The Aronsson equation for absolute minimizers of L^{∞}-functionals associated with vector fields satisfying Hörmander’s condition

Changyou Wang Department of Mathematics, University of Kentucky Lexington, KY 40506

Abstract. Given a Carnot-Carathéodory metric space (R^{n}, dX) generated by vector fields {X_{i}}_{i=1}^{m} satisfying Hörmander’s condition, we prove in theorem A that any absolute minimizer u ∈ WX^{1,∞}(Ω) to F (v, Ω) = esssup_{x∈Ω}f(x, Xv(x)) is a viscosity solution to the Aronsson equation (1.6), under suitable conditions on f. In particular, any AMLE is a viscosity solution to the subelliptic ∞-Laplacian equation (1.7). If the Carnot-Carathéodory space is a Carnot group G and f is independent of the x-variable, we establish in theorem C the uniquness of viscosity solutions to the Aronsson equation (1.13), under suitable conditions on f. As a consequence, the uniqueness of both AMLE and viscosity solutions to the subelliptic ∞-Laplacian equation is established on any Carnot group G.

§1. Introduction

Variational problems in L^{∞} are very important because of both their analytic difficulties and their frequent appearance in applications. (See the survey article by Barron [B]). The study began with Aronsson [A1, 2]. The simplest model is to consider minimal Lipschitz extensions (or MLE): for a bounded, Lipschitz domain Ω ⊂ R^{n} and g ∈ Lip(Ω), find u ∈ W^{1,∞}(Ω), with u|_{∂Ω} = g, such that

∥Du∥_{L^{∞}(Ω)} ≤ ∥Dw∥_{L^{∞}(Ω)}, ∀w ∈ W_{0}^{1,∞}(Ω) with w|_{∂Ω} = g.  (1.1)

Since MLE may be neither unique nor smooth, Aronsson [A1] introduced the notation of absolutely minimizing Lipschitz extensions(or AMLE for short), and proved that any C^{2} AMLE solves the ∞-Laplacian equation:

Δ_{∞}u := − ∑_{i,j=1}^{n} \frac{∂u}{∂x_{i}} \frac{∂u}{∂x_{j}} \frac{∂^{2}u}{∂x_{i}∂x_{j}} = 0, in Ω.  (1.2)

However, (1.2) is a fully nonlinear, degenerate PDE and may not have C^{2} solutions. The issue was finally settled by Jensen [J2], who not only established the equivalence between the AMLE and the viscosity solution to eqn.(1.2)(a notion of weak solutions first introduced by Crandall-Lions [CL], see also Crandall-Ishii-Lions [CIL]), but also proved the uniqueness of viscosity solutions to eqn.(1.2) with Dirichlet boundary data. The remarkable analysis of [J2] involves approximation by p-Laplacian and Jensen’s maximum principle for semiconvex functions [J1]. The reader can consult with Evans [E] and Lindqvist-Manfredi [LM] for qualitative estimates on ∞-harmonic functions. Crandall-Evans-Gariepy [CEG] developed the comparison principle for viscosity solutions of the eqn.(1.2) with cones, which are of the forms a + b|x − x_{0}|, and gave an alternative proof of the equivalence between AMLE and viscosity solution to eqn.(1.2). Furthermore, Crandall-Evans [CE] have utilized this property in their study on the regularity issue of ∞-harmonic functions. Recently, Barron-Jensen-Wang [BJW] considered general L^{∞}-functionals

F (u, Ω) := esssup_{x∈Ω}f(x, u(x), Du(x)), ∀u ∈ W^{1,∞}(Ω).

and proved, under suitable conditions on f, that any absolute minimizer of F (·, Ω) is a viscosity solution to the Aronsson equation:

− ∑_{i=1}^{n} f_{p_{i}}(x, u(x), Du(x))\frac{∂}{∂x_{i}} (f(x, u(x), Du(x))) = 0, in Ω.

Shortly after [BJW], Crandall [C] gave an elegant proof of an improved version of [BJW]. Through [BJW] and [C], it becomes clear that the classical solution to the Hamilton-Jacobi equation f(x, ϕ(x), Dϕ(x)) − c = 0 plays important roles in the analysis. We would like to mention that there is a very interesting survey paper available on this subject by Aronsson-Crandall-Juntinen [ACJ].

Since the notion of AMLE can easily be formulated in any metric space, it is a very natural and interesting problem to study AMLE in Carnot-Carathéodory metric spaces, which include Riemannian manifolds and Subriemannian manifolds arising from Hörmander’s vector fields (e.g. Heisenberg groups, Carnot groups). There have been several works in this direction. For example, Juutinen [J] extended the main theorems of [J2] into Riemannian manifolds. Bieske [B1,2] proved the uniqueness of viscosity solutions to the subelliptic ∞-Laplacian equation on the Heisenberg group H^{n} and the Grushin type space. Inspired by [C], Bieske-Capogna [BC] proved that any AMLE is a viscosity solution to the subelliptic ∞-Laplacian equation for any Carnot group G. In general, it was proved by [BC] that any horizontally C^{1} AMLE is a viscosity solution to the subelliptic ∞-Laplacian equation associated with free systems of Hörmander’s vector fields. Bieske [B3] recently extended [BC] to the Grushin type space.

In this paper, we are mainly interested in the Aronsson equation of AMLE’s and its uniqueness issue for any Carnot-Carathédory metric space generated by vector fields satisfying Hörmander’s condition. In this direction, we prove that any AMLE is a viscosity solution to the subelliptic ∞-Laplacian equation. Moreover, if the vector fields are horizontal vector fields of a Carnot group G, then both AMLE and viscosity solution to the Aronsson equation are unique. In fact some more general results are provided by theorems A and C below.

In order to state our results, we first recall some preliminary facts.

For a bounded domain Ω ⊂ R^{n} let X_{i} ∈ C^{0,1}(Ω, R^{n}), 1 ≤ i ≤ m, be given Lipschitz continous vector fields on Ω. We say that an absolutely continuous curve r : [0, δ] → Ω is admissible if there are measurable functions a_{j}(t) : [0, δ] → R, 1 ≤ j ≤ m, such that

∑_{j=1}^{m} a_{j}^{2}(t) ≤ 1, r′(t) = ∑_{j=1}^{m} a_{j}(t)X_{j}(r(t)), for a.e. t ∈ [0, δ].

For any x, y ∈ Ω, we define the Carnot-Carathéodory distance by

dX(x, y) = inf{T |∃ an admissible curve r : [0, T] → Ω, with r(0) = x, r(T) = y}. (1.3)

Definition 1.1 (Carnot-Carathédory space). For a bounded domain Ω ⊂ R^{n} and a family of Lipschitz continuous vector fields, {X_{i}}_{i=1}^{m}, on Ω. If the above defined d_{X}(·, ·) satisfies the three properties for the distance function, then we call (Ω, dX) a Carnot-Carathéodory space, i.e. there exists at least one admissible curve joining any two points in Ω. Furthermore, we assume throughout this paper that dX is uniformly continuous with respect to the euclidean norm ∥ · ∥, namely

ω(δ) = sup{dX(x, y) : x, y ∈ Ω, with ∥x − y∥ ≤ δ} → 0, as δ → 0.  (1.4)

We will focus on two classes of Carnot-Carathéodory spaces: (i) Hörmander’s vector fields, and (ii) Carnot groups.

Example 1.2 (Hörmander’s vector fields). Suppose that {X_{i}}_{i=1}^{m} ⊂ C^{∞}(Ω, R^{n}) satisfy Hörmander’s finite rank condition, i.e. there exists a step r ≥ 1 such that {X_{i}(x)}_{i=1}^{m} and their commutators up to at most order r span R^{n} for any x ∈ Ω. Then it follows from Nagel-Stein-Wainger [NSW] that dX, defined by (1.4), is a distance. Moreover, for each compact set K ⊂⊂ Ω there exists a C_{K} > 0 so that

C_{K}^{−1}∥x − y∥ ≤ dX(x, y) ≤ C_{K}∥x − y∥^{\frac{1}{r}} , ∀x, y ∈ K  (1.5)

where ∥ · ∥ is the Euclidean norm on R^{n}.

The description of Carnot groups will be given later in this section. We now recall the definition of horizontal Sobolev spaces and Lipschitz spaces.

For u : Ω → R, denote Xu := (X_{1}u, ⋯, X_{m}u) as the horizontal gradient of u. For 1 ≤ p ≤ ∞, the horizontal Sobolev space is defined by

WX^{1,p}(Ω) ≡ {u : Ω → R | ∥u∥_{WX^{1,p}(Ω)} ≡ ∥u∥_{L^{p}(Ω)} + ∥Xu∥_{L^{p}(Ω)} < ∞}.

The Lipschitz space is defined by

LipX(Ω) := \{u : Ω → R | ∥u∥_{LipX(Ω)} ≡ sup_{x,y∈Ω,x≠y} \frac{|u(x) − u(y)|}{dX(x, y)} < ∞\}.

It was proved by Garofalo-Nieu [GN] and Franchi-Serapioni-Serra [FSS] that for any Carnot-Carathédory space (Ω, dX) u ∈ LipX(Ω) iff u ∈ WX^{1,∞}(Ω).

We now give the definition of absolute minimizers of L^{∞}-functionals over WX^{1,∞}(Ω).

Definition 1.3 (absolute minimizers). For any nonnegative, continuous function f ∈ C(Ω × R^{m}), let

F (v, Ω) = esssup_{x∈Ω}f(x, Xv(x)), ∀v ∈ WX^{1,∞}(Ω).

A function u ∈ WX^{1,∞}(Ω) is an absolute minimizer of F (·, Ω), if for any open subset Ω̃ ⊂ Ω and w ∈ WX^{1,∞}(Ω̃), with w = u on ∂Ω̃, we have

F (u, Ω̃) ≤ F (w, Ω̃).  (1.6)

u ∈ WX^{1,∞}(Ω) is called an absolutely minimizing Lipschitz extension (or AMLE), if u is an absolute minimizer of F (·, Ω) when f(x, p) = |p|^{2}, for (x, p) ∈ Ω × R^{m}.

