Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 31, 2006, 363–379

EQUIVALENCE OF WEAK AND VISCOSITY SOLUTIONS TO THE p-LAPLACE EQUATION IN THE HEISENBERG GROUP

Thomas Bieske

University of South Florida, Department of Mathematics Tampa, FL 33620, U.S.A.; tbieske@math.usf.edu

Abstract. We prove weak and viscosity solutions to the p-Laplace equation in the Heisenberg group coincide by showing that the viscosity sub(super-) solutions coincide with the psub(super-)harmonic functions from potential theory. We are then able to obtain a comparison principle for viscosity solutions to the p-Laplace equation.

1. Introduction and motivation

In [JLM], Juutinen, Lindqvist, and Manfredi prove the equivalence of viscosity solutions and weak solutions to the p-Laplace equation in R^{n}, given by

−div(∥Du∥^{p−2}Du) = 0

for 1 < p < ∞. Here, Du denotes the gradient of the real-valued function u. The p-Laplace equation is a well-known example of a larger class of quasi-linear equations of the form

−div(𝒜_{p}(u, Du)) = 0

where 𝒜_{p} satisfies certain structure conditions. (See [HKM] and [HH] for complete details.) This class of equations plays a major role in non-linear potential theory and has been studied in the Euclidean environment [HKM], Carnot groups [HH], and general metric spaces [KM].

In this paper, we prove the equivalence of the potential-theoretic p-harmonic functions and viscosity solutions to the p-Laplace equation in the Heisenberg group. (See Section 3 for relevant definitions.) It should be noted that due to the geometry of the Heisenberg group, the method of proof in [JLM] cannot be used, for it relies upon the well-known C^{1,α} regularity of the weak solutions, which is unknown in the Heisenberg group. (Although, for p near 2, this was proved recently by Domokos and Manfredi [DM].) We therefore adopt a different strategy to obtain our results. We begin with a brief review of the Heisenberg group in Section 2, followed by the relevant definitions in Section 3. It is noted that this section highlights the nonlinear potential theory found in [HKM] and its extension to Carnot groups in [HH]. In Section 4, we prove the equivalence of viscosity solutions and p-harmonic functions and obtain a comparison principle for viscosity solutions to the p-Laplacian.

2000 Mathematics Subject Classification: Primary 31C45, 43A80; Secondary 31B05, 22E25.

2. The Heisenberg group

We begin with R^{2n+1} using the coordinates (x_{1}, x_{2}, …, x_{2n}, z) and consider the linearly independent vector fields {X_{i}, Z}, where the index i ranges from 1 to 2n, defined by

\begin{aligned}X_{i} & = \{ \begin{matrix}\frac{∂}{∂x_{i}} − \frac{x_{n+i}}{2} \frac{∂}{∂z} , & if 1 ≤ i ≤ n, \\ \frac{∂}{∂x_{i}} + \frac{x_{i−n}}{2} \frac{∂}{∂z} , & if n < i ≤ 2n, \end{matrix} \\ Z & = \frac{∂}{∂z} · \end{aligned}

For i ≤ j, these vector fields obey the relations

[X_{i}, X_{j}] = \{ \begin{matrix}Z, & if j = i + n, \\ 0, & otherwise, \end{matrix}

and for all i,

[X_{i}, Z] = 0.

We then have a Lie algebra denoted h_{n} that decomposes as a direct sum

h_{n} = V_{1} ⊕ V_{2}

where V_{1} is spanned by the X_{i} ’s and V_{2} is spanned by Z. We endow h_{n} with an inner product ⟨ · , · ⟩ and related norm ∥ · ∥ so that this basis is orthonormal. The corresponding Lie group is called the general Heisenberg group of dimension n and is denoted by H_{n}. With this choice of vector fields the exponential map can be used to identify elements of h_{n} and H_{n} with each other via

∑_{i=1}^{2n} x_{i}X_{i} + zZ ∈ h_{n} ↔ (x_{1}, x_{2}, …, x_{2n}, z) ∈ H_{n}.

In particular, for any P, Q in H_{n}, written as P = (x_{1}, x_{2}, …, x_{2n}, z_{1}) and Q = (y_{1}, y_{2}, …, y_{2n}, z_{2}) the group multiplication law is given by

P · Q = (x_{1} + y_{1}, x_{2} + y_{2}, …, x_{2n} + y_{2n}, z_{1} + z_{2} + \frac{1}{2} ∑_{i=1}^{n}(x_{i}y_{n+i} − x_{n+i}y_{i})).

The natural metric on H_{n} is the Carnot–Carathéodory metric given by

d_{C}(P, Q) = inf_{Γ} ∫_{0}^{1} ∥γ′(t)∥ dt

where the set Γ is the set of all curves γ such that γ(0) = P, γ(1) = Q and γ′(t) ∈ V_{1} . By Chow’s theorem (see, for example, [Be]) any two points can be connected by such a curve, which makes d_{C}(P, Q) a left-invariant metric on H_{n}. This metric induces a homogeneous norm on H_{n}, denoted | · |, by

|P | = d_{C}(0, P)

and we have the estimate

|P | ∼ ∑_{i=1}^{2n} |x_{i}| + |z|^{1/2}.

This estimate leads us to define the left-invariant gauge 𝒩 that is comparable to the Carnot–Carathéodory metric and is given by

𝒩 (P) = ((∑_{i=1}^{2n} x_{i}^{2})^{2} + 16z^{2})^{1/4}.

We define the Carnot–Carathéodory balls B(P, r) and the gauge balls B_{𝒩} (P, r) in the obvious way.

Given a smooth function u: H_{n} ↦ R, we define the horizontal gradient by ∇_{0}u = (X_{1}u, X_{2}u, …, X_{2n}u),

the full gradient by

∇u = (X_{1}u, X_{2}u, …, X_{2n}u, Zu),

and the symmetrized horizontal second derivative matrix (D^{2}u)^{⋆} by ((D^{2}u)^{⋆})_{ij} = \frac{1}{2}(X_{i}X_{j}u + X_{j}X_{i}u).

