Inhomogeneous Hopf-OleÄ­nik Lemma and regularity of semiconvex supersolutions via new barriers for the Pucci extremal operators

In this paper, we construct new barriers for the Pucci extremal operators with unbounded RHS. The geometry of these barriers is given by a Harnack inequality up to the boundary type estimate. Under the possession of these barriers, we prove a new quantitative version of the Hopf-Oleĭnik Lemma for quasilinear elliptic equations with g-Laplace type growth. Finally, we prove (sharp) regularity for ω-semiconvex supersolutions for some nonlinear PDEs. These results are new even for second order linear elliptic equations in nondivergence form. Moreover, these estimates extend and improve a classical a priori estimate proven by L. Caffarelli, J.J. Kohn, J. Spruck and L. Nirenberg in [13] in 1985 as well as a more recent result on the C1,1 regularity for convex supersolutions obtained by C. Imbert in [33] in 2006.