Formal calculations yield that an absolute minimizer u ∈ WX^{1,∞}(Ω) to F (·, Ω) satisfies the Aronsson equation

− ∑_{i=1}^{m} X_{i}(f(x, Xu(x)))f_{p_{i}}(x, Xu(x)) = 0, in Ω.  (1.7)

In particular, the Aronsson equation of an AMLE is the (subelliptic) ∞-Laplacian equation

Δ_{∞}^{(X)}u := − ∑_{i,j=1}^{m} X_{i}uX_{j}uX_{i}X_{j}u = 0, in Ω.  (1.8)

In order to show that an absolute minimizer is a solution to the eqn. (1.7), we need to recall the notion of viscosity solutions to (1.7), which is a natural extension of that developed by Crandall-Lions [CL] (see also [CIL]) of second order (degenerate) elliptic PDEs.

Let S^{m} denote the set of symmetric m×m matrices, equipped with the usual order. A function A ∈ C(R^{n} × R^{m} × S^{m}) is called horizontally elliptic, if, for any (x, p) ∈ R^{n} × R^{m}

A(x, p, M) ≤ A(x, p, N), ∀M, N ∈ S^{m}, with N ≤ M.  (1.9)

Let (D^{2}u)^{∗} ∈ S^{m} denote the horizontal hessian of u, defined by

(D^{2}u)_{ij}^{∗} = \frac{1}{2}(X_{i}X_{j} + X_{j}X_{i})u, ∀1 ≤ i, j ≤ m.

We now have

Definition 1.4. Consider a horizontally elliptic equation:

A(x, Xu(x), (D^{2}u)^{∗}(x)) = 0, in Ω.  (1.10)

A function u ∈ C(Ω) is called a viscosity subsolution to eqn.(1.10), if for any pair (x_{0}, ϕ) ∈ Ω × C^{2}(Ω) such that x_{0} is a local maximum point of (u − ϕ) then we have

A(x_{0}, Xϕ(x_{0}), (D^{2}ϕ)^{∗}(x_{0})) ≤ 0.  (1.11)

A function u ∈ C(Ω) is called a viscosity supersolution to eqn.(1.10) if −u is a viscosity subsolution to eqn.(1.10). Finally, a function u ∈ C(Ω) is a viscosity solution to eqn.(1.10) if it is both a viscosity subsolution and a viscosity supersolution to eqn.(1.10).

Remark 1.5. If {X_{i}}_{i=1}^{m} ∈ C^{2}(Ω, R^{n}), then we can define the horizontal C^{2} space, Γ^{2}(Ω), by

Γ^{2}(Ω) = {u : Ω → R | X_{i}u, X_{i}X_{j}u ∈ C(Ω), 1 ≤ i, j ≤ m}.

In [B1,2], [BC], Manfredi [M], the authors introduced a slightly different notion of viscosity solution to the eqn. (1.10) by requiring the test functions ϕ ∈ Γ^{2}(Ω) in the definition 1.4. Since C^{2}(Ω) ⊂ Γ^{2}(Ω), the notion of viscosity solution introduced here is weaker than that of [B1,2], [BC], and [M]. However, when the underlying Carnot-Carathéodory space is induced by a Carnot group, then these two notions are equivalent (see [B1]).

It is easy to check that both eqn. (1.7) and (1.8) are horizontally elliptic equation. Now we are ready to state our first theorem.

Theorem A. Assume that {X_{i}}_{i=1}^{m} ⊂ C^{2}(Ω, R^{n}) and (Ω, dX) is a Carnot-Carathéodory space. Suppose that u ∈ WX^{1,∞}(Ω) is an absolute minimizer of

F (v, Ω) = esssup_{x∈Ω}f(x, Xv(x)), v ∈ WX^{1,∞}(Ω)

where f ∈ C^{2}(Ω × R^{m}, R_{+}) satisfies (f1) for any x ∈ Ω, f(x, ·) is quasiconvex:

f(x, tp_{1} + (1 − t)p_{2}) ≤ max{f(x, p_{1}), f(x, p_{2})}, ∀p_{1} p_{2} ∈ R^{m}, 0 ≤ t ≤ 1.  (1.12)

(f2) f is homogeneous of degree α ≥ 1 and f_{p}(0, 0) = 0. Then u is a viscosity solution to the Aronsson equation (1.7).

The ideas to prove theorem A are based on: (1) the observation of rewrite eqn.(1.7) into an euclidean form which enables us to adopt Crandall’s construction [C] of solutions to the Hamilton-Jacobi equation as test functions (see also [BJW]), and (2) the comparison principle of viscosity solutions to Hamilton-Jacobi equations without u-dependence (see [CIL] or [BJW]).

Since f(x, p) ≡ |p|^{2} : Ω × R^{m} → R_{+} satisfies both (f1) and (f2), we have, as a consequence of theorem A,

Corollary B. Assume that {X_{i}}_{i=1}^{m} ⊂ C^{2}(Ω, R^{n}) and (Ω, dX) is a Carnot-Carathéodory space. Suppose that u ∈ WX^{1,∞}(Ω) is an AMLE. Then u is a viscosity solution to the (subelliptic) ∞-Laplacian equation (1.8).

Now we turn to the discussion on the uniqueness problem of absolute minimizers of F (·, Ω) and viscosity solutions to eqn.(1.7). For this purpose, we restrict our attention to the case where the vector fields generating the Carnot-Carathédory metric are the horizontal vector fields associated with a Carnot group G.

Recall that a Carnot group of step r ≥ 1 is a simply connected Lie group G whose Lie algebra g admits a vector space decomposition in r layers g = V^{1}+V^{2}+⋯+V^{r} having two properties: (i) g is stratified, i.e., [V^{1}, V^{j}] = V^{j+1}, j = 1, ⋯, r − 1; (ii) g is r-nilpotent, i.e. [V^{j}, V^{r}] = 0, j = 1, ⋯, r. V^{1} is called the horizontal layer and V^{j}, j = 2, ⋯, r, are vertical layers. It is well-known (cf. Folland-Stein [FS]) that the exponential map, exp : g → G, is a global differmorphism so that we can identify G with g ≡ R^{n} via exp. and G has an exponential coordinate system; here n = dim(G) is the topological dimension of G. More precisely, Let X_{i,j}, 1 ≤ i ≤ m_{j} = dim(V^{j}), be a basis of V^{j}, for 1 ≤ j ≤ r, which is orthonormal with respect to an arbitrarily chosen euclidean norm ∥ · ∥ on g, with respect to which the V^{j}’s are mutually orthogonal. Then p ∈ G has coordinate (p_{ij})_{1≤i≤m_{j},1≤j≤r} if p = exp(∑_{j=1}^{r} ∑_{i=1}^{m_{j}}(p_{ij}X_{i,j}). Let · denote the group multiplication on G. Then it is known ([FS]) that the group law (x, y) → x·y is a polynomial map with respect to the exponential map. From now on, we set m = m_{1} = dim(V^{1}) and denote X_{i} = X_{i,1} for 1 ≤ i ≤ m. Two bi-Lipschitz equivalent metrics, on G, we need are: (1) the Carnot-Carathédory metric dX on G generated by {X_{i}}_{i=1}^{m}; (2) the gauge metric d on G given as follows. For p = (p_{ij})_{1≤i≤m_{j},1≤j≤r},

∥p∥_{G}^{2r!} = ∑_{j=1}^{r}(∑_{i=1}^{m_{j}} |p_{ij}|^{2})^{\frac{r!}{j}} ,

with the induced gauge distance

d(x, y) = ∥x^{−1}y∥_{G}, ∀x, y ∈ G.

satisfying the invariant property:

d(z · x, z · y) = d(x, y), ∀x, y, z ∈ G.

Now we mention the Heisenberg group H^{n}, the simplest Carnot group of step two. H^{n} ≡ C^{n} × R endowed with the group law: for (z_{1}, ⋯, z_{n}, t), (z′_{1}, ⋯, z′_{n}, t′) ∈ C^{n} × R

(z_{1}, ⋯, z_{n}, t) · (z′_{1}, ⋯, z′_{n}, t′) = (z_{1} + z′_{1}, ⋯, z_{n} + z′_{n}, t + t′ + 2Im(∑_{i=1}^{n} z_{i}z̄′_{i})),

whose Lie algebra h = V_{1} + V_{2} with V_{1} = span{X_{i}, Y_{i}}_{1≤i≤n} and V_{2} = span{T }, where

T = 4\frac{∂}{∂t}, X_{i} = \frac{∂}{∂x_{i}} − 2y_{i} \frac{∂}{∂t}, Y_{i} = \frac{∂}{∂y_{i}} + 2x_{i} \frac{∂}{∂t}, 1 ≤ i ≤ n.

Recall that a function f ∈ C^{2}(R^{m}) is strictly convex if there is a C_{0} > 0 such that D^{2}f(x) ≥ C_{0} for any x ∈ R^{m}. Now we are ready to state the uniqueness theorem.

Theorem C. Let G be a Carnot group and Ω ⊂ G be a bounded domain. Assume that f ∈ C^{2}(R^{m}, R_{+}) is strictly convex, homogeneous of degree α ≥ 1, and f(p) > 0 for p ≠ 0. Then, for any ϕ ∈ WX^{1,∞}(Ω), the Dirichlet problem

A(Xu, (D^{2}u)^{∗}) := − ∑_{i,j=1}^{m} f_{p_{i}}(Xu)f_{p_{j}}(Xu)X_{i}X_{j}u = 0, in Ω,  (1.13)
u = ϕ, on ∂Ω.

has at most one viscosity solution in C(Ω̄).

