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.


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


[1] H Amann, M. G. Crandall, On some existence theorems for semilinear elliptic equations, Indiana Univ. Math. J. 27 (1978) 779-790. · Zbl 0391.35030
[2] A. Bakelman, Convex functions and elliptic equations, Book, to appear. · Zbl 0495.35038
[3] H. Berestycki, L. Nirenberg, Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations. J. Geometry and Physics 5 (1988) 237-275. · Zbl 0698.35031
[4] H. Berestycki, Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains, Analysis et cetera, ed. P. Rabinowitz et al., Academic Pr. (1990) 115-164. · Zbl 0705.35004
[5] H. Berestycki, Travelling front solutions of semilinear equations inn dimensions, to appear in volume dedicated to J. L. Lions. · Zbl 0780.35054
[6] H. Berestycki, KPP in higher dimensions, to appear.
[7] H. Berestycki, L. Nirenberg, S. R. S. Varadhan, On principal eigenvalues of second order elliptic operators in general domains, to appear. · Zbl 1316.35217
[8] J. M. Bony, Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris, Ser. A, 265 (1967) 333-336. · Zbl 0164.16803
[9] L. Caffarelli, L. Nirenberg, J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations I. Monge-Ampère equation, Comm. Pure Appl. Math. 38 (1984) 369-402. · Zbl 0598.35047
[10] W. Craig, P. Sternberg, Symmetry of solitary waves, Comm. P.D.E. 13 (1988) 603-633. · Zbl 0651.76008
[11] S. Y. Cheng, S. T. Yau, On the regularity of the Monge-Ampère equation det(?2 u/?x i ?x j =F(x,u), Comm. Pure Appl. Math. 30 (1977) 41-68. · Zbl 0347.35019
[12] B. Gidas, W. M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979) 209-243. · Zbl 0425.35020
[13] B. Gidas, Symmetry of positive solutions of nonlinear elliptic equations in ? n , Math. Anal. and Applic., Part A, Advances in Math. Suppl. Studies 7A, ed. L. Nachbin, Academic Press (1981) 369-402.
[14] D. Gilbarg, N. S. Trudinger,Elliptic Partial Differential Equations of Second Order, 2nd Ed’n. Springer Verlag, 1983. · Zbl 0562.35001
[15] Li, Cong Ming, Monotonicity and symmetry of solutions of fully nonlinear elliptic equation I, Bounded domains, Comm. Partial Diff. Eq’ns., to appear.
[16] Li, Cong Ming, Ibid., Monotonicity and symmetry of solutions of fully nonlinear elliptic equation II, Unbounded domains, Comm. Part. Diff. Eq’ns., to appear.
[17] P. L. Lions, A remark on Bony’s maximum principle, Proc. Amer. Math. Soc., 88 (1983) 503-508.
[18] J. Serrin, A symmetry problem in potential theory. Arch. Rat. Mech. 43 (1971) 304-318. · Zbl 0222.31007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.