Principal frequency of p-sub-Laplacians for general vector fields

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.


Introduction
Let M be a smooth manifold of dimension n with a volume form dx. Let {X k } N k=1 with n ≥ N be a family of vector fields defined on M. We consider the operator It is is locally hypoelliptic if the commutators of the vector fields {X k } N k=1 generate the tangent space of M as the Lie algebra.
In addition, we define the p-sub-Laplacian for general vector fields by the formula and the horizontal gradients ∇ X := (X 1 , . . ., X N ) and ∇ * X := (X * 1 , . . ., X * N ), where with the formal adjoint Let Ω ⊂ M be an open set.We define the functional spaces We also consider the following functional . (1.4) Now let us define the functional class S1,p (Ω) to be the completion of C 1 0 (Ω) in the norm generated by J p , see in [3].
We consider the following Dirichlet boundary value problem for L p : L p u = λ|u| p−2 u, in Ω, u = 0, on ∂Ω. (1.5) In the Euclidean setting, the first eigenvalue for the Dirichlet boundary value problem for L p was obtained by Lindqvist in [16].In the sub-elliptic setting, the study of boundary value problems started by the pioneering works of Bony [2], Gaveau [9], Kohn and Nirenberg [15].In 1981, the Dirichlet problem for the Kohn Laplacian on the Heisenberg group was studied by Jerison in [13,14] as a part of his dissertation under the supervision of E. Stein.Also, 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.[4], [5], [6], [7], [8] and references therein.
As usual, a weak solution of equation (1.5) means a function u ∈ S1,p (Ω) such that for all φ ∈ S1,p (Ω).Similarly, by the sup-solution and sub-solution of equation (1.5) we mean a function u ∈ S1,p (Ω) such that and for all φ ∈ S1,p (Ω), respectively.Here λ ∈ R and u are called a Dirichlet eigenvalue and eigenfunction of the operator L p in Ω ⊂ M, respectively.If we put u instead of φ in (1.6), we have which can be written for u = 0 as Thus, one can define the principal Dirichlet frequency λ 1 (Ω) of the operator L p as the minimum of the Rayleigh quotient Ω |∇ X u| p dx/ Ω |u| p dx taken over all nontrivial functions u ∈ S1,p (Ω), that is, The main aim of this paper is to prove uniqueness, simplicity, and a domain monotonicity of the principal frequency (or the first eigenvalue) of the Dirichlet p-sub-Laplacian for general vector fields (1.2).We also present the Caccioppoli inequality for this operator in the form , where v > 0 is a sub-solution of (1.5) in Ω ⊂ M. Picone's identity for general vector fields plays an important role in some of our computations.Recall Picone's identity for differentiable functions u and v with v(x) = 0 in Ω ⊂ R n : where ∇ and •, • are the standard gradient and the inner product in R n , respectively.The Picone identity is one of the important tools in the theory of partial differential equations (see, e.g.[24]).As consequences of Picone's identity, one can obtain, for instance, the simplicity of principal eigenvalues, Barta's inequalities, nonexistence of positive solutions, Hardy's inequalities, and Sturmian comparison.In the Euclidean setting, the Picone identity was investigated by many authors and generalised in different directions.For example, in [1] Allegato and Huang extended the Picone identity to the case of any p > 1.Recently, in [11] and [12] Jaros presented the Picone identity for the Finsler p-Laplacian and obtained the Caccioppoli inequality, which has the form where v > 0 is a weak solution of ∆v = 0 in Ω ⊂ R n and 0 ≤ φ ∈ C ∞ 0 (Ω) is a test function.The present paper is motivated by Jaros' results.The inequality (1.12) can be derived from (1.11) by integrating over Ω with u = vφ, then applying the Cauchy-Schwartz inequality and the Young inequality.The L p -version of inequality (1.12) was obtained in [10], [17], and [20].Note that the authors in [21] have obtained the weighted anisotropic Hardy and Rellich inequalities making use of the (first and second order) anisotropic Picone identities.We also refer to a recent open access book [22] for further discussions in this direction.Here, we establish a version of Picone's identity for general vector fields: Then we have (1.15)Moreover, we have L(u, v) = 0 a.e. in Ω if and only if u = cv a.e. in Ω with a positive constant c.
A Carnot group (stratified group) version of the Picone identity was obtained in [23] for a general case.Also, 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 Hörmander's condition.This idea was later extended in [22,Section 11.6].Indeed, all the extensions are a direct reworking of the proof [18,Lemma 2.1].Therefore, here we omit the proof of Lemma 1.1.
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.In Section 3, we obtain the Caccioppoli inequalities for general vector fields making use of Picone's identity.Then, we present some examples on the Grushin plane and the Heisenberg group.

