One dimensional symmetry in the Heisenberg group
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Isabeau Birindelli [1] ; Jyotshana Prajapat [2]
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[1] Università de Roma La Sapienza
Università de Roma La Sapienza
Roma Capitale, Italia
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[2] Indian Statistical Institute
Indian Statistical Institute
India
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- Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 30, Nº 2, 2001, págs. 269-284
- Idioma: inglés
- Enlaces
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Resumen
Let HI denote the Heisenberg space and let u be a solution of + M(l - M~) = 0 in H" satisfying 1. Let x be any variable orthogonal to the anisotropic direction t. Assume that for xl going to plus or minus infinity u converges uniformly to 1 and -1 respectively. Under these assumptions we prove that u is a function depending only on xl and that it is monotone increasing. , This result, which is the analogue for the Heisenberg space of the weak formulation of a conjecture by De Giorgi, is obtained for a wider class of equations;
it is a consequence of the invariance of the Heisenberg Laplacian with respect to Heisenberg group. The proof requires a Maximum Principle for unbounded domains which is interesting by itself. We also consider the case when u satisfies the limit condition in the t direction and we conclude that the solution is monotone in t.
