Well-posedness of a Neumann-type problem on a gauge ball in H-type groups

We discuss the existence and uniqueness of solution for the second boundary value problem of potential theory often referred to as the Neumann problem, on a gauge ball for the canonical sub-Laplacian in H-type groups. In this way we extend the classical results of the problem as well as its generalization to the Heisenberg group.


Introduction
In the study of partial differential equations, two boundary value problems associated with the Laplace equation occupy a special place, namely the Dirichlet and Neumann problems. While the Dirichlet problem asks to obtain a harmonic function in a domain whose value agrees with a prescribed (continuous) function on the boundary, the Neumann problem requires the normal derivative of the solution function to agree with a prescribed function [17]. If denotes the Laplacian on the Euclidean space R n , the Neumann problem for a smooth domain Ω in R n is finding a u ∈ C 2 (Ω) ∩ C 1 (Ω) such that where f is a prescribed function on ∂Ω and n is the outward normal at the boundary ∂Ω. For domains having C 1,α boundary where α is in (0, 1], for example, a unit ball, the Dirichlet problem is solvable for any continuous boundary value (see, for details, [10]) whereas the Neumann problem is solvable under the essential condition that the integral of the values assigned to the normal derivative vanishes over the boundary surface (see, for details, [14]), i.e., ∂Ω f ds = 0.
A straight generalization of these boundary value problems involving the Laplacian in the case of R n by replacing the operator with an arbitrary elliptic operator has been vastly discussed, and an extensive literature is available [7,10,14,18,19]. These problems become more interesting when the regularity of the differential operator involved is compromised. The conditions of being hypoelliptic and subelliptic, that are well exposed in the classical book [4], are weaker than ellipticity. In some sense, the first and nicest example of a subelliptic operator is the Kohn-Laplacian on the Heisenberg group. The discussions of boundary value problems involving the sub-Laplacian sometimes are parallel to the classical cases, while are considerably different in certain other cases. The Dirichlet problem for the Kohn-Laplacian on the Heisenberg groups was first discussed by Gaveau [9] and then settled completely by Jerison [11,12]. The Neumann boundary value problem on the Heisenberg group has been recently discussed by Dubey et al. [6]. The Heisenberg group case, however, comes with an explicit group law and the underlying manifold structure same as that of the Euclidean space R 2n+1 . Therefore, the case of Heisenberg group in [6], up to a certain extent, is same as dealing with a subelliptic operator on Euclidean spaces. The point where it shows contrast with the Euclidean setup is the Green's-type identity where, instead of the Lebesgue measure, one needs a different volume and surface element in the integrals.
The immediate generalization, namely the H-type groups, is by definition the range of the exponential map on an H-type Lie algebra. The group law on an H-type group, therefore, is not as explicit as in the case of Heisenberg group. Moreover, the manifold structure, which is still trivial, does not enjoy a natural coordinate system and there arises a need to find one that is most compatible with the Lie algebra structure. It is natural to see the extent to which the techniques can be generalized and equally natural to start with the H-type groups. The identification of characteristic points on the boundary and estimation of integrals involved are two focal areas in the work of Dubey et al. [6]. The case of H-type groups in this article has established that similar results are available in this setup and hence opens the scope to study Neumann boundary value problems for subelliptic operators on a more general class of 2-step nilpotent Lie groups [2,3].
The plan of our article is as follows. In Sect. 2, we refer to [3,5] and review some basic aspects of the H-type groups, sub-Laplacian, and the horizontal normal vectors. Analogous to the normal derivative, we define a similar operator in Sect. 3 to deal with the characteristic points and formulate the Neumann boundary value problem. In the subsequent sections, we prove a few results as a build-up to the main result in this article.
Then g is called an H-type Lie algebra if for all Z ∈ z, where I is the identity mapping. A connected and simply connected Lie group G with g as the associated Lie algebra is called an H-type group. The Heisenberg group H n is a trivial example of an H-type group and a nontrivial example will be discussed in the next section. We identify g with the corresponding simply connected Lie group G under the exponential map. From [16], the product in G is given by Denote by p and q the dimensions of v and z, respectively, so that p + 2q is the homogeneous dimension of G. When X is regarded as a left-invariant vector field on G at the point (X , Z ), it can be represented as We fix {X 1 , X 2 , . . . , X p } and {Z 1 , Z 2 , . . . , Z q } as orthonormal bases for v and z, respectively, so that {X 1 , X 2 , . . . X p , Z 1 , Z 2 , . . . , Z q } is an orthonormal basis of g. Then X = p j=1 X, X j X j and Z = q j=1 Z, Z j Z j for every X ∈ v and Z ∈ z. The canonical sub-Laplacian on G is given by Define p(α) = ( |X| is the fundamental solution of the sub-Laplacian, that is, where g β (α) = g(β -1 α) and δ β is the Dirac distribution with pole at β ∈ G.
Just for the sake of calculations, we use a slightly modified kernel Φ where Unless stated otherwise, all the derivatives of Φ would be taken with respect to the first variable α.
An infinitesimal metric that is consistent with the automorphisms of an H-type group happens to be a sub-Riemannian metric obtained by Korányi in [15]. We define for all Y , Y ∈ g where B denotes the killing form, θ is the Cartan involution, and 4b 2 = (p + 4q) -1 .

