An application of a global lifting method for homogeneous Hörmander vector fields to the Gibbons conjecture
| Autor: | Stefano Biagi |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
Conjecture Applied Mathematics 010102 general mathematics Order (ring theory) Gibbons conjecture Homogeneous Hörmander operators One-dimensional symmetry Function (mathematics) Lambda Differential operator 01 natural sciences 010101 applied mathematics Rank condition Vector field Linear independence 0101 mathematics Analysis Mathematics |
| Popis: | In this paper we exploit a global lifting method for homogeneous Hormander vector fields in order to extend the Gibbons conjecture to any second-order differential operator $$\mathcal {L}_X = \sum _{j = 1}^mX_j^2$$, where the $$X_j$$’s are linearly independent smooth vector fields on $$\mathbb {R}^n$$ satisfying Hormander’s rank condition and fulfilling a suitable homogeneity property with respect to a family of non-isotropic dilations. The class of these operators comprehends the sub-Laplacians on Carnot groups, the smooth Grushin-type operators and the smooth $$\Delta _\lambda $$-Laplacians studied by Franchi, Lanconelli and Kogoj. We also establish a comparison result for the solutions of the semi-linear equation $$\mathcal {L}_Xu+f(u) = 0$$ under suitable assumptions on the function f. |
| Databáze: | OpenAIRE |
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