On the method of moving planes and the sliding method
- Cite this article as:
- Berestycki, H. & Nirenberg, L. Bol. Soc. Bras. Mat (1991) 22: 1. doi:10.1007/BF01244896
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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.