Additionally, given a vector field F = ∑_{i=1}^{2n} f_{i}X_{i} + f_{2n+1}Z, we define the Heisenberg divergence of F, denoted div_{ℋ} F, by

div_{ℋ} F = ∑_{i=1}^{2n} X_{i}f_{i}.

A quick calculation shows that when f_{2n+1} = 0, we have

div_{ℋ} F = div_{eucl}F

where div_{eucl} is the standard Euclidean divergence. The main operator we are concerned with is the horizontal p-Laplacian for 1 < p < ∞ defined by

Δ_{p}f = div_{ℋ} (∥∇_{0}f∥^{p−2}∇_{0}f)

which is a specific type of operator in an important class of operators in potential theory as detailed in [HH] and [HKM].

A function f is 𝒞_{sub}^{1} if X_{i}f is continuous for all i and f is 𝒞_{sub}^{2} if f is 𝒞_{sub}^{1} and X_{i}X_{j}f is continuous for all i and j. Using the horizontal gradient, we may also define the Sobolev spaces W^{1,p} , W_{loc}^{1,p} , etc. in the obvious way. For a more complete treatment of the Heisenberg group, the interested reader is directed to [Be], [B], [F], [FS], [G], [H], [K], [S] and the references therein.

3. Notions of solution

In this section, we highlight some results from nonlinear potential theory as detailed in [HKM]. In [HH], many of the Euclidean results were extended into the general setting of Carnot groups. A more complete treatment of viscosity solutions in the Euclidean environment can be found in [CIL] and in the Heisenberg group in [B]. All the results below can be extended into general Carnot groups.

Our main goal is to relate three different notions of solutions to the equation

(3.1)  −Δ_{p}f = − div_{ℋ} (∥∇_{0}f∥^{p−2}∇_{0}f) = 0

in a bounded domain Ω.

3.1. Weak solutions. We begin by defining the concept of weak solutions to equation (3.1). We will actually do more, for we shall define weak solutions to a wider class of equations. Letting ε ≥ 0 be a real parameter, we consider equations of the form

(3.2)  −Δ_{p}f = − div_{ℋ} (∥∇_{0}f∥^{p−2}∇_{0}f) = ε

in a bounded domain Ω. Note that equation (3.1) corresponds to equation (3.2) with ε = 0. We now give the definition of weak solutions.

Definition 1. The function u ∈ W_{loc}^{1,p} is an ε-weak solution to equation (3.2) if

(3.3)  ∫_{Ω} ∥∇_{0}u∥^{p−2}⟨∇_{0}u, ∇_{0}ϕ⟩ = ε ∫_{Ω} ϕ

for all ϕ ∈ W_{0}^{1,p}(Ω).

A weak solution to equation (3.1) (i.e., a 0-weak solution) is called p-harmon-ic. It is well known that a p-harmonic function u has a continuous representative that satisfies

osc_{B_{r}}u ≤ C(\frac{r}{R})^{α}osc_{B_{R}}

when B_{R} ⊂ Ω and r ≤ R ([HH, Theorem 4.2]). We note that the constants C > 0 and α > 0 depend only on the group H_{n}. We therefore identify p-harmonic functions with their continuous representative.

In addition to weak solutions we may define weak supersolutions and weak subsolutions using the following definition (cf. [HKM]).

Definition 2. The function u ∈ W_{loc}^{1,p}(Ω) is an ε-weak supersolution to equation (3.2) if

∫_{Ω} ∥∇_{0}u∥^{p−2}⟨∇_{0}u, ∇_{0}ϕ⟩ ≥ ε ∫_{Ω} ϕ

for all non-negative ϕ ∈ W_{0}^{1,p}(Ω).

The function u ∈ W_{loc}^{1,p}(Ω) is an ε-weak subsolution to equation (3.2) if −u is an ε-weak supersolution, that is, if

∫_{Ω} ∥∇_{0}u∥^{p−2}⟨∇_{0}u, ∇_{0}ϕ⟩ ≤ ε ∫_{Ω} ϕ

for all non-negative ϕ ∈ W_{0}^{1,p}(Ω).

Using these definitions for ε_{1} > ε_{2} ≥ 0, we observe that an ε_{1} -weak solution is a ε_{2} -weak supersolution and an ε_{2} -weak solution is a ε_{1} -weak subsolution.

It is also well known that 0-weak subsolutions and supersolutions satisfy the following comparison principle

Lemma 3.1 ([HKM, Lemma 3.18]). Let u ∈ W^{1,p}(Ω) be a weak subsolution to equation (3.1) and let v ∈ W^{1,p}(Ω) be a weak supersolution to equation (3.1) in Ω. If γ ≡ min{v − u, 0} ∈ W_{0}^{1,p}(Ω) then u ≤ v almost everywhere in Ω.

We are then able to formulate the existence-uniqueness of p-harmonic functions (cf. [HKM, Theorem 3.17], [HH, Section 4.10]).

Theorem 3.2. Given a bounded domain Ω with boundary data Θ ∈ W^{1,p}(Ω), there is a unique p-harmonic function u that satisfies u−Θ ∈ W_{0}^{1,p}(Ω).

Using standard techniques in calculus of variations, one can show that εweak solutions exist and Lemma 3.1 can be extended to ε-weak solutions. In addition, ε-weak solutions have a continuous representative [CDG] and therefore such solutions will be identified with that representative.

3.2. p-superharmonic functions. The next class of solutions we wish to consider are p-superharmonic functions and p-subharmonic functions defined via the following definition.

Definition 3. The function u: Ω ↦ R^{N} ∪ {∞} is p-superharmonic if the following hold:

(1) u is lower semicontinuous.