Although the operator A is horizontally elliptic, one can check that the operator Ā(x, Du, D^{2}u) ≡ A(Xu, (D^{2}u)^{∗}) has x-dependence and is not elliptic in the euclidean sense (see [CIL] for its definition). Therefore, the uniqueness theorems, by [J1], Ishii [I], or Jensen-Lions-Souganidis [JLS], on viscosity solutions to 2nd order elliptic PDEs, are not applicable directly. Our ideas are: (i) We observe that eqn.(1.13) is invariant under group multiplications: for any a ∈ G, if u ∈ C(G) is a viscosity solution to eqn.(1.13), then u_{a}(x) = u(a · x) : G → R is also a viscosity solution to eqn.(1.13). This enables us to extend the sup/inf convoluation construction by [JLS] to G to convert viscosity subsolutions (or supersolutions, resp.) of eqn.(1.13) into semiconvex subsolutions (or semiconcave supersolutions, resp.). (ii) We modify the original arguments from [J1] and [J2] to prove a comparison principle between semiconvex subsolutions and semiconcave strict supersolutions to any 2nd order horizontally elliptic equations. (iii) We adopt the approximation scheme of p-Laplacians by [J2] to build viscosity solutions to two auxiliary equations having the property that any supersolution can be converted into strict supersolution under small perturbations. (iv) Finally, we apply the comparison principle for the two auxiliary equations to prove the uniqueness of eqn.(1.13).

As a consequence of theorem A and C, we have Corollary D. Let G be a Carnot group and Ω ⊂ G be a bounded domain. Assume that f ∈ C^{2}(R^{m}, R_{+}) is strictly convex, homogeneous of degree α ≥ 1, and f(p) > 0 for p ≠ 0. Then, for any ϕ ∈ WX^{1,∞}(Ω), there is a unique absolute minimizer u ∈ WX^{1,∞}(Ω), with u|_{∂Ω} = ϕ, to the functional F (v, Ω) = esssup_{x∈Ω}f(Xv(x)), and the eqn.(1.13) has a unique viscosity solution in C(Ω̄). In particular, ϕ has a unique AMLE in WX^{1,∞}(Ω) and the subelliptic ∞-Laplacian eqn. (1.8) has a unique viscosity solution.

We would like to remark that our method is of euclidean nature, which is considerably different from that by [B1,2] and [M], where a subelliptic approach was employed.

The paper is written as follows. In §2, we outline the proof of theorem A. In §3, we discuss the sup/inf convolution construction on any Carnot group G. In §4, we discuss the comparison principle between semiconvex subsolutions and strict semiconcave supersolutions to any horizontally elliptic equations. In §5, we study two auxiliary equations to eqn. (1.13), with horizontal gradient constraints. In §6, we prove theorem C.

Acknowledgement. The author would like to thank the referee for his many helpful comments. This work is partially supported by NSF grant 0400718.

§2. Proof of theorem A

This section is devoted to the proof of theorem A. It consists of two steps: (i) the construction of test functions by solving the Hamilton-Jacobi equation, motivated by [BJW] and [C], and (ii) the comparison between viscosity subsolution and classical strict supersolution of the Hamilton-Jacobi equation.

Proof of theorem A. It suffices to prove that if u is not a viscosity subsolution of eqn.(1.12) at the point x = 0 ∈ Ω, then u is not an absolute minimizer of F (·, Ω). This assumption implies that there exists r_{0} > 0 and ϕ ∈ C^{2}(Ω) such that B_{r_{0}}(0) ⊂⊂ Ω, and

0 = u(0) − ϕ(0) ≥ u(x) − ϕ(x), ∀x ∈ Ω  (2.1)

but

− ∑_{i=1}^{m} X_{i}(f(x, Xϕ(x)))f_{p_{i}}(x, Xϕ(x))|_{x=0} = C_{0} > 0.  (2.2)

Now we have

Lemma 2.1. There exist a neighborhood V of 0 and a function Φ ∈ C^{2}(V) such that

Φ(0) = ϕ(0), DΦ(0) = Dϕ(0), D^{2}Φ(0) > D^{2}ϕ(0),  (2.3)

and

f(x, XΦ(x)) = f(0, Xϕ(0)) > 0, ∀x ∈ V.  (2.4)

Proof. Since {X_{i}}_{i=1}^{m} ⊂ C^{2}(Ω, R^{n}), there is a matrix-valued function (a_{ij}) ∈ C^{2}(Ω, R^{mn}) such that for 1 ≤ i ≤ m,

X_{i}(x) = ∑_{j=1}^{n} a_{ij}(x)\frac{∂}{∂x_{j}} , ∀x ∈ Ω.

Define f̄: Ω × R^{n} → R by

f̄(x, q_{1}, ⋯, q_{n}) = f(x, ∑_{j=1}^{n} a_{1j}(x)q_{j}, ⋯, ∑_{j=1}^{n} a_{mj}(x)q_{j}), ∀(x, q_{1}, ⋯, q_{n}) ∈ Ω × R^{n}.

Since f ∈ C^{2}(Ω × R^{m}) and (a_{ij}) ∈ C^{2}(Ω, R^{mn}), it is easy to see that f̄ ∈ C^{2}(Ω × R^{n}). Moreover, for any (x, q) ∈ Ω × R^{n} and 1 ≤ i ≤ n, we have

\frac{∂f̄}{∂q_{i}}(x, q) = ∑_{k=1}^{m} a_{ki}(x)\frac{∂f}{∂p_{k}} (x, ∑_{j=1}^{n} a_{1j}(x)q_{j}, ⋯, ∑_{j=1}^{n} a_{mj}(x)q_{j}),

and

f̄(x, Dϕ(x)) = f(x, X_{1}ϕ(x), ⋯, X_{m}ϕ(x)) = f(x, Xϕ(x)), ∀x ∈ Ω.  (2.5)

Therefore, for any x ∈ Ω, we have

∑_{j=1}^{m} X_{j}(f(x, Xϕ(x)))f_{p_{j}}(x, Xϕ(x)) = ∑_{i=1}^{n} \frac{∂}{∂x_{i}} (f̄(x, Dϕ(x))\frac{∂f̄}{∂q_{i}}(x, Dϕ(x)).  (2.6)

This, combined with (2.2), implies

A(0, Dϕ(0), D^{2}ϕ(0)) := − ∑_{i=1}^{n} \frac{∂}{∂x_{i}} (f̄(x, Dϕ))\frac{∂f̄}{∂q_{i}}(x, Dϕ)(0) = C_{0} > 0.  (2.7)

Now the argument in [C] page 275-276 implies that there exist a neighborhood V of 0 and a function Φ ∈ C^{2}(V) such that

Φ(0) = ϕ(0), DΦ(0) = Dϕ(0), D^{2}Φ(0) > D^{2}ϕ(0),  (2.8)
f̄(x, DΦ(x)) = f̄(0, Dϕ(0)), ∀x ∈ V.  (2.9)

Now (2.8) and (2.9), combined with (2.5), give (2.4). For the convenience of readers, we briefly sketch Crandall’s argument as follows. Let E = D_{x}f̄(0, Dϕ(0)), F = −D_{q}f̄(0, Dϕ(0)), and G = E − D^{2}ϕ(0)F. Then (2.7) becomes G · F > 0. It is easy to check that X = P + D^{2}ϕ(0), where P is given by

P Z = \frac{Z · G}{F · G}G + Z − \frac{Z · F}{|F |^{2}} F, ∀Z ∈ R^{n}

has the property that X > D^{2}ϕ(0) and E − XF = 0. We now define

ψ(x) = ϕ(0) + x · Dϕ(0) + \frac{1}{2}(Xx · x)

and the hyperplane

Σ = {x ∈ R^{n} | x · {D_{x}f̄(0, Dϕ(0)) + D^{2}ϕ(0)D_{q}f̄(0, Dϕ(0))} = 0},

and solve the Cauchy problem:

f̄(x, DΦ(x)) = f̄(0, Dϕ(0)), and Φ = ψ on Σ

subject to

DΦ(0) = Dϕ(0)

in a neighhorhood of 0. Since Σ is not characteristic near 0, we have a local C^{2} solution Φ to the Cauchy problem. The classical theory also implies that D^{2}Φ(0) = X > D^{2}ϕ(0). This proves (2.8) and (2.9).

To see f(0, Xϕ(0)) > 0, we observe that (2.2) implies

f_{p}(0, Xϕ(0)) ≡ (\frac{∂f}{∂p_{1}} (0, Xϕ(0)), ⋯, \frac{∂f}{∂p_{m}} (0, Xϕ(0))) ≠ 0.

This implies Xϕ(0) ≠ 0, for f_{p}(0, 0) = 0. Note that the homogeneity of f implies that if f(0, p) = 0 then p = 0. Therefore, f(0, Xϕ(0)) > 0. This finishes the proof of Lemma 2.1.

It follows from Lemma 2.1 that there exists an open neighborhood V_{1} ⊂ V of 0 such that Φ(0) = ϕ(0) = u(0), and Φ(x) > ϕ(x) ≥ u(x) for any 0 ≠ x ∈ V_{1}. Therefore, for any small ϵ > 0, there exists another neighborhood V_{ϵ} ⊂ V_{1} of 0 such that

Φ(x) − ϵ < u(x), ∀x ∈ V_{ϵ}; Φ(x) − ϵ = u(x), ∀x ∈ ∂V_{ϵ}.  (2.10)

It follows from the absolute minimality of u to F (·, Ω) that

F (u, V_{ϵ}) ≤ F (Φ − ϵ, V_{ϵ}) = ess sup_{x∈V_{ϵ}}f(x, XΦ(x)) = f(0, Xϕ(0)).  (2.11)

Now we want to show that u is a viscosity subsolution of the Hamilton-Jacobi equation (2.4) on V_{ϵ}. For the reader’s convenience, we first recall from [CIL] the definition of viscosity solution to (2.4).

Definition 2.2. A function v ∈ C(V_{ϵ}) is a viscosity subsolution to the Hamilton-Jacobian equation:

f̄(x, Dv(x)) = K, x ∈ V_{ϵ}  (2.12)

where K > 0 is a constant, if for any (x, ϕ) ∈ V_{ϵ} × C^{1}(V_{ϵ}) such that ϕ touches v at x from above, we have

f(x, Dϕ(x)) ≤ K.

We remark that if v ∈ C^{1}(V_{ϵ}) is a classical subsolution to (2.12), then v is also a viscosity subsolution to (2.12).

We also recall the following compactness theorem and a comparison principle for eqn. (2.12).