Principal frequency of p-sub-Laplacians
Lemma 2.1.Assume that there exists a strictly positive sup-solution of (1.5).Then we have for all functions u ∈ S1,p (Ω).
We recall that a sup-solution v of (1.5) has the following form for all φ ∈ S1,p (Ω).
Proof of Lemma 2.1.Suppose that v is a strictly positive sup-solution of (1.5) in Ω and assume that u ∈ C ∞ 0 (Ω).Then for a given small ε > 0 in (1.7) we set In the last line, we have used the Picone identity.Now by taking the limit as ε → 0 + and making use of Fatou's lemma on the left-hand side and the Lebesgue dominated convergence theorem on the right-hand side of above expression, we arrive at By a density argument, we assert that the claim is valid for S1,p (Ω).
Theorem 2.2.Let Ω ⊂ M be a bounded open set.If there exists λ and a strictly positive v ∈ S1,p (Ω) such that for every nonnegative function φ ∈ S1,p (Ω), then we have and for all u ∈ S1,p (Ω).
Note that here and after we denote by λ 1 the principal frequency of L p and by u 1 the corresponding positive eigenfunction without the existence argument, that is, we assume that for every φ ∈ S1,p (Ω).
Proof of Theorem 2.4.Let both u and u 1 be eigenfunctions associated to the principal frequency Taking the limit as ε → 0 + and using the Lebesgue dominated convergence theorem and Fatou's lemma, we arrive at Since L(u, u 1 ) ≥ in Ω, it follows that L(u, u 1 ) = 0 a.e. in Ω.So it follows from Lemma 1.1 that u = Cu 1 with C > 0.
Then, by using the Picone identity, we show monotonicity of the principal frequency λ 1 (Ω) in Theorem 2.2 as the function of the set Ω.

Caccioppoli inequalities for general vector fields
Here we give the Caccioppoli inequalities for general vector fields by making use of Picone's identity, and we discuss some of their corollaries on the Grushin plane and the Heisenberg group.Theorem 3.1 (Caccioppoli inequality).Let v be a positive sub-solution of (1.5) in Ω ⊂ M. Then for every fixed q > p − 1, 1 < p < ∞, and λ ∈ R we have for all nonnegative functions φ ∈ C ∞ 0 (Ω).Note that in the case when q = p and λ = 0 in (3.1) we have Proof of Theorem 3.1.Setting u := v q/p φ in L(u, v) we have In the last line, we have used the equality and recalling Young's inequality in the form Here we have used Ω |∇ X u| p dx ≤ Ω λ|u| p dx for the sub-solution of (1.5).Thus, we arrive at Now we choose the suitable constant as s = q−p+1 p which leads to This proves Theorem 3.1.
Let us discuss some particular cases of the Caccioppoli inequalities on the Grushin plane and the Heisenberg group.One of the important examples of a sub-Riemannian manifold is the Grushin plane.Recall that the Grushin plane G is the space R 2 with vector fields and the gradient Then for every fixed q > p − 1 and 1 < p < ∞ we have and for q = p and λ = 0 we have for all nonnegative functions φ ∈ C ∞ 0 (Ω).
Let H n be the Heisenberg group, that is, R 2n+1 equipped with the group law where ξ := (x, y, t) ∈ R 2n+1 with x ∈ R n , y ∈ R n , t ∈ R. The dilation operation on the Heisenberg group with respect to the group law has the form δ λ (ξ) := (λx, λy, λ 2 t) for λ > 0.
The Heisenberg group is a basic example of step 2 stratified Lie groups (Carnot groups).
The Lie algebra h of the left-invariant vector fields on the Heisenberg group H n is spanned by with their (non-zero) commutator The horizontal gradient on H n is given by ∇ H := (X 1 , . . ., X n , Y 1 , . . ., Y n ), so the sub-Laplacian on H n is given by Note that in the case of the general stratified Lie groups G (Carnot groups) we have X k = −X * k , so the formula (1.2) can be rewritten as which is called the p-sub-Laplacian.Then for every fixed q > p − 1 and 1 < p < ∞ we have Ω v q−p φ p |∇ H v| p dx ≤ p q − p + 1 p Ω v q |∇ H φ| p dx + λp q − p + 1 Ω v q φ p dx, (3.11) and for q = p and λ = 0 we have for all nonnegative functions φ ∈ C ∞ 0 (Ω).Here the Caccioppoli inequality on the Heisenberg group seems to be also new.