(2) u is not identically infinity in each component of Ω.

(3) For each subdomain D ⊂⊂ Ω, a p-harmonic function g in D that is continuous in \overline{D} with g ≤ u on ∂D implies g ≤ u in D.

A function u is p-subharmonic if −u is p-superharmonic.

The key points of these definitions are that they are based on comparison with p-harmonic functions. We then are able to obtain the following comparison principle [KM, Theorem 7.2].

Lemma 3.3. Let Ω be a bounded domain in H_{n}. Let v be an p-superhar-monic function and u be a p-subharmonic function in Ω so that

lim sup_{Q→P} u(Q) ≤ lim inf_{Q→P} v(Q)

for all P ∈ ∂Ω with both sides not simultaneously −∞ or ∞. Then u ≤ v in Ω.

We are then able to conclude the following lemma ([HKM, Lemma 7.8]).

Lemma 3.4. A function is p-harmonic if an only if it is both p-subharmonic and p-superharmonic.

3.3. Viscosity solutions. In this subsection, we review the concept of viscosity solution and relate the first two notions of solution to viscosity solutions. Before we begin, we consider equation (3.2) in non-divergence form, namely,

(3.4) −(∥∇_{0}u∥^{p−2} tr((D^{2}u)^{⋆}) + (p − 2)∥∇_{0}u∥^{p−4}⟨(D^{2}u)^{⋆}∇_{0}u, ∇_{0}u⟩) = ε.

We note that equation (3.4) is degenerate elliptic and proper in the sense of [CIL]. Given the function u, we consider the set of functions ϕ that touch from below at the point P_{0} . That is, the set 𝒯 ℬ (u, P_{0}) given by

\begin{aligned}𝒯 ℬ (u, P_{0}) & = \{ϕ ∈ 𝒞_{sub}^{2} (Ω) : u(P_{0}) = ϕ(P_{0}), \\ & u(P) > ϕ(P) for P ≠ P_{0}, ∇_{0}ϕ(P_{0}) ≠ 0\}. \end{aligned}

We are now able to define the concept of viscosity solutions to equation (3.4).

Definition 4. The function u: Ω ↦ R^{N} ∪{∞} is an ε-viscosity supersolution to equation (3.4) if the following hold:

(1) u is lower semicontinuous.

(2) u is not identically infinity in each component of Ω.

(3) For P_{0} ∈ Ω, ϕ ∈ 𝒯 ℬ (u, P_{0}) satisfies

−Δ_{p}ϕ(P_{0}) ≥ ε.

A function u is an ε-viscosity subsolution to equation (3.4) if −u is an ε-viscosity supersolution. A function u is an ε-viscosity solution if it is both an ε-viscosity supersolution and an ε-viscosity subsolution.

We call the collection

{(∇ϕ(P_{0}), (D^{2}ϕ)^{⋆}(P_{0})) : ϕ ∈ 𝒯 ℬ (u, P_{0})}

the subjet of u at P_{0} and denote it J^{2,−}u(P_{0}). We define the superjet of u at P_{0} by J^{2,+}u(P_{0}) = −J^{2,−}(−u)(P_{0}). The set-theoretic closure J̄^{2,−}u(P_{0}) is defined by all pairs {(η, X)} so that there is a sequence {(P_{n}, ϕ_{n})} with (∇ϕ_{n}(P_{n}), (D^{2}ϕ_{n})^{⋆}(P_{n})) ∈ J^{2,−}u(P_{n}) so that P_{n} → P_{0}, u(P_{n}) → u(P_{0}), ∇ϕ_{n}(P_{n}) → η and (D^{2}ϕ_{n})^{⋆}(P_{n}) → X. For a more complete discussion of jets, the interested reader is directed to [CIL] for the Euclidean environment and [B] for Heisenberg groups.

Given the viscosity solutions, it is natural to ask how they relate to the previous notions of solutions. It was shown via Lemma 4.1 in [B] that upper(lower) semicontinuous ε-weak sub(super-)solutions are ε-viscosity sub(super-)solutions. In addition, we have the following lemma.

Lemma 3.5. A p-sub(super-)harmonic function is a 0-viscosity sub(super-) solution. Hence, a p-harmonic function is a 0-viscosity solution.

Proof. We will do only the p-superharmonic case. We let u be a p-superhar-monic function in Ω and choose P_{0} ∈ Ω. We let ϕ ∈ 𝒞_{sub}^{2} (Ω) be a function so that ϕ(P_{0}) = u(P_{0}), ∇_{0}ϕ(P_{0}) ≠ 0, and u(P) > ϕ(P) for P ≠ P_{0} . If −Δ_{p}ϕ(P_{0}) < 0 then by continuity, there is a small r > 0 so that −Δ_{p}ϕ < 0 in the ball B(P_{0}, r). Note we also have ∇_{0}ϕ(P) ≠ 0 in B(P_{0}, r). Define the strictly positive number m ≡ inf\{u(P) − ϕ(P) : d_{C}(P, P_{0}) = r\} and the 𝒞_{sub}^{2} function Φ ≡ ϕ + \frac{1}{2}m. We therefore have Φ is a 0-weak subsolution in B(P_{0}, r). Let g be the unique (continuous) p-harmonic function equal to Φ on ∂B(P_{0}, r) whose existence is guaranteed by Theorem 3.2. Using the comparison principle (Lemma 3.1), we conclude Φ ≤ g ≤ u in B(P_{0}, r), contrary to Φ(P_{0}) > u(P_{0}).

We now have existence of all three notions of solutions, but a comparison principle only for the first two. In the next section, we correct this deficiency.

4. Equivalence of notions of solution

In this section, our goal is to prove the equivalence of the various notions of solution. As a consequence, we obtain a comparison principle for viscosity solutions to equation (3.4) when ε = 0. We recall that Ω is a bounded domain. Our proof will rely heavily on the Heisenberg geometry.

