For a class of strongly elliptic, second order systems $L$ with rough coefficients on a Lipschitz domain $\Omega$, we show that if $Lu=0$ on $\Omega$ and $u$ vanishes on an open subset of the boundary, then weak a priori hypotheses on the nontangential maximal function of $u$ lead to strong estimates on $\nabla u$, in nontangential and Besov norms, near this subset.
"Local Regularity Results for Second Order Elliptic Systems on Lipschitz Domains." Funct. Approx. Comment. Math. 40 (2) 175 - 184, June 2009. https://doi.org/10.7169/facm/1246454027