Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2013 | 1 | 130-146
Tytuł artykułu

Nonexistence Results for Semilinear Equations in Carnot Groups

Treść / Zawartość
Warianty tytułu
Języki publikacji
In this paper, following [3], we provide some nonexistence results for semilinear equations in the the class of Carnot groups of type ★.This class, see [20], contains, in particular, all groups of step 2; like the Heisenberg group, and also Carnot groups of arbitrarly large step. Moreover, we prove some nonexistence results for semilinear equations in the Engel group, which is the simplest Carnot group that is not of type ★.
Słowa kluczowe
Opis fizyczny
  • Dipartimento di Matematica, Università di Bologna,
    Piazza di Porta S.Donato 5, 40126, Bologna, Italy
  • Dipartimento di Matematica, Università di Padova,
    Via Trieste 63, 35121, Bologna, Italy
  • L. Ambrosio, B. Kleiner, E. Le Donne, Rectifiability of sets of finite perimeter in Carnot groups: Existence of a tangent hyperplane, J. Geom. Anal. 19, 509-540 (2009) [WoS]
  • C. Bellettini, E. Le Donne, Regularity of sets with constant horizontal normal in the Engel group, to appear on Communications in Analysis and Geometry.
  • I. Birindelli, F. Ferrari, E. Valdinoci, Semilinear PDEs in the Heisenberg group: the role of the right invariant vector fields, Nonlinear Anal. 72,987-997 (2010)
  • I. Birindelli, E. Lanconelli, A negative answer to a one-dimensional symmetry problem in the Heisenberg group, Calc. Var. PDE 18, 357-372 (2003)
  • I. Birindelli, J. Prajapat, Monotonicity results for Nilpotent stratified groups, Pacific J. Math 204, 1-17 (2002)
  • A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer Monographs in Mathematics, (2007)
  • L. Capogna, D. Danielli, S. D. Pauls, J. T. Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Birkhäuser (2006).
  • W. L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordung, Math. Annalen 117, 98-105 (1939).
  • D. Danielli, N. Garofalo, D. M. Nhieu, Sub-Riemannian calculus on hypersurfaces in Carnot groups, Adv. Math. 215, 292-378 (2007). [WoS]
  • A. Farina, Propriétés qualitatives de solutions d’équations et systèmes d’équations non-linéaires. Habilitation à diriger des recherches, Paris VI (2002).
  • A. Farina, B. Sciunzi, E. Valdinoci, Bernstein and De Giorgi type problems: new results via a geometric approach, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7, 741-791 (2008).
  • A. Farina, E. Valdinoci, The state of the art for a conjecture of De Giorgi and related problems. In: Du, Y., Ishii, H., Lin, W.-Y. (eds.), Recent Progress on Reaction Diffusion System and Viscosity Solutions. Series on Advances in Mathematics for Applied Sciences, 372 World Scientific, Singapore (2008).
  • F. Ferrari, E. Valdinoci, A geometric inequality in the Heisenberg group and its applications to stable solutions of semilinear problems, Math. Annalen 343, 351-370 (2009). [WoS]
  • F. Ferrari, E. Valdinoci, Geometric PDEs in the Grushin plane: weighted inequalities and flatness of level sets, Int. Math. Res. Not. IMRN 22, 4232-4270 (2009).
  • F. Ferrari, E. Valdinoci, Some weighted Poincaré inequalities, Indiana Univ. Math. J. 58, 1619-1637 (2009).
  • B. Franchi, R. Serapioni, F. Serra Cassano, On the structure of finite perimeter sets in step 2 Carnot groups, J. Geom. Anal. 13, 421-466 (2003).
  • N. Garofalo, E. Lanconelli, Existence and nonexistence results for semilinear equations on the Heisenberg group, Indiana Univ. Math. J. 41, 71-98 (1992).
  • M. Gromov, Carnot-Carathéodory spaces seen from within, in Subriemannian Geometry, Progress in Mathematics, 144, Bellaiche, A. and Risler, J., Eds., Birkäuser Verlang, Basel 1996.
  • L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119, 147-171 (1967).
  • M. Marchi, Rectifiability of sets of finite perimeter in a class of Carnot groups of arbitrary step available at
  • R. Montgomery, A tour of Subriemannian geometries, their geodesics and applications, mathematical surveys and monographs, 91, Amer. Math. Soc., Providence 2002.
  • R. Monti, Distances, boundaries and surface measures in Carnot-Carathéodory spaces, UTM PhDTS, Department of Mathematics, University of Trento (2001), available at monti/pubblicazioni.html.
  • A. Pinamonti, E. Valdinoci, A geometric inequality for stable solutions of semilinear elliptic problems in the Engel group, Ann. Acad. Sci. Fenn. Math. 37, 357-373 (2012). [WoS]
  • P. Sternberg, K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math. 503, 63-85 (1998).
  • P. Sternberg, K. Zumbrun, Connectivity of phase boundaries in strictly convex domains, Arch. Ration. Mech. Anal. 141, 375-400 (1998).
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.