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

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Abstract

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 Święch in Nonlin. Diff. Eq. Appl. 11, 491–509, 2004) for fully nonlinear elliptic equations. Our results extend those of (Koike and Święch 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 Święch in Lecture Notes in Pure and Applied Mathematics, vol. 234. Dekker, New York, 2003) in the parabolic case.

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Authors and Affiliations

  1. Department of Mathematics, Saitama University, Sakura, Saitama, 338-8570, Japan

    Shigeaki Koike

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Andrzej Święch

Authors
  1. Shigeaki Koike
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  2. Andrzej Święch
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Correspondence to Shigeaki Koike.

Additional information

S. Koike was supported by Grant-in-Aid for Scientific Research (no. 16340032) of Japan Society for the Promotion of Science. A. Święch was supported by NSF grant DMS 0500270.

Dedicated to Hitoshi Ishii on the occasion of his 60th birthday.

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Koike, S., Święch, A. Maximum principle for fully nonlinear equations via the iterated comparison function method. Math. Ann. 339, 461–484 (2007). https://doi.org/10.1007/s00208-007-0125-z

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