Proposition 2.3 ([CIL]). (1) Assume that {v_{k}} ⊂ C(V_{ϵ}) are a family of viscosity subsolutions to eqn.(2.12) with K = C_{k}. If v_{k} → v uniformly on V_{ϵ} and C_{k} → C > 0, then v ∈ C(V_{ϵ}) is a viscosity subsolution to (2.12) with K = C. (2). If v ∈ C(V̄_{ϵ}) is a viscosity subsolution to eqn. (2.12) and w ∈ C^{1}(V̄_{ϵ}) is a strict, classical supersolution to eqn. (2.12), i.e.

f̄(x, Dw(x)) > K, ∀x ∈ V_{ϵ},

then we have

sup_{V_{ϵ}}(v − w) = sup_{∂V_{ϵ}}(v − w).  (2.13)

Now we have

Lemma 2.4. Under the same notations as above, u ∈ WX^{1,∞}(V_{ϵ}) is a viscosity subsolution to the Hamilton-Jacobi equation (2.4).

Proof. For any subdomain U ⊂⊂ V_{ϵ} and 0 < δ < dist(U, ∂V_{ϵ}) = inf{∥x − y∥, x ∈ U, y ∈ ∂V_{ϵ}}, let g_{δ} : U → R be the standard δ-mollifier of g for any function g on V_{ϵ}. Since u ∈ WX^{1,∞}(V_{ϵ}), u_{δ} converges uniformly to u on U as δ → 0. Since (f1) implies f(x, ·) is quasiconvex, it follows from the Jensen inequality for quasiconvex functions (cf. [BJW] theorem 1.1) that for any x ∈ U, we have

f(x, (Xu)_{δ}(x)) ≤ ess sup_{y∈B_{δ}(x)}f(x, Xu(y))
≤ F (u, V_{ϵ}) + Cδ
≤ f(0, Xϕ(0)) + Cδ,  (2.14)

where C = sup{|D_{x}f|(x, p) : x ∈ U, |p| ≤ 2∥Xu∥_{L^{∞}(Ω)}} < ∞. Therefore we have

max_{x∈U} f(x, (Xu)_{δ}(x)) ≤ f(0, Xϕ(0)) + Cδ.  (2.15)

On the other hand, for any 1 ≤ i ≤ m and x ∈ U, we estimate (X_{i}u)_{δ}(x) − X_{i}(u_{δ})(x) as follows

(X_{i}u)_{δ}(x) − X_{i}(u_{δ})(x)

= ∫_{R^{n}} η_{δ}(x − y)(∑_{j=1}^{n} a_{ij}(y)\frac{∂}{∂y_{j}})(u(y) − u(x)) dy

− ∫_{R^{n}} ∑_{j=1}^{n} a_{ij}(x)\frac{∂η_{δ}(x − y)}{∂x_{j}}(u(y) − u(x)) dy

= ∑_{j=1}^{n} ∫_{R^{n}}\{−\frac{∂}{∂y_{j}} (a_{ij}(y)η_{δ}(x − y)) − a_{ij}(x)\frac{∂(η_{δ}(x − y))}{∂x_{j}}\}(u(y) − u(x)) dy

= ∑_{j=1}^{n} ∫_{R^{n}}(a_{ij}(y) − a_{ij}(x))\frac{∂η_{δ}(x − y)}{∂x_{j}}(u(y) − u(x)) dy

+ ∑_{j=1}^{n} ∫_{R^{n}} \frac{∂a_{ij}(y)}{∂y_{j}}η_{δ}(x − y)(u(y) − u(x)) dy.

Therefore we have, for any x ∈ U,

|(X_{i}u)_{δ}(x) − X_{i}(u_{δ})(x)| ≤ C max_{1≤j≤n} ∥Da_{ij}∥_{L^{∞}(Ω)} ∫_{R^{n}}{|η_{δ}(x − y)|u(y) − u(x)|

+ |y − x||Dη_{δ}(x − y)||u(y) − u(x)|} dy

≤ C∥X_{i}∥_{C^{1}(Ω)} sup_{∥y−x∥≤δ} |u(y) − u(x)|

≤ C∥X_{i}∥_{C^{1}(Ω)}∥u∥_{WX^{1,∞}(V_{ϵ})}ω(δ)

where ω(δ) = sup{dX(x, y) : ∥y − x∥ ≤ δ} is the modular of continuity of dX with respect to the euclidean distance ∥ · ∥. This implies

f(x, X(u_{δ})(x)) ≤ esssup_{x∈U}f(x, (Xu)_{δ}(x)) + ∥f_{p}∥_{L^{∞}}∥X(u_{δ}) − (Xu)_{δ}∥_{L^{∞}(U)}
≤ f(0, Xϕ(0)) + C(δ + ω(δ)), ∀x ∈ U.  (2.16)

Hence u_{δ} is a classical subsolution to eqn. (2.12), with K = K_{δ} ≡ f(0, Xϕ(0)+C(δ+ω(δ)), on U. Since (1.4) implies lim_{δ→0} ω(δ) = 0, we have that K_{δ} → f(0, Xϕ(0)) as δ → 0. Notice also that u_{δ} → u uniformly on U, Proposition 2.3 (1) implies that that u is a viscosity subsolution to the eqn. (2.4) in U. Since U exhausts V_{ϵ} as δ → 0, we have that u is a viscosity subsolution of the eqn. (2.4) in V_{ϵ}. The proof of Lemma 2.4 is complete.

Now we return to the proof of theorem A. It follows from (f2) that

f(x, (1 + t)p) = (1 + t)^{α}f(x, p) = (1 + g(t))f(x, p), ∀t > 0, ∀(x, p) ∈ Ω × R^{n},

where g(t) ≡ (1 + t)^{α} − 1 > 0 for t > 0 and α ≥ 1. This, combined with (2.8), implies that for any t > 0 we have

f(x, X((1 + t)Φ_{ϵ})(x)) = (1 + g(t))f(0, Xϕ(0)) = f(0, Xϕ(0)) + δ(t), ∀x ∈ V_{ϵ}, (2.17)

where Φ_{ϵ} ≡ Φ − ϵ and δ(t) = g(t)f(0, Xϕ(0)) > 0. Therefore, for any t > 0, (1 + t)Φ_{ϵ} is a strict, classical supersolution of eqn. (2.4). Therefore, Proposition 2.3 (2) implies

sup_{V_{ϵ}}(u − (1 + t)Φ_{ϵ}) ≤ sup_{∂V_{ϵ}}(u − (1 + t)Φ_{ϵ}), ∀t > 0.  (2.18)

Taking t into zero, we have

sup_{V_{ϵ}}(u − Φ_{ϵ}) ≤ sup_{∂V_{ϵ}}(u − Φ_{ϵ}) = 0.

This implies

u(x) ≤ Φ_{ϵ}(x), ∀x ∈ V_{ϵ}.

This clearly contradicts with (2.10). Therefore the proof of theorem A is complete.

§3. The construction of sup/inf convolutions on G

This section is devoted to the construction of sup/inf convolutions on any Carnot group G, which is the necessary extension of Jensen-Lions-Souganidis [JLS] and plays an important role in the proof of theorem C.

Let Ω ⊂ G be a bounded domain and d : G × G → R_{+} be the gauge distance defined in §1. For any ϵ > 0, define

Ω_{ϵ} = {x ∈ Ω| inf_{y∈G\Ω} d(x^{−1}, y^{−1})^{2r!} ≥ ϵ}.

Definition 3.1. For any u ∈ C(Ω̄) and ϵ > 0, the sup convolution, u_{ϵ}, of u is defined by

u^{ϵ}(x) = sup_{y∈Ω̄}\{u(y) − \frac{1}{2ϵ}d(x^{−1}, y^{−1})^{2r!}\}, ∀x ∈ Ω.  (3.1)

Similarly, the inf convolution, v_{ϵ}, of v ∈ C(Ω̄) is defined by

v_{ϵ}(x) = inf_{y∈Ω̄}\{v(y) + \frac{1}{2ϵ}d(x^{−1}, y^{−1})^{2r!}\}, ∀x ∈ Ω.  (3.2)

For x ∈ G, let ∥x∥ := (∑_{j=1}^{r} ∑_{i=1}^{m_{j}} x_{ij}^{2})^{\frac{1}{2}} be the euclidean norm. We recall the notion of semiconvexity in the Eucliean sense.

Definition 3.2. A function u ∈ C(Ω̄) is semiconvex, if there is a constant C > 0 such that u(x) + C∥x∥^{2} is convex; u is semiconcave if −u is semiconvex. Notice that if u ∈ C^{2}(Ω) satisfies D^{2}u(x) ≥ −C for x ∈ Ω, then u is semiconvex, here D^{2}u = (\frac{∂^{2}u}{∂x_{i}∂x_{j}})_{1≤i,j≤n} is the euclidean hessian of u.

We now have the generalized version of [JLS].

Proposition 3.3. For u, v ∈ C(Ω̄), denote R_{0} = 2 max{∥u∥_{L^{∞}(Ω)}, ∥v∥_{L^{∞}(Ω)}}. Then, for any ϵ > 0, u^{ϵ}, v_{ϵ} ∈ WX^{1,∞}(Ω) satisfy:

(1) u^{ϵ} is semiconvex and v_{ϵ} is semiconcave.

(2) u^{ϵ} (or v_{ϵ}, resp.) is monotonically nondecreasing (or nonincreasing, resp.) w.r.t. ϵ and converges uniformly to u on Ω_{(1+4R_{0})ϵ}.

(3) if u (or v, resp.) is a viscosity subsolution (or supersolution, resp.) to the horizontally elliptic equation:

B(Xu, (D^{2}u)^{∗}) = 0 in Ω  (3.3)

then u^{ϵ} (or v_{ϵ}, resp.) is a viscosity subsolution (or supersolution, resp.) to eqn.(3.3) in Ω_{(1+4R_{0})ϵ}.

Proof. Since the proof for v_{ϵ} is parallel to that for u^{ϵ}, we only consider u^{ϵ}. (1) Since Ω ⊂ G is bounded, it is easy to see from the formula of d that

C_{d}(Ω) ≡ ∥D_{x}^{2}{d(x^{−1}, y^{−1})^{2r!}}∥_{L^{∞}(Ω̄×Ω̄)} < ∞.