We begin with a technical lemma whose Euclidean version is Lemma 3.2 in [JLM].

Lemma 4.1. Let v ∈ W_{loc}^{1,p} be a continuous ε-weak solution. Let P_{0} ∈ Ω and let ϕ ∈ 𝒞_{sub}^{2} (Ω) be a function such that v − ϕ has a strict local minimum at P_{0} . Then

lim sup_{{P →P_{0} \atop P ≠P_{0}}}(−Δ_{p}ϕ(P)) ≥ ε

provided that ∇_{0}ϕ(P_{0}) ≠ 0 or P_{0} is an isolated critical point.

Proof. By left translation we may assume P_{0} = 0 and by adding constants, ϕ(0) = v(0). If the conclusion is false, then there is an r_{1} > 0 so that ∇_{0}ϕ(P) ≠ 0 and −Δ_{p}ϕ(P) < ε when 0 < 𝒩 (P) < r_{1} . Because v − ϕ has a strict local minimum at 0, there is an r_{2} > 0 so that v(P) > ϕ(P) when 0 < 𝒩 (P) < r_{2} . Let r = min\{\frac{1}{2}r_{1}, \frac{1}{2}r_{2}\}. Then ∇_{0}ϕ(P) ≠ 0, −Δ_{p}ϕ(P) < ε and v > ϕ when 0 < 𝒩 (P) ≤ r. We next define the strictly positive constant m = min{v(P) − ϕ(P) : 𝒩 (P) = r\} and consider the function ϕ̃ = ϕ + \frac{1}{2}m. By construction, ϕ̃ ∈ 𝒞_{sub}^{2} (Ω) and

(−Δ_{p}ϕ̃(P)) = (−Δ_{p}ϕ(P)).

Let ψ ∈ C_{0}^{∞} be a non-negative test function with compact support contained in the ball B_{𝒩} (r). Let 0 < ϱ < r and consider the annulus A ≡ B_{𝒩} (r) \ B_{𝒩} (ϱ). Using the identity

div_{ℋ} (ψ∥∇_{0}ϕ̃∥^{p−2}∇_{0}ϕ̃) = ψΔ_{p}ϕ̃ + ∥∇_{0}ϕ̃∥^{p−2}⟨∇_{0}ϕ̃, ∇_{0}ψ⟩

we obtain

∫_{A} ∥∇_{0}ϕ̃∥^{p−2}⟨∇_{0}ϕ̃, ∇_{0}ψ⟩ dV = − ∫_{A} ψΔ_{p}ϕ̃ dV + ∫_{A} div_{ℋ} (ψ∥∇_{0}ϕ̃∥^{p−2}∇_{0}ϕ̃) dV
≤ ε ∫_{A} ψ dV + ∫_{∂A} ψ∥∇_{0}ϕ̃∥^{p−2}∇_{0}ϕ̃ · ν dS
≤ ε ∫_{B_{𝒩} (r)} ψ dV − ∫_{B_{𝒩} (ϱ)} ψ∥∇_{0}ϕ̃∥^{p−2}∇_{0}ϕ̃ · ν dS

where ν is the outward unit (Euclidean) normal. We now estimate the last integral.

|∫_{B_{𝒩} (ϱ)} ψ∥∇_{0}ϕ̃∥^{p−2}∇_{0}ϕ̃ · ν dS| ≤ ∥ψ∥_{∞}∥∇_{0}ϕ̃∥_{∞}^{p−1} ∫_{B_{𝒩} (ϱ)} dS ≲ ϱ^{2n+1}.

Letting ϱ → 0, we have

∫_{B_{𝒩} (r)} ∥∇_{0}ϕ̃∥^{p−2}⟨∇_{0}ϕ̃, ∇_{0}ψ⟩ dV ≤ ε ∫_{B_{𝒩} (r)} ψ dV.

Thus, ϕ̃ is an ε-weak subsolution.

By the comparison principle for ε-weak solutions, ϕ̃ ≤ v in B_{𝒩} (r) because ϕ̃ ≤ v on ∂B_{𝒩} (r) by construction. However, we have

ϕ̃(0) = ϕ(0) + \frac{1}{2}m = v(0) + \frac{1}{2}m > v(0).

This contradiction finishes the proof.

Note that in the case when p ≥ 2, by continuity we have −Δ_{p}ϕ(P_{0}) ≥ ε and so ∇_{0}ϕ(P) ≠ 0 near P_{0} .

We next consider the function φ: H_{n} × H_{n} ↦ R given by

\begin{matrix}φ(P, Q) & = \frac{1}{m} ∑_{i=1}^{2n} |x_{i} − y_{i}|^{m} + \frac{1}{m}|z_{1} − z_{2} + \frac{1}{2} ∑_{i=1}^{n}(x_{n+i}y_{i} − x_{i}y_{n+i})|^{m} \\ ≡^{def} \frac{1}{m} ∑_{i=1}^{2n} |x_{i} − y_{i}|^{m} + \frac{1}{m}|ζ(P, Q)|^{m} \end{matrix}

for some large positive integer m ≥ 4. The important properties of φ are found in the following lemma.

Lemma 4.2. As above, let m ≥ 4. Let the vector η be given by

η = ( |x_{1} − y_{1}|^{m−2}(x_{1} − y_{1}) ; |x_{2} − y_{2}|^{m−2}(x_{2} − y_{2}) ; ⋮ ; |x_{2n} − y_{2n}|^{m−2}(x_{2n} − y_{2n}) ; |ζ(P, Q)|^{m−2}ζ(P, Q) ) .

