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On the method of moving planes and the sliding method. (English) Zbl 0784.35025
Summary: The method of moving planes and the sliding method are used in proving monotonicity or symmetry in, say, the $$x_ 1$$ direction for solutions of nonlinear elliptic equations $$F(x,u,Du,D^ 2u)=0$$ in a bounded domain $$\Omega$$ in $$\mathbb{R}^ n$$ which is convex in the $$x_ 1$$ 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.

##### MSC:
 35J60 Nonlinear elliptic equations 35B50 Maximum principles in context of PDEs 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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