Therefore, for any y ∈ Ω̄, the function

ũ_{y}^{ϵ}(x) := u(y) − \frac{1}{2ϵ}d(x^{−1}, y^{−1})^{2r!} + \frac{C_{d}(Ω)}{2ϵ}|x|^{2}, ∀x ∈ Ω

has nonnegative hessian and is convex. Since the supremum of a family of convex functions is convex, we have that

u_{ϵ}(x) + \frac{C_{d}(Ω)}{2ϵ}|x|^{2} = sup_{y∈Ω̄} ũ_{y}^{ϵ}(x)

is convex so that u_{ϵ} is semiconvex. Since any semiconvex function is Lipschitz with respect to the euclidean metric, we have u^{ϵ} ∈ WX^{1,∞}(Ω).

(2) It is easy to see that for any ϵ_{1} < ϵ_{2}, u^{ϵ_{1}}(x) ≤ u^{ϵ_{2}}(x) and u(x) ≤ u^{ϵ}(x) ≤ R_{0} for any x ∈ Ω. Observe that for any x ∈ Ω we have

u^{ϵ}(x) = sup_{Ω̄∩\{d(x^{−1},y^{−1})^{2r!}≤4R_{0}ϵ\}}(u(y) − \frac{1}{2ϵ}d(x^{−1}, y^{−1})^{2r!}).

Therefore, for any x ∈ Ω_{(1+4R_{0})ϵ}, u^{ϵ}(x) is attained at a point y ∈ Ω. To see u^{ϵ} → u uniformly on Ω_{(1+4R_{0})ϵ}, we observe that if u^{ϵ}(x) is attained by x_{ϵ} then

u\frac{ϵ}{2} (x) ≥ u(x_{ϵ}) − \frac{1}{ϵ}d(x^{−1}, x_{ϵ}^{−1})^{2r!} = u_{ϵ}(x) − \frac{1}{2ϵ}d(x^{−1}, x_{ϵ}^{−1})^{2r!}.

This implies

lim_{ϵ→0} \frac{1}{ϵ}d(x^{−1}, x_{ϵ}^{−1})^{2r!} = 0

so that x_{ϵ} → x, and lim_{ϵ→0} u^{ϵ}(x) = u(x). Since

|u^{ϵ}(x_{1}) − u^{ϵ}(x_{2})| ≤ |u(x_{1}) − u(x_{2})|, ∀x_{1}, x_{2} ∈ Ω,

u^{ϵ} → u uniformly on Ω_{(1+4R_{0})ϵ}.

(3) For any x_{0} ∈ Ω_{(1+4R_{0})ϵ}, let ϕ ∈ C^{2}(Ω_{(1+4R_{0})ϵ}) be such that

u^{ϵ}(x_{0}) − ϕ(x_{0}) ≥ u^{ϵ}(x) − ϕ(x), ∀x ∈ Ω_{(1+4R_{0})ϵ}.

It follows from the proof of (2) above that there exists a y_{0} ∈ Ω such that

u^{ϵ}(x_{0}) = u(y_{0}) − \frac{1}{2ϵ}d(x_{0}^{−1}, y_{0}^{−1})^{2r!}.

Therefore we have

u(y_{0}) − \frac{1}{2ϵ}d(x_{0}^{−1}, y_{0}^{−1})^{2r!} − ϕ(x_{0}) ≥ u(y) − \frac{1}{2ϵ}d(x^{−1}, y^{−1})^{2r!} − ϕ(x), ∀x, y ∈ Ω_{(1+4R_{0})ϵ}.

For y near y_{0}, we have x = x_{0} · y_{0}^{−1} · y ∈ Ω_{(1+4R_{0})ϵ} and hence

u(y_{0}) − ϕ(x_{0} · y_{0}^{−1} · y_{0}) ≥ u(y) − ϕ(x_{0} · y_{0}^{−1} · y).

Set ϕ̃(y) = ϕ(x_{0} ·y_{0}^{−1} ·y) for y ∈ Ω_{(1+4R_{0})ϵ} near y_{0}. Then ϕ̃ touches u from above at y = y_{0} so that we have

B(Xϕ̃, (D^{2}ϕ̃)^{∗})(y_{0}) ≤ 0.  (3.4)

It follows from the left-invariance of X_{i} that

Xϕ̃(y) = (Xϕ)(x_{0} · y_{0}^{−1} · y), (D^{2}(ϕ̃))^{∗}(y) = (D^{2}ϕ)^{∗}(x_{0} · y_{0}^{−1} · y).

This implies

B(Xϕ(x_{0}), (D^{2}ϕ)^{∗}(x_{0})) ≤ 0.

Hence u^{ϵ} is a viscosity subsolution of eqn.(3.3) on Ω_{(1+4R_{0})ϵ}. The proof is complete.

§4. Comparison principle between semiconvex subsolutions and semiconcave supersolutions

In this section, we establish the comparison principle between semiconvex subsolutions and semiconcave strict supersolutions for any 2nd order horizontally elliptic PDE on the Carnot-Carathédory space. The argument is inspired by Jensen’s maximum principle for semiconvex functions (see [J1] [J2]) on 2nd order elliptic PDEs. Here we assume that \{X_{i} = ∑_{j=1}^{n} a_{ij}(x)\frac{∂}{∂x_{j}} \}_{i=1}^{m}\} are C^{1} vector fields on R^{n}. The main proposition of this section is

Proposition 4.1. For any bounded domain Ω ⊂ R^{n}, suppose that B ∈ C(Ω × S^{m}) is horizontally elliptic. Assume that u ∈ C(Ω̄) is a semiconvex subsolution to

B(Xw, (D^{2}w)^{∗}) = 0 in Ω,  (4.1)

and v ∈ C(Ω̄) is a semiconcave supersolution to

B(Xw, (D^{2}w)^{∗}) − μ = 0 in Ω  (4.2)

for some μ > 0. Then

sup_{Ω}(u − v) ≤ sup_{∂Ω}(u − v).  (4.3)

Proof. Suppose that (4.3) were false. Then

sup_{Ω}(u − v) > sup_{∂Ω}(u − v).

Hence u − v achieves its maximum on Ω̄ at a point x_{0} ∈ Ω. Since u − v is semiconvex, it is well-known (cf. [J2] page 67) that

Du(x_{0}), Dv(x_{0}) both exist and are equal,

u(x) − u(x_{0}) − ⟨Du(x_{0}), x − x_{0}⟩ = O(∥x − x_{0}∥^{2}),  (4.4)
v(x) − v(x_{0}) − ⟨Dv(x_{0}), x − x_{0}⟩ = O(∥x − x_{0}∥^{2}),  (4.5)

where ⟨·, ·⟩ and ∥·∥ denote the Euclidean inner product and norm. Denote R_{0} = inf{∥x_{0} − y∥ : y ∈ ∂Ω} and let R_{1} > 0 be such that both (4.4) and (4.5) hold with ∥x − x_{0}∥ < R_{1}.

Set R_{2} = min{R_{0}, R_{1}} > 0 and define, for any ρ > 0, the rescaled maps u^{ρ}, v^{ρ} : B_{R_{2}ρ^{−1}} = {x ∈ R^{n}|∥x∥ ≤ R_{2}ρ^{−1}} → R by

u^{ρ}(x) = \frac{1}{ρ^{2}} (u(x_{0} + ρx) − u(x_{0}) − ρ⟨Du(x_{0}), x⟩),
v^{ρ}(x) = \frac{1}{ρ^{2}} (v(x_{0} + ρx) − v(x_{0}) − ρ⟨Dv(x_{0}), x⟩),

where the addition and scalar multiplication are the euclidean ones. Then it is easy to see that

0 = (u^{ρ} − v^{ρ})(0) ≥ (u^{ρ} − v^{ρ})(x), ∀x ∈ B_{R_{2}ρ^{−1}}.

It follows from (4.4) and (4.5) that, for any R > 0, there exists an ρ_{0} = ρ_{0}(R) > 0 such that (i) {u^{ρ}}_{{0<ρ≤ρ_{0}}} are uniformly bounded, uniformly semiconvex, and uniformly Lipschitz continuous in B_{R}; (ii) {v^{ρ}}_{{0<ρ≤ρ_{0}}} are uniformly bounded, uniformly semiconcave, and uniformly Lipschitz continuous in B_{R}. Therefore, by the Cauchy diagonal process, we find that there is ρ_{i} ↓ 0 such that u^{ρ_{i}} → u^{∗}, v^{ρ_{i}} → v^{∗} locally uniformly in R^{n}. Moreover, it is not difficult to see that u^{∗} is locally bounded, semiconvex in R^{n}, v^{∗} is locally bounded, semiconcave in R^{n}, and

0 = (u^{∗} − v^{∗})(0) ≥ (u^{∗} − v^{∗})(x), ∀x ∈ R^{n}.

Now we need

Claim 4.2. u^{∗} is a viscosity subsolution to

B_{1}(D^{2}w) = 0 in R^{n},  (4.6)

and v^{∗} is a viscosity supersolution to

B_{2}(D^{2}w) + μ = 0 in R^{n},  (4.7)

where B_{1}, B_{2} : S^{n} → R are defined by

B_{1}(M) = B(Xu(x_{0}), \{ ∑_{k,l=1}^{n}(a_{ik}(x_{0})a_{jl}(x_{0})M_{kl} + a_{ik}(x_{0})\frac{∂a_{jl}}{∂x_{k}} (x_{0})\frac{∂u}{∂x_{l}} (x_{0}))\}_{1≤i,j≤m}),

B_{2}(M) = B(Xv(x_{0}), \{ ∑_{k,l=1}^{n}(a_{ik}(x_{0})a_{jl}(x_{0})M_{kl} + a_{ik}(x_{0})\frac{∂a_{jl}}{∂x_{k}} (x_{0})\frac{∂v}{∂x_{l}} (x_{0})\}_{1≤i,j≤m})).