Recall that the differential of left multiplication with respect to P, denoted DL_{P} , is given by

( \begin{matrix}I_{2n×2n} & 𝒫 \\ 0_{1×2n} & 1 \end{matrix} )

where the 2n × 1 vector 𝒫 is given by

(−\frac{1}{2}x_{n+1}, −\frac{1}{2}x_{n+2}, …, −\frac{1}{2}x_{2n}, \frac{1}{2}x_{1}, \frac{1}{2}x_{2}, …, \frac{1}{2}x_{n})^{T}

with a similar definition for DL_{Q} using the 2n × 1 vector 𝒬 given by

(−\frac{1}{2}y_{n+1}, −\frac{1}{2}y_{n+2}, …, −\frac{1}{2}y_{2n}, \frac{1}{2}y_{1}, \frac{1}{2}y_{2}, …, \frac{1}{2}y_{n})^{T} .

Define the (2n + 1) × (2n + 1) matrix ℳ by

ℳ_{ij} = \{ \begin{matrix}(m − 1)|x_{i} − y_{i}|^{m−2}, & i = j, 1 ≤ i ≤ 2n, \\ (m − 1)|ζ(P, Q)|^{m−2}, & i = j = 2n + 1, \\ 0, & i ≠ j, \end{matrix}

and denote Euclidean differentiation with respect to the point R by D_{R}. We then have the following properties:

(1) D_{P} φ(P, Q) = DL_{Q}η, D_{Q}φ(P, Q) = −DL_{P} η, (2) D_{P} η = ℳ DL_{Q}^{T}, D_{Q}η = −ℳ DL_{P}^{T} ,

(3) DL_{P} DL_{Q} = DL_{Q}DL_{P} ,

(4) DL_{P} D_{P} φ(P, Q) = −DL_{Q}D_{Q}φ(P, Q) ≡ Υ(P, Q),

(5)  \begin{matrix}DL_{P} (D_{P P} φ(P, Q)DL_{P}^{T} + D_{P Q}φ(P, Q)DL_{Q}^{T}) \\ = \frac{1}{2}|ζ(P, Q)|^{m−2}ζ(P, Q) ( \begin{matrix}0_{n×n} & −I_{n×n} & 0_{n×1} \\ I_{n×n} & 0_{n×n} & 0_{n×1} \\ 0_{1×n} & 0_{1×n} & 0cr \end{matrix} ) \end{matrix} ,

(6)  \begin{matrix}DL_{Q}(D_{QQ}φ(P, Q)DL_{Q}^{T} + D_{QP} φ(P, Q)DL_{P}^{T}) \\ = \frac{1}{2}|ζ(P, Q)|^{m−2}ζ(P, Q) ( \begin{matrix}0_{n×n} & I_{n×n} & 0_{n×1} \\ −I_{n×n} & 0_{n×n} & 0_{n×1} \\ 0_{1×n} & 0_{1×n} & 0 \end{matrix} ) \end{matrix} .

(7) Let ξ ∈ h_{n} be a vector. Then

\begin{matrix}⟨D_{P P} φ(P, Q)DL_{P}^{T} ξ, DL_{P}^{T} ξ⟩ + ⟨D_{P Q}φ(P, Q)DL_{Q}^{T}ξ, DL_{P}^{T} ξ⟩ \\ + ⟨D_{QP} φ(P, Q)DL_{P}^{T} ξ, DL_{Q}^{T}ξ⟩ + ⟨D_{QQ}φ(P, Q)DL_{Q}^{T}ξ, DL_{Q}^{T}ξ⟩ = 0 \end{matrix}

(8) Let ξ ∈ h_{n} be a vector. We define ξ̄ to be the projection of ξ onto V_{1} . That is, if ξ = (ξ_{1}, ξ_{2}, …, ξ_{2n+1}), then ξ̄ = (ξ_{1}, ξ_{2}, …, ξ_{2n}). We then have

∥( \begin{matrix}D_{P P} φ(P, Q) & D_{P Q}φ(P, Q) \\ D_{QP} φ(P, Q) & D_{QQ}φ(P, Q) \end{matrix} ) ( \begin{matrix}DL_{P}^{T} ξ \\ DL_{Q}^{T}ξ \end{matrix} )∥^{2} = \frac{1}{2}∥ξ̄∥^{2}|ζ(P, Q)|^{2m−2}.

Proof. The first three properties are elementary calculations and left to the reader. The fourth follows from the first three. We therefore turn our attention to the last four. Let M_{P Q} be the left-hand side of Property (5). Then,

M_{P Q} = DL_{P} (D_{P} (DL_{Q}η)DL_{P}^{T} + D_{Q}(DL_{Q}η)DL_{Q}^{T})
= DL_{P} (DL_{Q}D_{P} ηDL_{P}^{T} + D_{Q}(DL_{Q})ηDL_{Q}^{T} + DL_{Q}D_{Q}ηDL_{Q}^{T})
= DL_{P} (DL_{Q}ℳ DL_{Q}^{T}DL_{P}^{T} + D_{Q}(DL_{Q})ηDL_{Q}^{T} − DL_{Q}ℳ DL_{P}^{T} DL_{Q}^{T})
= DL_{P} (D_{Q}(DL_{Q})ηDL_{Q}^{T})

and so we are left to compute only the derivative of the matrix DL_{Q}. Knowing the definition of η above and the formula for DL_{Q} as given above, we see that when 1 ≤ i ≤ n, we have D_{y_{i}}(DL_{Q}) is a matrix with every entry 0 except for the (i + n, 2n + 1) entry, which is \frac{1}{2} . When n < i < 2n, we have D_{y_{i}}(DL_{Q}) has all entries 0 except for the (i − n, 2n + 1) entry, which is −\frac{1}{2} . Clearly, D_{z_{2}}(DL_{Q}) is the 0 matrix. We then compute

D_{Q}(DL_{Q})η = \frac{1}{2}|ζ(P, Q)|^{m−2}ζ(P, Q) ( \begin{matrix}0_{n×n} & −I_{n×n} & 0_{n×1} \\ I_{n×n} & 0_{n×n} & 0_{n×1} \\ 0_{1×n} & 0_{1×n} & 0 \end{matrix} ) .

