Resumen de Maximum principle for fully nonlinear equations via the iterated comparison function method

Shigeaki Koike, Andrzej Swiech

• We present various versions of generalized Aleksandrov-Bakelman-Pucci (ABP) maximum principle for L p -viscosity solutions of fully nonlinear second-order elliptic and parabolic equations with possibly superlinear-growth gradient terms and unbounded coefficients. We derive the results via the "iterated" comparison function method, which was introduced in our previous paper (Koike and Swiech in Nonlin. Diff. Eq. Appl. 11, 491-509, 2004) for fully nonlinear elliptic equations. Our results extend those of (Koike and Swiech in Nonlin. Diff. Eq. Appl. 11, 491-509, 2004) and (Fok in Comm. Partial Diff. Eq. 23(5-6), 967-983) in the elliptic case, and of (Crandall et al. in Indiana Univ. Math. J. 47(4), 1293-1326, 1998; Comm. Partial Diff. Eq. 25, 1997-2053, 2000; Wang in Comm. Pure Appl. Math. 45, 27-76, 1992) and (Crandall and Swiech in Lecture Notes in Pure and Applied Mathematics, vol. 234. Dekker, New York, 2003) in the parabolic case.