Before we prove this claim, we would like to remark that the horizontal ellipticity (1.9) of B implies that B_{1}, B_{2} ∈ C(S^{n}) is elliptic in the sense that

B_{i}(M) ≤ B_{i}(N), ∀M, N ∈ S^{n}, with M ≥ N

for i = 1, 2. This is also a key observation we need to use in the following proof. To prove it, we first observe that u^{ρ} is a viscosity subsolution to

B(Xu(x_{0})+ρX^{ρ}w, \{ ∑_{k,l=1}^{n} a_{ik}^{ρ}a_{jl}^{ρ} \frac{∂^{2}w}{∂x_{k}∂x_{l}} +a_{ik}^{ρ}(\frac{∂a_{jl}}{∂x_{k}})^{ρ}(\frac{∂u}{∂x_{l}} (x_{0})+ρ\frac{∂w}{∂x_{l}})\}_{1≤i,j≤m}) = 0 (4.8)

and v^{ρ} is a viscosity supersolution to

B(Xv(x_{0})+ρX^{ρ}w, \{ ∑_{k,l=1}^{n} a_{ik}^{ρ}a_{jl}^{ρ} \frac{∂^{2}w}{∂x_{k}∂x_{l}} +a_{ik}^{ρ}(\frac{∂a_{jl}}{∂x_{k}})^{ρ}(\frac{∂v}{∂x_{l}} (x_{0})+ρ\frac{∂w}{∂x_{l}})\}_{1≤i,j≤m}) = 0 (4.9)

where X^{ρ} = (X_{1}^{ρ}, ⋯, X_{m}^{ρ}), X_{i}^{ρ}(x) = X_{i}(x_{0} + ρx), a_{ik}^{ρ}(x) = a_{ik}(x_{0} + ρx), and (\frac{∂a_{jl}}{∂x_{k}})^{ρ}(x) = \frac{∂a_{jl}}{∂x_{k}} (x_{0} + ρx).

For simplicity, we only indicate (4.8). Let x_{1} ∈ B_{R_{2}ρ^{−1}} and ϕ ∈ C^{2}(B_{R_{2}ρ^{−1}}) be such that

0 = u^{ρ}(x_{1}) − ϕ(x_{1}) ≥ u^{ρ}(x) − ϕ(x), ∀x ∈ B_{R_{2}ρ^{−1}}.

It is straightforward to see that

ϕ^{ρ}(x) ≡ u(x_{0}) + ⟨Du(x_{0}), x − x_{0}⟩ + ρ^{2}ϕ(\frac{x − x_{0}}{ρ})

satisfies

0 = u(x_{0} + ρx_{1}) − ϕ^{ρ}(x_{0} + ρx_{1}) ≥ u(x) − ϕ^{ρ}(x), ∀x ∈ B_{R_{0}}(x_{0}).

This, combined with the fact that u is a viscosity subsolution to (4.6), implies

B(Xϕ^{ρ}, (D^{2}ϕ^{ρ})^{∗})(x_{0} + ρx_{1}) ≥ 0.  (4.10)

Direct calculations yield

\frac{∂ϕ^{ρ}}{∂x_{k}} (x) = \frac{∂u}{∂x_{k}} (x_{0}) + ρ\frac{∂ϕ}{∂x_{k}} (\frac{x − x_{0}}{ρ}), \frac{∂^{2}ϕ^{ρ}}{∂x_{k}∂x_{l}} (x) = \frac{∂^{2}ϕ}{∂x_{k}∂x_{l}} (\frac{x − x_{0}}{ρ}).

Hence (4.10) implies that u^{ρ} is a viscosity solution to (4.8). Since X^{ρ} → X(x_{0}), a_{ij}^{ρ} → a_{ij}(x_{0}), and (\frac{∂a_{ik}}{∂x_{l}})^{ρ} → \frac{∂a_{ik}}{∂x_{l}} (x_{0}) locally uniformly on R^{n}, we observe that the family of elliptic operator B^{ρ} : R^{n} × R^{n} × S^{n} → R defined by

B^{ρ}(x, p, M) : = B[(X_{i}u(x_{0}) + ρ ∑_{j=n}^{n} a_{ij}^{ρ}(x)p_{j})_{1≤i≤n},

\{ ∑_{k,l=1}^{n} a_{ik}^{ρ}(x)a_{jl}^{ρ}(x)M_{kl} + a_{ik}^{ρ}(x)(\frac{∂a_{jl}}{∂x_{k}})^{ρ}(x)(\frac{∂u}{∂x_{l}} (x_{0}) + ρp_{l})\}_{1≤i,j≤m}]

coverges to B_{1}(M) locally uniformly as ρ → 0. Therefore the compactness theorem for viscosity subsolutions (see, e.g. Caffarelli-Cabré [CC] proposition 2.9, page 14) implies that u^{∗} is a viscosity subsolution of (4.6). This proves claim 4.2.

Since u^{∗} −v^{∗} is semiconvex and achieves its maximum at x = 0, we can apply Jensen’s maximum principle for semiconvex functions (see [J1] and also [CIL] Lemma A.3, page 60) to conclude that there exists x_{∗} ∈ R^{n} such that D^{2}u^{∗}(x_{∗}), D^{2}v^{∗}(x_{∗}) both exist and D^{2}(u^{∗} − v^{∗})(x_{∗}) ≤ 0. Let M_{1}, M_{2} : S^{m} → R be given by

M_{1}^{ij} = ∑_{k,l=1}^{n}\{a_{ik}(x_{0})a_{jl}(x_{0})\frac{∂^{2}u^{∗}}{∂x_{k}∂x_{l}} (x_{∗}) + a_{ik}(x_{0})\frac{∂a_{jl}}{∂x_{k}} (x_{0})\frac{∂u}{∂x_{l}} (x_{0})\}, 1 ≤ i, j ≤ m,

M_{2}^{ij} = ∑_{k,l=1}^{n}\{a_{ik}(x_{0})a_{jl}(x_{0})\frac{∂^{2}v^{∗}}{∂x_{k}∂x_{l}} (x_{∗}) + a_{ik}(x_{0})\frac{∂a_{jl}}{∂x_{k}} (x_{0})\frac{∂v}{∂x_{l}} (x_{0})\}, 1 ≤ i, j ≤ m.

Since Du(x_{0}) = Dv(x_{0}), we have Xu(x_{0}) = Xv(x_{0}) and, for any p ∈ R^{m},

∑_{1≤i,j≤m}(M_{1}^{ij} − M_{2}^{ij})p_{i}p_{j} = ∑_{k,l=1}^{n}\{∑_{i=1}^{m} p_{i}a_{ik}(x_{0})\}\{∑_{j=1}^{m} p_{j}a_{jl}(x_{0})\}\frac{∂^{2}(u^{∗} − v^{∗})}{∂x_{k}∂x_{l}}(x_{∗}) ≤ 0.

Hence M_{1} ≤ M_{2}. Therefore the horizontal ellipticity of B implies

B(Xu(x_{0}), M_{1}) − B(Xv(x_{0}), M_{2}) ≥ 0.  (4.11)

On the other hand, we have

B_{1}(D^{2}u^{∗}(x_{∗})) − B_{2}(D^{2}v^{∗}(x_{∗})) = B(Xu(x_{0}), M_{1}) − B(Xv(x_{0}), M_{2}) ≤ −μ < 0. (4.12)

This contradicts (4.11). Hence the proof of proposition 4.3 is complete.

§5. Auxiliary equations with horizontal gradient constraints

Due to the degenerancy of eqn.(1.13), we establish a comparison principle for its viscosity solutions via the L^{p}-approximation scheme by [J2] to construct two auxiliary equations with horizontal gradient constraints, to which supersolutions can be deformed into strict supersolutions under small perturbations. This section is valid for any Carnot-Carathédory metric space generated by a family of Hörmander’s vector fields {X_{i}}_{i=1}^{m} ⊂ C^{∞}(Ω, R^{n}).

Lemma 5.1. Suppose that f ∈ C^{2}(R^{m}, R_{+}) is homogeneous of degree α ≥ 1. Let v ∈ C(Ω̄) be a viscosity supersolution to

min{f(Xw) − ϵ, − ∑_{i,j=1}^{m} f_{p_{i}}(Xw)f_{p_{j}}(Xw)X_{i}X_{j}w} = 0, in Ω  (5.1)

where ϵ > 0. Then, for any δ > 0, there exist μ = μ(α, ϵ, δ) > 0 and v_{δ} ∈ C(Ω̄), with ∥v_{δ} − v∥_{L^{∞}(Ω)} ≤ δ, such that v_{δ} is a viscosity supersolution to

min{f(Xw) − ϵ, − ∑_{i,j=1}^{m} f_{p_{i}}(Xw)f_{p_{j}}(Xw)X_{i}X_{j}w} − μ = 0, in Ω.  (5.2)

Proof. It is similar to that by [J2] (see also [J] and [B1]). We sketch it here. We look for v_{δ} = g_{δ}(v), where g_{δ} ∈ C^{∞}(R) is monotonically increasing such that g_{δ}^{−1} ∈ C^{∞}(R). To find g_{δ}, let x_{0} ∈ Ω and ϕ ∈ C^{2}(Ω) touches v_{δ} from below at x_{0}. Let ϕ_{δ} = g_{δ}^{−1}(ϕ). Then ϕ_{δ} touches v from below at x_{0} and

min{f(Xϕ_{δ}) − ϵ, − ∑_{i,j=1}^{m} f_{p_{i}}(Xϕ_{δ})f_{p_{j}}(Xϕ_{δ})X_{i}X_{j}ϕ_{δ}}|_{x=x_{0}} ≥ 0.

