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Inhomogeneous Hopf-Oleĭnik Lemma and Applications. Part IV: Sharp Krylov Boundary Gradient Type Estimates for Solutions to Fully Nonlinear Differential Inequalities with unbounded coefficients and $C^{1,Dini}$ boundary data

In this paper we provide another application of the Inhomogeneous Hopf-Ole\uınik Lemma (IHOL) proved in \cite{BM-IHOL-PartI} or \cite{Boyan-2}. As a matter of fact, we also provide a new and simpler proof of a slightly weaker version IHOL for the uniformly elliptic fully nonlinear case which is sufficient for most purposes. The paper has essentially two parts. In the first part, we use IHOL for unbounded RHS to develop a Caffarelli's "Lipschitz implies $C^{1,α}$" approach to prove Ladyzhenskaya-Uraltseva boundary gradient type estimates for functions in $S^{*}(γ, f)$ that vanishes on the boundary. Here, unbounded RHS means that $f\in L^{q}$ with $q>n$. This extends the celebrated Krylov's boundary gradient estimate proved in \cite{Krylov}. A Phragmén-Lindelöf classification result for solutions in half spaces is recovered from these estimates. Moreover, a Hölder estimate up to the boundary (in the half-ball) for $u(x)/x_{n}$ is obtained. In the second part, we extend the previous results for functions in $S^{*}(γ, σ, f)$ where $γ,f\in L^{q}$ with $q>n$ that have a $C^{1,Dini}$ boundary data on a $W^{2,q}$ domain. Here, we use an "improvement of flatness" strategy suited to the unbounded coefficients scenario. As a consequence of that, a quantitative version of IHOL under pointwise $C^{1,Dini}$ boundary regularity is obtained.