Abstract
In this paper, we prove the uniqueness and simplicity of the principal frequency (or the first eigenvalue) of the Dirichlet p-sub-Laplacian for general vector fields. As a byproduct, we establish the Caccioppoli inequalities and also discuss the particular cases on the Grushin plane and on the Heisenberg group.
Highlights
Let M be a smooth manifold of dimension n with a volume form dx
We define the p-sub-Laplacian for general vector fields by the formula
The main aim of this paper is to prove uniqueness, simplicity, and a domain monotonicity of the principal frequency of the Dirichlet p-subLaplacian for general vector fields (1.2)
Summary
Let M be a smooth manifold of dimension n with a volume form dx. Let {Xk}Nk=1 with n ≥ N be a family of vector fields defined on M. Semilinear equations on the Heisenberg group for sums of vector fields were studied in [25] and [19] Those works attracted significant attention to boundary value problems for the sub-elliptic operators, see e.g. Niu, Zhang, and Wang in [18] obtained the Picone identity on the Heisenberg group and remarked that it could be extended to general vector fields satisfying Hormander’s condition. This idea was later extended in [22, Section 11.6]. The paper is organised in the following way: In Section 2, we prove the uniqueness and simplicity of the principal frequency (or the first eigenvalue) of the Dirichlet p-sub-Laplacian for general vector fields. We present some examples on the Grushin plane and the Heisenberg group
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