We then have

( \begin{matrix}I_{2n×2n} & 𝒫 \\ 0_{1×2n} & 1 \end{matrix} ) ( \begin{matrix}0_{n×n} & −I_{n×n} & 0_{n×1} \\ I_{n×n} & 0_{n×n} & 0_{n×1} \\ 0_{1×n} & 0_{1×n} & 0 \end{matrix} ) ( \begin{matrix}I_{2n×2n} & 0_{2n×1} \\ 𝒬^{T} & 1 \end{matrix} )

= ( \begin{matrix}I_{2n×2n} & 𝒫 \\ 0_{1×2n} & 1 \end{matrix} ) ( \begin{matrix}0_{n×n} & −I_{n×n} & 0_{n×1} \\ I_{n×n} & 0_{n×n} & 0_{n×1} \\ 0_{1×n} & 0_{1×n} & 0 \end{matrix} )

= ( \begin{matrix}0_{n×n} & −I_{n×n} & 0_{n×1} \\ I_{n×n} & 0_{n×n} & 0_{n×1} \\ 0_{1×n} & 0_{1×n} & 0 \end{matrix} )

and Property (5) follows. To prove Property (6), we let M_{P Q} be the left-hand side of the identity. Then,

M_{P Q} = DL_{Q}(−D_{Q}(DL_{P} η)DL_{Q}^{T} + D_{P} (−DL_{P} η)DL_{P}^{T} )
= DL_{Q}(−DL_{P} D_{Q}ηDL_{Q}^{T} − D_{P} (DL_{P})ηDL_{P}^{T} − DL_{P} D_{P} ηDL_{P}^{T} )
= DL_{Q}(DL_{P} ℳ DL_{P}^{T} DL_{Q}^{T} − D_{P} (DL_{P})ηDL_{P}^{T} − DL_{P} ℳ DL_{Q}^{T}DL_{P}^{T} )
= −DL_{Q}(D_{P} (DL_{P})ηDL_{P}^{T} )

and we compute D_{P} (DL_{P})η in the same way as the above computation for D_{Q}(DL_{Q})η and arrive at Property (6).

To prove Property (7), we note that the right-hand side can be written as

\begin{matrix}⟨DL_{P} (D_{P P} φ(P, Q)DL_{P}^{T} + D_{P Q}φ(P, Q)DL_{Q}^{T})ξ, ξ⟩ \\ + ⟨DL_{Q}(D_{QQ}φ(P, Q)DL_{Q}^{T} + D_{QP} φ(P, Q)DL_{P}^{T})ξ, ξ⟩. \end{matrix}

Using Properties (5) and (6) we see this is zero. Property (7) then follows.

Using the proofs of Properties (5) and (6), we have

( \begin{matrix}D_{P P} φ(P, Q) & D_{P Q}φ(P, Q) \\ D_{QP} φ(P, Q) & D_{QQ}φ(P, Q) \end{matrix} ) ( \begin{matrix}DL_{P}^{T} ξ \\ DL_{Q}^{T}ξ \end{matrix} )

= ( \begin{matrix}(D_{P P} φ(P, Q)DL_{P}^{T} + D_{P Q}φ(P, Q)DL_{Q}^{T})ξ \\ (D_{QP} φ(P, Q)DL_{P}^{T} + D_{QQ}φ(P, Q)DL_{Q}^{T})ξ \end{matrix} )

= \frac{1}{2}|ζ(P, Q)|^{m−2}ζ(P, Q) [{( \begin{matrix}0_{n×n} & −I_{n×n} & 0_{n×1} \\ I_{n×n} & 0_{n×n} & 0_{n×1} \\ 0_{1×n} & 0_{1×n} & 0 \end{matrix} ) ξ \atop ( \begin{matrix}0_{n×n} & I_{n×n} & 0_{n×1} \\ −I_{n×n} & 0_{n×n} & 0_{n×1} \\ 0_{1×n} & 0_{1×n} & 0 \end{matrix} ) ξ} ] .

We then see that

∥( \begin{matrix}D_{P P} φ(P, Q) & D_{P Q}φ(P, Q) \\ D_{QP} φ(P, Q) & D_{QQ}φ(P, Q) \end{matrix} ) ( \begin{matrix}DL_{P}^{T} ξ \\ DL_{Q}^{T}ξ \end{matrix} )∥^{2} = \frac{1}{2}∥ξ̄∥^{2}|ζ(P, Q)|^{2m−2}

and Property (8) is proved.

We will use the function φ above as a penalty function in our proof of a preliminary comparison principle. A key step in the proof will be the twisting of the Euclidean jets into Heisenberg jets.

Lemma 4.3 ([B, Lemma 3.4]). Let DL_{P_{0}} be the differential of the left multiplication map at the point P_{0} , let J_{eucl}^{2,+}u(P_{0}) be the traditional Euclidean superjet of u at the point P_{0} and let (η, X) ∈ R^{2n+1} × S^{2n+1} . Then,

(η, X) ∈ J̄_{eucl}^{2,+}u(P_{0})

gives the element

(DL_{P_{0}}η, (DL_{P_{0}} X (DL_{P_{0}})^{T})_{2n}) ∈ J̄^{2,+}u(P_{0})

with the convention that for any matrix M, M_{n} is the n × n principal minor.

We now prove a preliminary comparison principle.

Theorem 4.4. Fix ε > 0 and 1 < p < ∞. Let v be a continuous ε-weak solution and let u be a 0-viscosity subsolution so that u ≤ v on ∂Ω. Then u ≤ v in Ω.

