On the method of moving planes and the sliding method

  • H. Berestycki
  • L. Nirenberg
Article

DOI: 10.1007/BF01244896

Cite this article as:
Berestycki, H. & Nirenberg, L. Bol. Soc. Bras. Mat (1991) 22: 1. doi:10.1007/BF01244896

Abstract

The method of moving planes and the sliding method are used in proving monotonicity or symmetry in, say, thex1 direction for solutions of nonlinear elliptic equationsF(x, u, Du, D2u)=0 in a bounded domain Ω in ℝn which is convex in thex1 direction. Here we present a much simplified approach to these methods; at the same time it yields improved results. For example, for the Dirichlet problem, no regularity of the boundary is assumed. The new approach relies on improved forms of the Maximum Principle in “narrow domains”. Several results are also presented in cylindrical domains—under more general boundary conditions.

Copyright information

© Sociedade Brasileira de Matemática 1991

Authors and Affiliations

  • H. Berestycki
    • 1
  • L. Nirenberg
    • 2
  1. 1.Lab. d'Analyse NumeriqueUniv. Paris VIParisFrance
  2. 2.Courant InstituteNew York UniversityNew York

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