Krylov's Boundary Gradient Type Estimates for Solutions to Fully Nonlinear Differential Inequalities with Quadratic Growth on the Gradient

In this paper we prove Krylov's boundary gradient type estimates (and regularity) for solutions to fully nonlinear differential inequalities with unbounded coefficients and quadratic growth on the gradient with 𝐶1,𝑑⁢𝑖⁢𝑛⁢𝑖 boundary data. This means that drift coefficients and the right-hand side (RHS) are in 𝐿𝑞with 𝑞 >𝑛. We also show that in the case the RHS is in 𝐿𝑛 the result does not hold and solutions may fail to be even Lipschitz in (tiny) neighborhoods of the boundary. Our approach is based on a new improvement of flatness argument together with an iteration process based on scaling arguments and perturbation by linear functions. Our results can be seen as an extension of the corresponding ones obtained in [L. Silvestre and B. Sirakov, Comm. Partial Differential Equations, 39 (2014), pp. 1694--1717] in the case of bounded coefficients and no quadratic term.