Boundary weak Harnack estimates and regularity for elliptic PDE in divergence form

We obtain a global extension of the classical weak Harnack inequality which extends and quantifies Hopf–Oleinik boundary-point lemma, for uniformly elliptic equations in divergence form. Among consequences is boundary gradient estimate, due to Krylov well-studied non-divergence form equations, but completely novel framework. Another consequence new more general version lemma.