at the boundary in low dimensions.

Findings

Almost every boundary point is regular for the gradient in dimensions 2, 3, and 4.

Boundary regularity holds under smooth boundary data and coefficients.

Results apply to systems with inhomogeneities satisfying natural growth conditions.

Abstract

We consider weak solutions $u \in u_0 + W^{1,2}_0(\Omega,R^N) \cap L^{\infty}(\Omega,R^N)$ of second order nonlinear elliptic systems of the type $- div a (\cdot, u, Du) = b(\cdot,u,Du)$ 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.