Since

X_{i}ϕ = g′_{δ}(ϕ_{δ})X_{i}ϕ_{δ}, X_{i}X_{j}ϕ = g′_{δ}(ϕ_{δ})X_{i}X_{j}ϕ_{δ} + g′′_{δ}(ϕ_{δ})X_{i}ϕ_{δ}X_{j}ϕ_{δ},

we have, by the α-homogeneity of f,

f(Xϕ(x_{0})) = f(g′_{δ}(ϕ_{δ}(x_{0}))Xϕ_{δ}(x_{0})) = (g′_{δ}(ϕ_{δ}(x_{0})))^{α}f(Xϕ_{δ}(x_{0})) ≥ (g′_{δ}(ϕ_{δ}(x_{0})))^{α}ϵ, (5.3)

and

− ∑_{i,j=1}^{m} f_{p_{i}}(Xϕ)f_{p_{j}}(Xϕ)X_{i}X_{j}ϕ|_{x=x_{0}}

= {g′_{δ}(ϕ_{δ})^{3α}(− ∑_{ij=1}^{m} f_{p_{i}}(Xϕ_{δ})f_{p_{i}}(Xϕ_{δ})X_{i}X_{j}ϕ_{δ})

− g′_{δ}(ϕ_{δ})^{2α}g′′_{δ}(ϕ_{δ})(∑_{i=1}^{m} f_{p_{i}}(Xϕ_{δ})X_{i}ϕ_{δ})^{2}}|_{x=x_{0}}

≥ −g′_{δ}(ϕ_{δ}(x_{0}))^{2α}g′′_{δ}(ϕ_{δ}(x_{0}))α^{2}ϵ^{2},  (5.4)

provided that g′′_{δ}(ϕ_{δ}) < 0, here we have used (5.3) and the identity ∑_{i=1}^{m} f_{p_{i}}(p)p_{i} = αf(p) in the last step. Let C_{0} = 4∥v∥_{L^{∞}(Ω)} < ∞ and define

g_{δ}(t) = (1 + δ)t − \frac{δ}{4C_{0}} t^{2}, |t| ≤ 2C_{0},

and extend it suitably to a monotonically increasing function on R. Since g′(t) ≥ 1 + \frac{δ}{2} and g′′(t) = −\frac{δ}{2C_{0}} for |t| ≤ C_{0}, we have

f(Xϕ)(x_{0}) ≥ (1 + \frac{δ}{2})ϵ,

and

− ∑_{i,j=1}^{m} f_{p_{i}}(Xϕ)f_{p_{j}}(Xϕ)X_{i}X_{j}ϕ|_{x=x_{0}} ≥ \frac{δα^{2}ϵ^{2}}{2C_{0}} .

Therefore, if we choose μ = min\{\frac{δϵ}{2}, \frac{δα^{2}ϵ^{2}}{2C_{0}} \} > 0, then

min{f(Xϕ) − ϵ, − ∑_{i,j=1}^{m} f_{p_{i}}(Xϕ)f_{p_{j}}(Xϕ)X_{i}X_{j}ϕ}|_{x=x_{0}} ≥ μ.

The proof of Lemma 5.1 is complete.

Since the argument is similar, we state without proof the analogous Lemma on viscosity subsolutions.

Lemma 5.2. Suppose that f ∈ C^{2}(R^{m}, R_{+}) is of homogeneous of degree α ≥ 1. Let u ∈ C(Ω̄) be a viscosity subsolution to

max{ϵ − f(Xw), − ∑_{i,j=1}^{m} f_{p_{i}}(Xw)f_{p_{j}}(Xw)X_{i}X_{j}w} = 0, in Ω,  (5.5)

where ϵ > 0. Then, for any δ > 0, there are an μ = μ(α, ϵ, δ) > 0 and u_{δ} ∈ C(Ω̄), with ∥u_{δ} − u∥_{L^{∞}(Ω)} ≤ δ, such that u_{δ} is a viscosity subsolution to the equation

max{ϵ − f(Xw), − ∑_{i,j=1}^{m} f_{p_{i}}(Xw)f_{p_{j}}(Xw)X_{i}X_{j}w} = −μ, in Ω.  (5.6)

We end this section with existences of viscosity solutions to eqn. (1.13), (5.3), (5.5). For this, we need both convexity of f and f(p) > 0 for p ≠ 0. More precisely,

Theorem 5.3. Suppose that f ∈ C^{2}(R^{m}, R_{+}) is strictly convex, homogeneous of degree α ≥ 1, and f(p) > 0 for p ≠ 0. Then, for any g ∈ Wcc^{1,∞}(Ω), we have (1). There exists a viscosity solution u ∈ Wcc^{1,∞}(Ω) to eqn.(1.13) such that u|_{∂Ω} = g. (2). There exists a viscosity solution u_{ϵ} ∈ Wcc^{1,∞}(Ω) of eqn. (5.3) such that u_{ϵ}|_{∂Ω} = g. (3). There exists a viscosity solution v_{ϵ} ∈ Wcc^{1,∞}(Ω) of the eqn. (5.5) such that v_{ϵ}|_{∂Ω} = g. (4). There exists a continuous, nondecreasing function β : R_{+} → R_{+}, with β(0) = 0, such that

∥u_{ϵ} − v_{ϵ}∥_{L^{∞}(Ω)} ≤ β(ϵ).  (5.7)

Proof. The proof is inspired by the L^{k}-approximation scheme, which has been used by Bhatthacharya-Dibenedetto-Manfredi [BDM], [J2], [J], [B1] in the euclidean setting. For the completeness, we outline it here. Since (1) follows from (2) with ϵ = 0 and (3) can be done exactly in the way as (2), we only sketch (2) and (4) as follows. For 1 < k < ∞, let u_{k} ∈ WX^{1,k}(Ω) be the unique minimizer to the functional

F_{k}(v) = ∫_{Ω}(f(Xv)^{k} − ϵ^{k−1}v), ∀v ∈ WX^{1,k}(Ω), with v|_{∂Ω} = g.

The existence of u_{k} can be obtained by the direct method, due to both the convexity of f and α-homogeneity of f, i.e. f(p) = |p|^{α}f(\frac{p}{|p|}) ≥ |p|^{α} min_{|z|=1} f(z) ≥ C|p|^{α}. It is easy to verify that u_{k} satisfies the (subelliptic) k-Laplacian: equation

− ∑_{i=1}^{m} X_{i}^{∗}(kf(Xu_{k})^{k−1}f_{p_{i}}(Xu)) = ϵ^{k−1}, in Ω  (5.8)

in the sense of distributions, where X_{i}^{∗}w = ∑_{j=1}^{m} \frac{∂}{∂x_{j}} (a_{ij}(x)w) is the adjoint operator of X_{i}. Let Q denote the homogeneous dimension of R^{n}, with respect to the vector fields {X_{i}}_{i=1}^{m}. Then it follows from the Sobolev inequality (see Hajlasz-Koskela [HK]) that {u_{k}}_{k≥Q+1} is bounded and equicontinuous. Therefore we may assume, after taking possible subsequences, that there exists a u_{ϵ} ∈ WX^{1,∞}(Ω) such that

u_{k} → u_{ϵ} in C^{0}(Ω̄) ∩_{Q+1≤k<∞} WX^{1,k}(Ω).

It is easy to see that u_{ϵ}|_{∂Ω} = g. To show that u_{ϵ} is a viscosity solution to the eqn. (5.3), we need

Claim 5.4. For k ≥ Q + 1, u_{k} ∈ C(Ω̄) is a viscosity solution to the eqn.(5.8).

For simplicity, we only indicate that u_{k} is a viscosity subsolution. For, otherwise, there are x_{0} ∈ Ω and ϕ ∈ C^{2}(Ω) such that

0 = u_{k}(x_{0}) − ϕ(x_{0}) ≥ u_{k}(x) − ϕ(x), ∀x ∈ Ω,

but

− ∑_{i=1}^{m} X_{i}^{∗}(kf(Xϕ)^{k−1}f_{p_{i}}(Xϕ))(x_{0}) − ϵ^{k−1} = C_{0} > 0.  (5.9)

Then there exists δ_{0} > 0 such that

− ∑_{i=1}^{m} X_{i}^{∗}(kf(Xϕ)^{k−1}f_{p_{i}}(Xϕ))(x) − ϵ^{k−1} ≥ \frac{C_{0}}{2} , ∀x ∈ B_{δ_{0}}(x_{0}).  (5.10)

For any small δ > 0, there is a neighborhood V_{δ}(⊂ B_{δ_{0}}(x_{0})) of x_{0} such that ϕ_{δ} ≡ ϕ − δ satisfies

ϕ_{δ}(x) < u_{k}(x), ∀x ∈ V_{δ}; ϕ_{δ}(x) = u_{k}(x), ∀x ∈ ∂V_{δ}.

Note that ϕ_{δ} also satisfies (5.10). Multiplying (5.8) by u_{k} − ϕ_{δ} and integrating over V_{δ}, we have

∑_{i=1}^{m} ∫_{V_{δ}} kf(Xu_{k})^{k−1}f_{p_{i}}(Xu_{k})X_{i}(u_{k} − ϕ_{δ}) = ϵ^{k−1} ∫_{V_{δ}}(u_{k} − ϕ_{δ}).  (5.11)

On the other hand, multiplying (5.10) by (u_{k} − ϕ_{δ})(≥ 0) and integrating over V_{δ}, we have

∑_{i=1}^{m} ∫_{V_{δ}} k(f(Xϕ_{δ}))^{k−1}f_{p_{i}}(Xϕ_{δ})X_{i}(u_{k} − ϕ_{δ}) > ϵ^{k−1} ∫_{V_{δ}}(u_{k} − V_{δ}).  (5.12)

Subtracting (5.11) from (5.12), we obtain

0 > k ∫_{V_{δ}} ∑_{i=1}^{m}(f(Xu_{k})^{k−1}f_{p_{i}}(Xu_{p}) − f(Xϕ_{δ})^{k−1}f_{p_{i}}(Xϕ_{δ}))X_{i}(u_{k} − ϕ_{δ})),

this contradicts with the convexity of f. This finishes the proof of Claim 5.4.

Now we show that u_{ϵ} is a viscosity subsolution to the eqn. (5.3). Let x ∈ Ω and ϕ ∈ C^{2}(Ω) be such that

0 = u_{ϵ}(x) − ϕ(x) ≥ u_{ϵ}(y) − ϕ(y), ∀y ∈ Ω.

We need to show

min{f(Xϕ) − ϵ, − ∑_{i,j=1}^{m} f_{p_{i}}(Xϕ)f_{p_{j}}(Xϕ)X_{i}X_{j}ϕ}|_{x} ≤ 0.