Definition 2.2 A vector at any point
. The length of a horizontal vector is given by X α 0 = |X| where | · | denotes the norm induced by ·, · on v.
Vectors that are not horizontal are said to have infinite length. We call the unique horizontal vector ∇ 0 f , the horizontal gradient of a function f on G which is defined as is an orthonormal basis in v. The outward horizontal normal unit vector at each point of the boundary of a domain {F < 0} where ∇ 0 F 0 = 0 is given by

Formulation of the problem
The unit horizontal normal vector in Eq. (2.5) is undefined at the "characteristic" points, i.e., the points where ∇ 0 F vanishes. When F is smooth, the set of characteristic points forms a lower-dimensional subset of the boundary. In the following example, we explicitly calculate the set of characteristic points on an H-type group.
Example 3.1 Consider the group with the following binary operation: Here, R 2 serves as the center of G. We consider the Euclidean inner product on G. The left-invariant vector field X j on G that equals ∂ ∂x j at the origin is given by where t k (j,i) is the (j, i)th entry in T k . The left-invariant vector field Z j on G that equals ∂ ∂z j at the origin is given by Z j = ∂ ∂z j . Denote by g the associated Lie algebra of G and represent it as We also denote {X j } 4 j=1 and {Z k } k=1,2 as the orthonormal bases for v and z, respectively. Clearly, [g, z] = {0} and [g, g] ⊆ z. For any X ∈ v and Z ∈ z, we define a map J Z : v → v as Now, let F(X, Z) = |X| 4 16 + |Z| 2 -1 = 0 be a smooth surface in G. Substituting ∇ 0 F = 0, we obtain which further gives X = 0. Hence, the set of characteristic points is given by {(0, Z) : |Z| 2 -1 = 0} which is a sphere of dimension (q -1). Thus, the set of characteristic points is a lower-dimensional subset of the boundary, which finishes the example.
Now we formulate the Neumann problem. Let Ω be a bounded domain in G whose boundary is given as a level set of a smooth function T, i.e., ∂Ω = {γ ∈ G : T(γ ) = 0}. To deal with the set of characteristic points in G, we define a new class of functions f (x) exists for all characteristic points where the limit is taken with respect to the relative topology ofΩ. Define the operator ∂f ∂n 0 (γ 0 ) i fγ 0 is non-characteristic point on ∂Ω.
As the set of characteristic points is a lower-dimensional subset of ∂Ω, the Green's formula [16,Eq. (1.9)] can be rewritten as and ds is the surface element on ∂Ω determined by the Euclidean measure. The homogeneous Neumann problem for a domain Ω in G is to find a function u ∈ C(Ω) such that where g ∈ C(∂Ω).

