On the alexandroff‐bakelman‐pucci estimate and the reversed hölder inequality for solutions of elliptic and parabolic equations

Abstract

Abstract The constant appearing in the classical Alexandroff‐Bakelman‐Pucci estimate 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 Trudinger, we show that the dependence on the diameter may be replaced by dependence on a more precise geometric quantity of the domain. As a consequence, we get dependence on the measure instead of the diameter. We also give new bounds for subsolutions in some unbounded domains, such as domains contained in cones. We apply the Fabes and Stroock reversed Hölder inequality for the Green's function to improve our estimates. We also give a new proof of the reversed Hölder inequality for the Green's function based on the Krylov‐Safonov Harnack inequality. Finally, we find new bounds for subsolutions of uniformly parabolic equations in cylindrical and noncylindrical domains. The constant in the (Alexandroff‐Bakelman‐Pucci‐) Krylov‐Tso estimate was known to depend on the diameter of the base of the cylinder. We get dependence either on the measure of the base or on the height of the cylinder. We also give bounds for subsolutions in noncylindrical domains. ©1995 John Wiley & Sons, Inc.