On the maximum principle for general linear elliptic equations

We consider maximum principles and related estimates for linear second order elliptic partial differential operators in n-dimensional Euclidean space, which improve previous results, with H-J Kuo, through sharp Lp dependence on the drift coefficient b. As in our previous work, the ellipticity is determined through the principal coefficient matrix A lying in sub-cones of the positive cone, which are dual cones of the Garding k-cones. Our main results are maximum principles for bounded domains, which extend those of Aleksandrov in the case k = n, together with extensions to unbounded domains, depending on appropriate integral norms of A, and corresponding local maximum principles. We also consider applications to local estimates in the uniformly elliptic case, including extensions of the Krylov-Safonov Holder and Harnack estimates.