The Uniform Hopf Inequality for discontinuous coefficients and optimal regularity in BMO for singular problems

We consider some singular second order semilinear problems which include, among many other special cases, the boundary layer equations such as they were treated by O.A. Oleinik in her pioneering works. We consider diffusion linear operator with possible discontinuous coefficients and prove an optimal criterion to get a quantitative strong maximum principle that we call Uniform Hopf Inequality UHI. Since the solutions of the singular semilinear problems under consideration are not Lipschitz continuous we carry out a careful study of the regularity of solutions when the coefficients of the diffusion matrix are merely in the vmo space and bounded. We prove that the gradient of the solution is still p-integrable, in absence of any continuity assumption on the spatial potential coefficient in the singular term. To this end, the UHI property is used several times. We also apply and improve previous a priori estimates due to S. Campanato in 1965.