ABP inequality and weak Harnack inequality for fully nonlinear elliptic operators with coefficients in weighted spaces (Q2837914)

Let \(\Omega\) a bounded domain of \(\mathbb{R}^n\). This paper deals with an Aleksandrov-Bakel'mann-Pucci (ABP) inequality for \(L^{n}\)-viscosity solutions of fully nonlinear equations in non-divergence form; the model case that the author considers is: NEWLINE\[NEWLINE P(D^{2}u)+b(x)\left|Du\right|+c(x)u=f(x),NEWLINE\]NEWLINE where \(P\) is a Pucci extremal operator, assuming that the coefficients are in weighted Lebesgue spaces, denoted by \(L^{p}(d^{\epsilon},\Omega)\). The author firstly obtains the weighted ABP estimate for strong solutions, assuming \(f\in L^{n}(d^\epsilon,\Omega)\). Indeed he obtains the result applying the method of Escauriaza, for \(f\in L^{n}(d^{\epsilon},\Omega)\). As a consequence of the ABP inequality, the author shows a maximum principle in small domains, the weak Harnak inequality for non-negative \(L^{n}\) viscosity solutions of extremal PDEs with \(L^{p}(d^{\epsilon},\Omega)\) coefficients, and, using the last one, a version of Hopf's Lemma.