Stable solutions of elliptic equations on Riemannian manifolds with Euclidean coverings

Farina, Alberto; Sire, Yannick; Valdinoci, Enrico

doi:10.1090/S0002-9939-2011-11241-3

Stable solutions of elliptic equations on Riemannian manifolds with Euclidean coverings
HTML articles powered by AMS MathViewer

by Alberto Farina, Yannick Sire and Enrico Valdinoci PDF

Proc. Amer. Math. Soc. 140 (2012), 927-930 Request permission

Abstract:

We investigate the rigidity properties of stable, bounded solutions of semilinear elliptic partial differential equations in Riemannian manifolds that admit a Euclidean universal covering, finding conditions under which the level sets are geodesics or the solution is constant.

References

L. Dupaigne and A. Farina, Stable solutions of $-\Delta u=f(u)$ in $\Bbb R^N$, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 4, 855–882. MR 2654082, DOI 10.4171/JEMS/217
Ennio De Giorgi, Convergence problems for functionals and operators, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978) Pitagora, Bologna, 1979, pp. 131–188. MR 533166
Manuel del Pino, MichałKowalczyk, and Juncheng Wei, A counterexample to a conjecture by De Giorgi in large dimensions, C. R. Math. Acad. Sci. Paris 346 (2008), no. 23-24, 1261–1266 (English, with English and French summaries). MR 2473304, DOI 10.1016/j.crma.2008.10.010
Louis Dupaigne, Stable solutions of elliptic partial differential equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 143, Chapman & Hall/CRC, Boca Raton, FL, 2011. MR 2779463, DOI 10.1201/b10802
Alberto Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\Bbb R^N$, J. Math. Pures Appl. (9) 87 (2007), no. 5, 537–561 (English, with English and French summaries). MR 2322150, DOI 10.1016/j.matpur.2007.03.001
D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three-manifolds, Invent. Math. 82 (1985), no. 1, 121–132. MR 808112, DOI 10.1007/BF01394782
Doris Fischer-Colbrie and Richard Schoen, The structure of complete stable minimal surfaces in $3$-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), no. 2, 199–211. MR 562550, DOI 10.1002/cpa.3160330206
Alberto Farina, Berardino Sciunzi, and Enrico Valdinoci, Bernstein and De Giorgi type problems: new results via a geometric approach, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7 (2008), no. 4, 741–791. MR 2483642
Alberto Farina, Yannick Sire, and Enrico Valdinoci, Stable solutions of elliptic equations on Riemannian manifolds, preprint (2008), www.mat.uniroma2.it/ $\sim$valdinoc/manifold.pdf.
Alberto Farina and Enrico Valdinoci, The state of the art for a conjecture of De Giorgi and related problems, Recent progress on reaction-diffusion systems and viscosity solutions, World Sci. Publ., Hackensack, NJ, 2009, pp. 74–96. MR 2528756, DOI 10.1142/9789812834744_{0}004
Sylvestre Gallot, Dominique Hulin, and Jacques Lafontaine, Riemannian geometry, 2nd ed., Universitext, Springer-Verlag, Berlin, 1990. MR 1083149, DOI 10.1007/978-3-642-97242-3
Shuichi Jimbo, On a semilinear diffusion equation on a Riemannian manifold and its stable equilibrium solutions, Proc. Japan Acad. Ser. A Math. Sci. 60 (1984), no. 10, 349–352. MR 778527
Frank Pacard and Juncheng Wei, Stable solutions of the Allen-Cahn equation in dimension $8$ and minimal cones, preprint (2011), http://arxiv.org/abs/1102.3446.