Proof. Suppose that sup(u − v) > 0 occurs at the interior point P_{0} . For each positive integer j, we consider the function ψ_{j}: H_{n} × H_{n} ↦ R defined by

ψ_{j}(P, Q) = u(P) − v(Q) − jφ(P, Q)

where φ(P, Q) is the function from Lemma 4.2, with m chosen so that m > max{4, p/(p − 1), p}. Following the scheme of [B] and [CIL], we let the maximum of ψ_{j} occur at (P_{j}, Q_{j}) and observe for large j, these are interior points. In addition, these points tend to P_{0} as j → ∞. Using the Euclidean results of [CIL] and the above twisting lemma, we have

(jΥ(P_{j}, Q_{j}), 𝒳_{j}) ∈ J̄^{2,+}u(P_{j}) and (jΥ(P_{j}, Q_{j}), 𝒴_{j}) ∈ J̄^{2,−}v(Q_{j})

where Υ(P, Q) = DL_{P} D_{P} φ(P, Q) = −DL_{Q}D_{Q}φ(P, Q) as detailed in Lemma 4.2.

Claim 4.5. By passing to a subsequence if needed, we may assume P_{j} ≠ Q_{j}.

Proof. Fix j > 0. By definition, we have for any P and Q,

u(P) − v(Q) − jφ(P, Q) ≤ u(P_{j}) − v(Q_{j}) − jφ(P_{j}, Q_{j})

and so when P_{j} = P, we have

v(Q) ≥ v(Q_{j}) + jφ(P_{j}, Q_{j}) − jφ(P_{j}, Q).

Defining the function β(Q) by

β(Q) = v(Q_{j}) + jφ(P_{j}, Q_{j}) − jφ(P_{j}, Q) − φ(Q_{j}, Q)

we see that v − β has a strict local minimum at Q_{j} and Q_{j} is an isolated critical point. Applying Lemma 4.1, we have

(4.1)  lim sup_{Q→Q_{j}}(−Δ_{p}β(Q)) ≥ ε.

Suppose now that P_{j} = Q_{j}. Then β(Q) = v(Q_{j}) − (j + 1)φ(Q_{j}, Q). We then need to estimate Δ_{p}β(Q). Using the non-divergence form of the p-Laplacian (equation (3.4)) and the definition of β(Q), we have

|Δ_{p}β(Q)| ≲ ∥∇_{0}φ(Q_{j}, Q)∥^{p−2}|tr(D^{2}φ)^{⋆}(Q_{j}, Q) + ∥(D^{2}φ)^{⋆}(Q_{j}, Q)∥ |.

Using Lemma 4.2, we have

∥∇_{0}φ(Q_{j}, Q)∥ ∼ ∥η∥ ∼ φ(Q_{j}, Q)^{(m−1)/m}.

376 Thomas Bieske We note that given the standard vectors e_{k} with every entry 0 except for the k th entry which is equal to 1, we see that for any matrix A,

tr(A) = ∑⟨Ae_{k}, e_{k}⟩

and so

| tr(D^{2}φ)^{⋆}(Q_{j}, Q)| ≲ ∥(D^{2}φ)^{⋆}(Q_{j}, Q)∥.

We then conclude via Lemma 4.2

|tr(D^{2}φ)^{⋆}(Q_{j}, Q) + ∥(D^{2}φ)^{⋆}(Q_{j}, Q)∥ | ≲ ∥ℳ ∥ ∼ φ(Q_{j}, Q)^{(m−2)/m} so that

|Δ_{p}β(Q)| ≲ (φ(Q_{j}, Q)^{1/m})^{(m−1)(p−2)+(m−2)}.

Since m > p/(p − 1), we would have

lim_{{Q→Q_{j} \atop Q≠Q_{j}}} (−Δ_{p}β(Q)) = 0.

This contradicts equation (4.1).

Proceeding as in [B], u is a viscosity subsolution to equation (3.4) with ε = 0. That is,

\begin{aligned}0 & ≥ −(∥jΥ(P_{j}, Q_{j})∥^{p−2} tr(𝒳_{j})^{⋆} \\ & + (p − 2)∥jΥ(P_{j}, Q_{j})∥^{p−4}⟨𝒳_{j} jΥ(P_{j}, Q_{j}), jΥ(P_{j}, Q_{j})⟩). \end{aligned}

Using Lemmas 3.5 and 4.1 along with the definition of J̄^{2,−}, we have

\begin{aligned}ε & ≤ −(∥jΥ(P_{j}, Q_{j})∥^{p−2} tr(𝒴_{j})^{⋆} \\ & + (p − 2)∥jΥ(P_{j}, Q_{j})∥^{p−4}⟨𝒴_{j} jΥ(P_{j}, Q_{j}), jΥ(P_{j}, Q_{j})⟩). \end{aligned}

Subtracting these two inequalities, we have (4.2)

0 < ε < j^{p−2}∥Υ(P_{j}, Q_{j})∥^{p−2}(tr(𝒳_{j}) − tr(𝒴_{j}))
+ (p − 2)j^{p−2}∥Υ(P_{j}, Q_{j})∥^{p−4}(⟨𝒳_{j}Υ(P_{j}, Q_{j}), Υ(P_{j}, Q_{j})⟩
− ⟨𝒴_{j}Υ(P_{j}, Q_{j}), Υ(P_{j}, Q_{j})⟩).

As in the proof of the above claim, we have ∥Υ(P_{j}, Q_{j})∥ ∼ φ(P_{j}, Q_{j})^{(m−1)/m}.

