Abstract
In the Heisenberg group framework, we obtain a geometric inequality for stable solutions of \({\Delta_{\mathbb{H}} u=f(u)}\) in a domain \({\Omega\subseteq\mathbb{H}}\) . More precisely, if we denote the horizontal intrinsic Hessian by Hu, the mean curvature of a level set by h, its imaginary curvature by p, the intrinsic normal by ν and the unit tangent by υ, we have that
for any \({\phi\in C^\infty_0(\Omega)}\) . Stable solutions in the entire \({{\mathbb{H}}}\) satisfying a suitably weighted energy growth and such that \({\langle T\nu,v \rangle_{\mathbb{H}}\ge0}\) are then shown to have level sets with vanishing mean curvature.
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References
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)
Arcozzi N., Ferrari F.: Metric normal and distance function in the Heisenberg group. Math. Z. 256(3), 661–684 (2007)
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
Arcozzi, N., Ferrari, F.: A variational approximation of the perimeter with second order penalization in the Heisenberg group (in preparation)
Balogh Z.: Size of characteristic sets and functions with prescribed gradient. J. Reine Angew. Math. 564, 63–83 (2003)
Berestycki H., Caffarelli L., Nirenberg L.: 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 (1997)
Birindelli I., Lanconelli E.: 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 (2002)
Birindelli I., Lanconelli E.: A negative answer to a one-dimensional symmetry problem in the Heisenberg group. Calc. Var. Partial Differ. Equ. 18(4), 357–372 (2003)
Birindelli I., Prajapat J.: Monotonicity and symmetry results for degenerate elliptic equations on nilpotent Lie groups. Pacific J. Math. 204, 1–17 (2002)
Birindelli, I., Valdinoci, E.: The Ginzburg-Landau equation in the Heisenberg group. Commun. Contemp. Math. 10(5) (2008)
Cabré X., Capella A.: Regularity of radial minimizers and extremal solutions of semilinear elliptic equations. J. Funct. Anal. 238(2), 709–733 (2006)
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)
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)
Cheng J.-H., Hwang J.-F., Malchiodi A., Yang P.: Minimal surfaces in pseudohermitian geometry. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4(1), 129–177 (2005)
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)
Farina, A.: Propriétés qualitatives de solutions d’équations et systèmes d’équations non-linéaires. Habilitation à diriger des recherches, Paris VI (2002)
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)
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)
Ferrari, F.: Curvature formulae of noncharacteristic smooth sets in the Heisenberg group, preprint (2006)
Ferrari, F., Valdinoci, E.: Some weighted Sobolev-Poincaré inequalities, preprint (2008)
Garofalo N., Danielli D., Nhieu D.-M.: Notion of convexity in Carnot groups. Comm. Anal. Geom. 11, 263–341 (2003)
Ghoussoub N., Gui C.: On a conjecture of De Giorgi and some related problems. Math. Ann. 311(3), 481–491 (1998)
Monti R., Serra Cassano F.: Surface measures in Carnot-Carathéodory spaces. Calc. Var. Partial Differ. Equ. 13(3), 339–376 (2001)
Pauls S.: Minimal surfaces in the Heisenberg group. Geom. Dedicata 104, 201–231 (2004)
Sire, Y., Valdinoci, E.: Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, preprint (2007)
Sternberg P., Zumbrun K.: A Poincaré inequality with applications to volume-constrained area-minimizing surfaces. J. Reine Angew. Math. 503, 63–85 (1998)
Sternberg P., Zumbrun K.: Connectivity of phase boundaries in strictly convex domains. Arch. Rational Mech. Anal. 141(4), 375–400 (1998)
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F. Ferrari is partially supported by GALA project Geometric Analysis in Lie groups and Applications, supported by the European Commission within the 6th Framework Programme and by the PRIN project Viscosity, metric and control theoretic methods in nonlinear partial differential equations, MIUR (Italy). E. Valdinoci is partially supported by the PRIN project Variational Methods and Nonlinear Differential Equations, MIUR (Italy).
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Ferrari, F., Valdinoci, E. A geometric inequality in the Heisenberg group and its applications to stable solutions of semilinear problems. Math. Ann. 343, 351–370 (2009). https://doi.org/10.1007/s00208-008-0274-8
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DOI: https://doi.org/10.1007/s00208-008-0274-8