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The Liouville property and a conjecture of De Giorgi. (English) Zbl 1072.35526
The authors study bounded entire solutions of the equation \(\Delta u+u-u^3=0\) in \({\mathbb R}^d\) and prove that under certain monotonicity conditions these solutions must be constant on hyperplanes. The proof uses a Liouville theorem for harmonic functions associated with a nonuniformly elliptic divergence form operator.

MSC:
35J60 Nonlinear elliptic equations
35C05 Solutions to PDEs in closed form
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[1] Barlow, Canad J Math 50 pp 487– (1998) · Zbl 0912.31004
[2] Probabilistic techniques in analysis. Probability and Its Applications. Springer, New York, 1995.
[3] Diffusions and elliptic operators. Probability and Its Applications. Springer, New York, 1998.
[4] Bass, Ann of Math (2) 134 pp 253– (1991)
[5] ; ; One-dimensional symmetry of bounded entire solutions of some elliptic equations. Preprint.
[6] Caffarelli, Comm Pure Appl Math 47 pp 1457– (1994)
[7] Carbou, Ann Inst H Poincaré Anal Non Linéaire 12 pp 305– (1995)
[8] Carlen, Ann Inst H Poincaré Probab Statist 23 pp 245– (1987)
[9] Dang, Z Angew Math Phys 43 pp 984– (1992)
[10] Convergence problems for functionals and operators. Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), 131-188. Pitagora, Bologna, 1979.
[11] Fabes, Arch Rational Mech Anal 96 pp 327– (1986)
[12] Symmetry for solutions of semilinear elliptic equations in \(\mathbb{R}\)N and related conjectures. Preprint.
[13] ; ; Dirichlet forms and symmetric Markov processes. de Gruyter Studies in Mathematics, 19. de Gruyter, Berlin, 1994. · Zbl 0838.31001
[14] Ghoussoub, Math Ann 311 pp 481– (1998)
[15] Gidas, Comm Math Phys 68 pp 209– (1979)
[16] ; Elliptic partial differential equations of second order. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224. Springer, Berlin-New York, 1993.
[17] Heat kernel of a noncompact Riemannian manifold. Stochastic analysis (Ithaca, N.Y., 1993), 239-263. Proc. Sympos. Pure Math., 57. Amer. Math. Soc., Providence, R.I., 1995.
[18] ; Stochastic differential equations and diffusion processes. North-Holland Mathematical Library, 24. North-Holland, Amsterdam-New York; Kodansha, Tokyo, 1981.
[19] Jacod, Z Wahrscheinlichkeitstheorie und Verw Gebiete 38 pp 83– (1977)
[20] Ggr;-convergence to minimal surfaces problem and global solutions of {\(\Delta\)}u = 2(u3 - u). Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978) 223-244. Pitagora, Bologna, 1979.
[21] Modica, Comm Pure Appl Math 38 pp 679– (1985)
[22] Modica, Boll Un Mat Ital B (5) 17 pp 614– (1980)
[23] ; Diffusions, Markov processes, and martingales. Vol. 1. Foundations. Second edition. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, Chichester, 1994.
[24] Stroock, Ann Inst H Poincaré Probab Statist 33 pp 619– (1997)
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