Since this is true if f(Xϕ(x)) ≤ ϵ, we may assume that f(Xϕ(x)) ≥ (1 + 2δ)ϵ for some δ > 0. We know that there exist x_{k} ∈ Ω such that (u_{k} − ϕ) achieves its maximum at x_{k} and x_{k} → x. We may also assume that, for k sufficiently large,

f(Xϕ(x_{k})) ≥ (1 + δ)ϵ.

It follows from claim 5.4 that

− ∑_{i=1}^{m} X_{i}^{∗}(kf(Xϕ)^{k−1}f_{p_{i}}(Xϕ))(x_{k}) ≤ ϵ^{k−1}.

After expansion and dividing both sides by k(k − 1)f(Xϕ)^{k−2}(x_{k}), this gives

− ∑_{i,j=1}^{m} f_{p_{i}}(Xϕ)f_{p_{j}}(Xϕ)X_{i}X_{j}ϕ(x_{k}) ≤ [\frac{ϵ}{k(k − 1)}\{\frac{ϵ}{f(Xϕ(x_{k}))}\}^{k−2}

− \frac{f(Xϕ(x_{k}))}{(k − 1)} ∑_{i=1}^{m} X_{i}^{∗}(f_{p_{i}}(Xϕ))(x_{k})].

This, after letting k tend to ∞, gives

− ∑_{i,j=1}^{m} f_{p_{i}}(Xϕ)f_{p_{j}}(Xϕ)X_{i}X_{j}ϕ(x) ≤ 0.

One can show similarly that u_{ϵ} is also a viscosity supersolution to the eqn. (5.3). This finishes the proof of (2).

Since v_{ϵ} is a limit, as k → ∞, of the minimizers v_{k} to

G_{k}(v) = ∫_{Ω} f(Xv)^{k} + ϵ^{k−1}v, ∀v ∈ WX^{1,k}(Ω), with u|_{∂Ω} = g,

v_{k} satisfies

− ∑_{i=1}^{m} X_{i}^{∗}(kf(Xv_{k})^{k−1}f_{p_{i}}(Xv_{k})) = −ϵ^{k−1}, in Ω.  (5.13)

Multiplying (5.11) and (5.13) by (u_{k} − v_{k}), integrating over Ω, and subtracting each other, we get

∫_{Ω} k(f(Xu_{k})^{k−1}f_{p_{i}}(Xu_{k}) − f(Xv_{k})^{k−1}f_{p_{i}}(Xv_{k}))X_{i}(u_{k} − v_{k})
≤ 4ϵ^{k−1}∥u_{k} − v_{k}∥_{L^{1}(Ω)}.  (5.14)

Now we need

Claim 5.5. If f ∈ C^{2}(R^{m}) is strictly convex, then for any p, q ∈ R^{m}

(f^{k−1}(p)f_{p}(p) − f^{k−1}(q)f_{p}(q)) · (p − q) ≥ C|p − q|^{α(k−1)+2}.  (5.15)

To see (5.15), we observe that

(f^{k−1}(p)f_{p}(p) − f^{k−1}(q)f_{p}(q)) · (p − q)

= k^{−1} ∫_{0}^{1} \frac{d}{dt}(f^{k})_{p}(tp + (1 − t)q) dt · (p − q)

≥ ∑_{ij=1}^{m} ∫_{0}^{1} f^{k−1}(tp + (1 − t)q)f_{p_{i}p_{j}}(tp + (1 − t)q) dt(p_{i} − q_{i})(p_{j} − q_{j})

≥ C^{k−1} ∫_{0}^{1} |tp + (1 − t)q|^{α(k−1)} dt|p − q|^{2},  (5.16)

where we have used the strict convexity of f:

∑_{ij=1}^{m} f_{p_{i}p_{j}}(v)p_{i}p_{j} ≥ C_{0}|p − q|^{2}, ∀p, q, v ∈ R^{m},

the α-homogeneity of f and the fact that

f(v) = |v|^{α}f(\frac{v}{|v|}) ≥ min_{|z|=1} f(z)|v|^{α} ≥ C|v|^{α}, ∀v ∈ R^{m}

for some C > 0 depending only on f. Since

∫_{0}^{1} |tp + (1 − t)q|^{α(k−1)} dt ≥ C|p − q|^{α(k−1)},

(5.16) implies (5.15). Putting (5.15) into (5.14), we obtain

kC^{k−1} ∫_{Ω} |Xu_{k} − Xv_{k}|^{α(k−1)+2} ≤ Cϵ^{k−1}.

This, combined with the Hölder inequality, implies

∫_{Ω} |Xu_{k} − Xv_{k}| ≤ k^{− \frac{1}{α(k−1)+2}} (Cϵ)^{\frac{k−1}{α(k−1)+2}} |Ω|^{\frac{α(k−1)+1}{α(k−1)+2}} .

Letting k tend to ∞, we have

∥Xu_{ϵ} − Xv_{ϵ}∥_{L^{1}(Ω)} ≤ Cϵ^{\frac{1}{α}} .  (5.17)

For u_{ϵ}, v_{ϵ} ∈ WX^{1,∞}(Ω), (5.17), the usual interpolation inequality, and the Sobolev inequality yield that the function β must exist as asserted in (4).

§6. Proof of theorem C

This section is devoted to the proof of theorem C. In this section, the {X_{i}}_{i=1}^{m} are horizontal vector fields of a Carnot group G. Since we can identify G with R^{n}, n = dim(G), via the exponential map, the results in §4 and §5 are all applicable to G. The idea to prove theorem C is based on the sup/inf convolution and the comparison principle for both equations (5.1) and (5.5).

Lemma 6.1. Under the same assumptions as theorem C, for any ϵ > 0, if v ∈ C(Ω̄) is a viscosity subsolution to the eqn.(5.1) and w ∈ C(Ω̄) is a viscosity supersolution to the eqn.(5.1), then

sup_{x∈Ω}(v − w)(x) = sup_{x∈∂Ω}(v − w)(x).  (6.1)

Proof. Suppose that (6.1) were false. Then there is an δ_{0} > 0 such that

sup_{x∈Ω}(v − w)(x) ≥ sup_{x∈∂Ω}(v − w)(x) + δ_{0}.

For any δ ∈ (0, \frac{δ_{0}}{2} ), it follows from Lemma 5.1 that there are w_{δ} ∈ C(Ω̄), with ∥w_{δ} − w∥_{L^{∞}(Ω)} ≤ δ, and μ = μ(δ, ϵ, α) > 0, such that w_{δ} is a viscosity supersolution to the eqn.(5.2). Moreover, we have

sup_{x∈Ω}(v − w_{δ}) ≥ sup_{x∈∂Ω}(v − w_{δ})(x) + \frac{δ_{0}}{4} .  (6.2)

Now we apply proposition 3.3 to conclude that for any δ ∈ (0, \frac{δ_{0}}{2} ) there are a semiconvex v^{δ} ∈ WX^{1,∞}(Ω) and a semiconcave w̃_{δ} ∈ WX^{1,∞}(Ω) such that

lim_{δ→0} max{∥v^{δ} − v∥_{L^{∞}(Ω_{Cδ})}, ∥w̃_{δ} − w_{δ}∥_{L^{∞}(Ω_{Cδ})}} = 0,  (6.3)

where Ω_{Cδ} is defined in §3. Moreover, v^{δ} is a viscosity subsolution to the eqn. (5.1) and w̃_{δ} is a viscosity supersolution to the eqn. (5.2) on Ω_{Cδ}. Therefore, we can apply proposition 4.1 to conclude that

sup_{Ω_{Cδ}}(v^{δ} − w̃_{δ}) = sup_{∂Ω_{Cδ}}(v^{δ} − w̃_{δ}).  (6.4)

Letting δ tend to zero, this yields

lim_{δ→0} sup_{Ω_{Cδ}}(v − w) = lim_{δ→0} sup_{Ω_{Cδ}}[(v − v^{δ}) + (v^{δ} − w̃_{δ}) + (w̃_{δ} − w_{δ}) + (w_{δ} − w)]

= lim_{δ→0} sup_{Ω_{Cδ}}(v^{δ} − w̃_{δ})

= lim_{δ→0} sup_{∂Ω_{Cδ}}(v^{δ} − w̃_{δ})

= sup_{∂Ω}(v − w).

This yields the desired contradiction. The proof is complete.

Similarly, we have the comparison principle for the equation (5.5).

Lemma 6.2. Under the same assumptions as theorem C, for any ϵ > 0, let v ∈ C(Ω̄) be a viscosity subsolution to the eqn.(5.5) and w ∈ C(Ω̄) be a viscosity supersolution to the eqn.(5.5). Then

sup_{x∈Ω}(v − w)(x) = sup_{x∈∂Ω}(v − w)(x).

We are ready to prove the maximum principle for the eqn. (1.13).

Lemma 6.3. Under the same assumptions as theorem C, for a given ϕ ∈ WX^{1,∞}(Ω), assume that v ∈ C(Ω̄) is a viscosity subsolution to the eqn.(1.13) and w ∈ C(Ω̄) is a viscosity supersolution to the eqn.(1.13) such that v|_{∂Ω} = w|_{∂Ω} = ϕ. Then

v(x) ≤ w(x), ∀x ∈ Ω.  (6.5)

Proof. Let v^{+} be a viscosity solution to the eqn.(5.1) and w^{−} be a viscosity solution to the eqn.(5.5), with v^{+}|_{∂Ω} = w^{−}|_{∂Ω} = ϕ, obtained by theorem 5.3. Since subsolutions of eqn.(1.13) are also subsolutions to eqn.(5.1) and supersolutions to eqn.(1.13) are also supersolutions to eqn.(5.5), we can apply Lemma 6.1, 6.2 to conclude that

sup_{Ω}(v − v^{+}) = sup_{∂Ω}(v − v^{+}) = 0, sup_{Ω}(w^{−} − w) = sup_{∂Ω}(w^{−} − w) = 0.

Hence we have

sup_{Ω}(v − w) ≤ sup_{Ω}(v^{+} − w^{−}) ≤ β(ϵ),

where β is given by theorem 5.3. Since ϵ is arbitrary, this implies

sup_{Ω}(v − w) ≤ 0.

This finishes the proof of Lemma 6.3.

It is clear that theorem C follows from Lemma 6.3 and hence the proof is complete.

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