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A Note on the Maximum Principle for Second-Order Elliptic Equations in General Domains

A Note on the Maximum Principle for Second-Order Elliptic Equations in General Domains
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摘要 We make a further advance concerning the maximum principle for second-order elliptic operators. We investigate in particular a geometric condition, first considered by Berestycki Nirenberg Varadhan, that seems to be natural in view of the application of the boundary weak Harnack inequality, on which our argument is based. Setting it free from some technical assumptions, apparently needed in earlier papers, we significantly enlarge the class of unbounded domains where the maximum principle holds, compatibly with the first-order term. We make a further advance concerning the maximum principle for second-order elliptic operators. We investigate in particular a geometric condition, first considered by Berestycki Nirenberg Varadhan, that seems to be natural in view of the application of the boundary weak Harnack inequality, on which our argument is based. Setting it free from some technical assumptions, apparently needed in earlier papers, we significantly enlarge the class of unbounded domains where the maximum principle holds, compatibly with the first-order term.
作者 Antonio VITOLO
机构地区 Department of Mathematics and Informatics
出处 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2007年第11期1955-1966,共12页 数学学报（英文版）
基金 Supported by MURST ex 60% 2005-06
关键词 elliptic equations ABP estimate maximum principle elliptic equations, ABP estimate, maximum principle
分类号 O175.25 [理学—基础数学]
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参考文献12

  • 1Cabre, X.: On the Alexandroff-Bakelman-Pucci estimate and reversed Holder inequality for solutions of elliptic and parabolic equations. Comm. Pure Appl. Math., 4,8, 539-570 (1995)
  • 2Meyers, N., Serrin, J.: The exterior Dirichlet problem for second order elliptic partial differential equations. J. Math. Mech., 9(4), 539-570 (1960)
  • 3Vitolo, A.: Remarks on the maximum principle and uniqueness estimates. Int. J. Pure Appl. Math., 9(1), 77-87 (2003)
  • 4Protter, M. H., Weinberger, H. F.: Maximum Principles in Differential Equations, Springer-Verlag, New York, 1997
  • 5Cafagna, V., Vitolo, A.: On the maximum principle for second-order elliptic operators in unbounded domains. U.R. Acad. Sci. Paris Ser.I, 334, 1-5 (2002)
  • 6Vitolo, A.: On the maximum principle for complete second-order elliptic operators in general domains. J. Differential Equations, 194, 166-184 (2003)
  • 7Berestycki, H., Nirenberg, L., Varadhan,S. R. S.: The principal eigenvalue and maximum principle for second-order elliptic operators in general domains. Comm. Pure Appl. Math., 4,7, 47-92 (1994)
  • 8Berestycki, H., Caffarelli, L., Nirenberg, L.: Monotonicity for elliptic equations in unbounded Lipscbitz domains. Comm. Pure Appl. Math., 50, 1089-1111 (1997)
  • 9Chaojiang, X.: Dirichlet problems for the quasilinear second order subelliptic equations. Acta Mathermatica Sinica, English Series, 12(1), 18-32 (1996)
  • 10Ma, L., Chen, D. Z., Yang, Y.: Some results on subelliptic equations. Acta Mathematica Sinica, English Series, 22(6), 1695-1704 (2006)
Acta Mathematica Sinica,English Series
2007年 第11期

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