A geometric inequality in the Heisenberg group and its applications to stable solutions of semilinear problems

Mathematische Annalen

Volumn 343, Issue 2, 2009, Pages 351-370

  Ferrari, Fausto ^a,b   Valdinoci, Enrico ^c  

^a UNIVERSITY OF BOLOGNA   (Italy)

^b CIRAM   (Italy)

^c Dipartimento di Matematica   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 57049170765     PISSN: 00255831     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00208-008-0274-8     Document Type: Article

References (27)

1. Alberti, G., Ambrosio, L., Cabré, X.: On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property. Special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday, Acta Appl. Math. 65(1-3), 9-33 (2001)
2. N. Arcozzi F. Ferrari 2007 Metric normal and distance function in the Heisenberg group Math. Z. 256 3 661 684
3. Arcozzi, N., Ferrari, F.: The Hessian of the distance from a surface in the Heisenberg group. Ann. Acad. Sci. Fenn. Math. 33(1), 35-63 (2008). http://mathstat.helsinki.fi/Annales/Vol33/vol33.html
4. Arcozzi, N., Ferrari, F.: A variational approximation of the perimeter with second order penalization in the Heisenberg group (in preparation)
5. Z. Balogh 2003 Size of characteristic sets and functions with prescribed gradient J. Reine Angew. Math. 564 63 83
6. H. Berestycki L. Caffarelli L. Nirenberg 1997 Further qualitative properties for elliptic equations in unbounded domains Dedicated to Ennio De Giorgi, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 1-2 69 94
7. I. Birindelli E. Lanconelli 2002 A note on one dimensional symmetry in Carnot groups Atti Accad. Naz. Lincei Cl. Sci. Fis. (9) Mat. Natur. Rend. Lincei Mat. Appl. 13 1 17 22
8. I. Birindelli E. Lanconelli 2003 A negative answer to a one-dimensional symmetry problem in the Heisenberg group Calc. Var. Partial Differ. Equ. 18 4 357 372
9. I. Birindelli J. Prajapat 2002 Monotonicity and symmetry results for degenerate elliptic equations on nilpotent Lie groups Pacific J. Math. 204 1 17
10. Birindelli, I., Valdinoci, E.: The Ginzburg-Landau equation in the Heisenberg group. Commun. Contemp. Math. 10(5) (2008)
11. X. Cabré A. Capella 2006 Regularity of radial minimizers and extremal solutions of semilinear elliptic equations J. Funct. Anal. 238 2 709 733
12. Capogna, L., Danielli, D., Pauls, S., Tyson, J.: An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem. Progress in Mathematics, 259. Birkhäuser Verlag, Basel (2007)
13. Capogna, L., Han, Q.: Pointwise Schauder estimates for second order linear equations in Carnot groups. Harmonic analysis at Mount Holyoke, Contemp. Math. vol. 320, pp. 45-69. Amer. Math. Soc., Providence (2003)
14. J.-H. Cheng J.-F. Hwang A. Malchiodi P. Yang 2005 Minimal surfaces in pseudohermitian geometry Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 1 129 177
15. De Giorgi, E.: Convergence problems for functionals and operators. Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), Pitagora, Bologna, pp. 131-188 (1979)
16. Farina, A.: Propriétés qualitatives de solutions d'équations et systèmes d'équations non-linéaires. Habilitation à diriger des recherches, Paris VI (2002)
17. Farina, A., Sciunzi, B., Valdinoci, E.: Bernstein and De Giorgi type problems: new results via a geometric approach. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2008, in press)
18. Farina, A., Valdinoci, E.: The state of the art for a conjecture of De Giorgi and related problems. Ser. Adv. Math. Appl. Sci., World Sci. Publ., Hackensack (2008, in press)
19. Ferrari, F.: Curvature formulae of noncharacteristic smooth sets in the Heisenberg group, preprint (2006)
20. Ferrari, F., Valdinoci, E.: Some weighted Sobolev-Poincaré inequalities, preprint (2008)
21. N. Garofalo D. Danielli D.-M. Nhieu 2003 Notion of convexity in Carnot groups Comm. Anal. Geom. 11 263 341
22. N. Ghoussoub C. Gui 1998 On a conjecture of De Giorgi and some related problems Math. Ann. 311 3 481 491
23. R. Monti F. Serra Cassano 2001 Surface measures in Carnot- Carathéodory spaces Calc. Var. Partial Differ. Equ. 13 3 339 376
24. S. Pauls 2004 Minimal surfaces in the Heisenberg group Geom. Dedicata 104 201 231
25. Sire, Y., Valdinoci, E.: Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, preprint (2007)
26. P. Sternberg K. Zumbrun 1998 A Poincaré inequality with applications to volume-constrained area-minimizing surfaces J. Reine Angew. Math. 503 63 85
27. P. Sternberg K. Zumbrun 1998 Connectivity of phase boundaries in strictly convex domains Arch. Rational Mech. Anal. 141 4 375 400