Given a vector η ∈ V_{1} , we denote its extension to h_{n} by η̃. That is, η = (η_{1}, η_{2}, …, η_{2n}) yields η̃ = (η_{1}, η_{2}, …, η_{2n}, 0). Using the formulas for the matrices 𝒳_{j} and 𝒴_{j} given by Lemma 4.3 and the standard estimate on the matrix ordering ([CIL, Theorem 3.2]) produces

⟨𝒳_{j}Υ(P_{j}, Q_{j}), Υ(P_{j}, Q_{j})⟩ − ⟨𝒴_{j}Υ(P_{j}, Q_{j}), Υ(P_{j}, Q_{j})⟩
= ⟨X(DL_{P_{j}}^{T} Υ̃(P_{j}, Q_{j})), DL_{P_{j}}^{T} Υ̃(P_{j}, Q_{j})⟩
− ⟨Y (DL_{Q_{j}}^{T} Υ̃(P_{j}, Q_{j})), DL_{Q_{j}}^{T} Υ̃(P_{j}, Q_{j})⟩
≤ j⟨𝒟ξ, ξ⟩

where the matrix 𝒟 is the Euclidean second derivative of φ given by

𝒟 = [ \begin{matrix}D_{P P} φ(P_{j}, Q_{j}) & D_{P Q}φ(P_{j}, Q_{j}) \\ D_{QP} φ(P_{j}, Q_{j}) & D_{QQ}φ(P_{j}, Q_{j}) \end{matrix} ]

and the vector

ξ = (DL_{P_{j}}^{T} Υ̃(P_{j}, Q_{j}) ⊕ DL_{Q_{j}}^{T} Υ̃(P_{j}, Q_{j})).

We then conclude via Properties (7) and (8) that

⟨𝒳_{j}Υ(P_{j}, Q_{j}), Υ(P_{j}, Q_{j})⟩ − ⟨𝒴_{j}Υ(P_{j}, Q_{j}), Υ(P_{j}, Q_{j})⟩
≲ j∥Υ(P_{j}, Q_{j})∥^{2}φ(P_{j}, Q_{j})^{(2m−2)/m}
≲ jφ(P_{j}, Q_{j})^{(2m−2)/m}φ(P_{j}, Q_{j})^{(2m−2)/m}
= jφ(P_{j}, Q_{j})^{(4m−4)/m}.

As in the proof of the claim, we write the trace difference as

tr(𝒳_{j}) − tr(𝒴_{j}) = ∑_{k=1}^{2n}⟨𝒳_{j}e_{k}, e_{k}⟩ − ⟨𝒴_{j}e_{k}, e_{k}⟩

and following the previous calculation, we obtain

tr(𝒳_{j}) − tr(𝒴_{j}) ≲ j(φ(P_{j}, Q_{j})^{(2m−2)/m})

so that with equation (4.2), we obtain

0 < ε ≲ j^{p−1}(φ(P_{j}, Q_{j})^{(m−1)/m})^{p−2}φ(P_{j}, Q_{j})^{(2m−2)/m}
+ j^{p−1}(φ(P_{j}, Q_{j})^{(m−1)/m})^{p−4}(φ(P_{j}, Q_{j})^{(4m−4)/m})
∼ j^{p−1}(φ(P_{j}, Q_{j})^{1/m})^{p(m−1)}.

Since m > p, we have (p(m − 1))(1/m) > p − 1. We arrive at a contradiction as j → ∞.

It is here where we stray significantly from the Euclidean proof in [JLM]. That proof relies on the C^{1,α} regularity of the solutions. This is not known for the Heisenberg group, although a recent result [DM] has proved this regularity for p near 2. We therefore adopt a completely different approach beginning with the next lemma. The first difference is that we only have a weaker version of Lemma 3.1 in [JLM], in that our sequence converges pointwise instead of locally uniformly.

Lemma 4.6. Let v be a p-harmonic function in Ω. For each ε ≥ 0, let v_{ε} be the continuous ε-weak solution equal to v on the boundary. Then v_{ε} → v pointwise as ε → 0.

Proof. Arguing as in [JLM], we see that v_{ε} → v in L^{p}. We may assume that ε ≤ 1 and observe that as noted above, if ε_{1} > ε_{2} , then v_{ε_{2}} is a weak subsolution to equation (3.2) for ε_{1} . By the comparison principle (Lemma 3.1), we have that v_{ε_{2}} ≤ v_{ε_{1}} when ε_{1} > ε_{2} . In particular for all ε > 0, v ≤ v_{ε}. We then conclude that

w = lim_{ε→0} v_{ε} = inf_{ε>0}{v_{ε}}

exists and v ≤ w. Since v_{ε} → w pointwise, we have |v_{ε}|^{p} → |w|^{p}. By the Lebesque dominated convergence theorem, (using v_{1} as dominator) we have v_{ε} → w in L^{p} so that actually, v = w.

Combining the previous theorem and lemma, we obtain the following consequence.

Lemma 4.7. Let 1 < p < ∞. 0-viscosity subsolutions are p-subharmonic. 0-viscosity supersolutions are p-superharmonic and 0-viscosity solutions are pharmonic.

Proof. The last statement follows from the first two and the second follows from the first by replacing u with −u. We let u be a 0-viscosity subsolution that is not p-subharmonic. Then there is a p-harmonic function v so that u ≤ v on ∂Ω but for some P ∈ Ω, we have u(P) > v(P). For ε ≤ 1, we let v_{ε} be ε-weak solutions equal to v on ∂Ω so that u ≤ v_{ε} on ∂Ω. By Lemma 4.6 we conclude for some ε near 0, u(P) > v_{ε}(P), contrary to Theorem 4.4.

Combining Lemmas 3.5 and 4.7, we have the following corollary.

Corollary 4.8. Let 1 < p < ∞. Then 0-viscosity sub(super-)solutions to equation (3.4) and p-sub(super-)harmonic functions coincide. In particular, for 1 < p < ∞, a function is p-harmonic if and only if it is a 0-viscosity solution to equation (3.4).

We are then able to conclude the following comparison principle.

Equivalence of solutions 379

Corollary 4.9. Let ε = 0. Let v be a viscosity supersolution to equation (3.4) and let u be a viscosity subsolution of equation (3.4) so that u ≤ v on ∂Ω. Then u ≤ v in Ω.

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Received 20 April 2005