ON THE ALEXANDROFF-BAKELMAN-PUCCI ESTIMATE AND THE REVERSED HOLDER INEQUALITY FOR SOLUTIONS OF ELLIPTIC AND PARABOLIC EQUATIONS

The constant appearing in the classical Alexandroff-Bakelman-Pucci est imate for subsolutions of second-order uniformly elliptic equations in nondivergence form was known to depend on the diameter of the domain. Using the Krylov-Safonov boundary weak Harnack inequality due to Trud inger, we show that the dependence on the diameter may be replaced by dependence on a more precise geometric quantity of the domain. As a co nsequence, we get dependence on the measure instead of the diameter. W e also give new bounds for subsolutions in some unbounded domains, suc h as domains contained in cones. We apply the Fabes and Stroock revers ed Holder inequality for the Green's function to improve our estimates . We also give a new proof of the reversed Holder inequality for the G reen's function based on the Krylov-Safonov Harnack inequality. Finall y, we find new bounds for subsolutions of uniformly parabolic equation s in cylindrical and noncylindrical domains. The constant in the (Alex androff-Bakelman-Pucci-) Krylov-Tso estimate was known to depend on th e diameter of the base of the cylinder. We get dependence either on th e measure of the base or on the height of the cylinder. We also give b ounds for subsolutions in noncylindrical domains. (C) 1995 John Wiley and Sons, Inc.