Uniqueness
Both V andṼ are G -harmonic and are respectively called the single-and the doublelayer potentials with density φ.

Lemma 4.2 (Green's first identity) Let Ω be any bounded domain in G having boundary of class C ∞ and u, v be C 1 functions onΩ, then
Proof The Divergence theorem [8, Corollary 3.11] when applied to v∇ 0 u proves the above lemma. Proof The difference u = u 1u 2 of two solutions of the Neumann problem is a harmonic function in Ω, continuous up to boundary, satisfying the homogeneous boundary condition ∂ ⊥ u = 0 on ∂Ω. Using Lemma 4.2, we get Thus, ∇ 0 u = 0. Further using Lemma 4.3, u must be a constant.
In the rest of the article, Ω will denote the open unit ball {α = (X, Z) ∈ G : p(α) < 1} in G.

∂Ω φ(α)Φ(α, β) dS(α) exists and V is continuous throughout G.
Proof From Theorem 2.1 and Eq. (2.3), we have It can be observed from [16] that dS = 4|X|.|p(α)| 2 (|X| 4 +64) 1 2 ds, where ds is the Euclidean surface element. As Ω β (R) is bounded, we have for some positive constant c 1 . Let n denote the unit inward normal to the surface ∂Ω at the point β. We denote by Π the map Ω β (R) α → αα, n n. Clearly, Π is a projection of the set Ω β (R) onto the tangent space T β to ∂Ω at the point β. The range of Π lies inside P β := Ω β (R) ∩ T β and for sufficiently small R, the map Π is a bijection on its range. Moreover, R may be chosen small enough so that the surface element ds satisfies for some positive constant M. Let P e denote the set {βα : α ∈ P β }. Then P e = Ω e (R) ∩ L where L is the hyperplane n · g = 0 in G. By translation invariance of the surface measure ds induced by the Lebesgue measure, we have where the equation of the hyperplane L 0 is x 1 = 0 when in standard local coordinates, X is represented as (x 1 , x 2 , . . . , x p ). From [1, Theorem 5.12], we parameterize Ω e (R) ∩ L 0 using the polar coordinates (a, ϕ,x,z) ∈ (0, R) × (0, π 2 ] × S p-2 × S q-1 in the following manner: . ,x p ),z =Z Z , and the measure ds is given by where dx and dz denote the usual surface area measures on S p-2 and S q-1 , respectively. By the substitution relations, we have a = p(X,Z) = p(X, Z). If (X,Z) is a point that corresponds to Π(α) for α ∈ Ω β (R) then a ≤ p(α) so that a m ≥ p(α) m as m < 0. Using [1, Hence, the integral exists on Ω β (R). Moreover, we have Hence, the integral exists for all β ∈ ∂Ω. Further, since dS is a Radon measure, a routine proof may be given to establish the uniform continuity of convolution of two integrable functions with respect to dS over a compact set. In particular, V is continuous.
whereβ denotes the unit normal in the direction of β and limit is in the sense of uniform convergence over the compact neighborhoods of β. In what follows, we show that the limits exist and hence determine their values.

Theorem 4.10
Let φ ∈ C(∂Ω). Then it is possible to extendṼ from Ω toΩ and from G \Ω to G \ Ω in a continuous fashion with the following limiting values: Proof By Corollary 4.8, the above integral is a continuous function on ∂Ω. Take N ∂Ω (h 0 ) as in the previous lemma and writeṼ in the following form: and Now using Lemma 4.9, the proof follows.
Proof The proof follows along similar lines to the proof of Theorem 4.10. As ∂Ω is of class C 1 and by using the estimate (4.5), we can easily conclude that W 1 , W 2 are compact. Also with respect to the dual system C(∂Ω), C(∂Ω) , defined by φ, ψ := ∂Ω φψ dS φ, ψ ∈ C(∂Ω), the operators W 1 and W 2 are adjoint.

Theorem 5.1
The operators I + W 1 , I + W 2 have one-dimensional nullspaces.