On the method of moving planes and the sliding method

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Abstract

The method of moving planes and the sliding method are used in proving monotonicity or symmetry in, say, thex 1 direction for solutions of nonlinear elliptic equationsF(x, u, Du, D 2 u)=0 in a bounded domain Ω in ℝ n which is convex in thex 1 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.

dedicated to Shmuel Agmon