volume 71 no. 3 2011, 509-524 - ijpam · [14]), and conversely there are continuously di erentiabl...

International Journal of Pure and Applied Mathematics————————————————————————–Volume 71 No. 3 2011, 509-524

EXISTENCE OF POTENTIAL FUNCTIONS OF

INTRINSIC GRADIENTS IN HEISENBERG GROUPS

Francesco Bigolin

Department of MathematicsUniversity of Trento

14, Via Sommarive, 38050, Povo (Trento), ITALY

Abstract: We give explicit conditions equivalent to the existence of a potentialfunction φ : ω ⊂ W ≡ R

2n → R for the intrinsic gradient ∇φφ, introducedby [2],[9] and studied in [4],[5] to characterize intrinsic regular hypersrfaces inHeisenberg groups.

AMS Subject Classification: 53C17, 35L60, 49Q15Key Words: Heisenberg Group, intrinsic regular hypersurfaces, Burgersequation

1. Introduction and Motivation of Paper

In the last years the study of H-regular intrinsic graphs in the Heisenberggroup H

n has been studied and developed by many authors (see [2], [4], [5], [9],[10]). H-regular intrinsic graphs are a class of intrinsic regular hypersurfacesin the setting of the Heisenberg group H

n = Cn × R ≡ R

2n+1, endowed witha left-invariant not euclidean metric d∞. Here hypersurface simply means atopological codimension 1 surface and by the words ”intrinsic” and ”regular”we will mean of notions involving respectively the group structure of H

n and itsdifferential structure. Let us point out that the class of H- regular surfaces isdeeply different from the class of Euclidean regular surfaces, in the sense thatthere are H-regular surfaces in H

1 ≡ R3 that are (Euclidean) fractal sets (see

[14]), and conversely there are continuously differentiable 2-submanifolds in R3

that are not H- regular hypersurfaces (see [10], Remark 6.2).As we will explain below, H-regular graphs have been described in [2], [9],

Received: July 8, 2011 c© 2011 Academic Publications, Ltd.

510 F. Bigolin

[10] by an intrinsic viewpoint with a parametric function φ : ω ⊂ W ≡ R2n →

R, where W is a subgroup of Hn homeomorphic to R

2n. In particular theexistence of parametrizations φ, which are not continuously differentiable butonly continuous, has been showed in [14].

Important characterizations of H-regular graphs have been given in [2], [4],[5], [9] using the intrinsic gradient ∇φφ (see (7) as definition). The parametriza-tion φ has been seen as a ”weak” solution of the problem ∇φφ = w, where w isa vector-function associated with the horizontal normal of the H-regular graph.The problem ∇φφ = w looks like a non linear PDE’s system, in particular theequation W φφ = wn+1, where W φφ := φy1

+φφt, looks like a conservation laws,the traditional Burgers’ equation, see [2], [4], [5], [15].

In the present paper we study the problem of the existence of a potentialfunction φ for a given vector-function w, i.e. we give explicite conditions onw so that there exists a function φ such that ∇φφ = w. In Theorem 10 weconsider the problem in H

1. In this case the system ∇φφ = w consists onlyin the scalar Burgers’ equation φy1

+ φφt = w. According to the results andproving strategies of [4], [5], [6], we cannot use the classical PDE’s theory forconservation laws, because the solution φ of the problem ∇φφ = w is ”a priori”only continuous. Indeed we need to introduce a new notion of weak solution ofconservation laws, the broad* solution, see (9) for definition and [4], [5], [6], [7],[15]. Intuitively a broad solution is a continuous function which is a solution ofthe conservation law along the characteristic lines, in our case the exponentialmaps of ∇φφ.

In Hn, n ≥ 2 the problem is more difficult. Given a regular vector function

w, Theorem 11 gives explicite conditions among the components wi’s so thatthere exists a potential function φ such that ∇φφ = w. In this case we askmore regularity on w with respect to the case of H

1 because the conditions of

Theorem 11 need the existence of the derivative∂φ

∂t. Indeed the strategy of the

proof is to linearize the problem ∇φφ = w using a function ψ =∂φ

∂t, so that it

is possible to apply classical ODEs’ theory.

The structure of the paper is the following: in Section 2 we present notationand a short introduction to the H-regular hypersurfaces in H

n. In particularwe recall in Theorem 4 the problem of the existence of a potential functionf : Ω ⊂ H

n → R for the problem of the horizontal gradient ∇Hf . In Section 3we present and prove the main results of the paper in Theorems 10 and 11. Atthe end of Section 3 we present some examples and remarks to explain clearlythe conditions of Theorem 11.

EXISTENCE OF POTENTIAL FUNCTIONS OF... 511

2. Notation and Preliminary Results

We shall denote the points of Hn by P = [z, t] = [x+ iy, t], z ∈ C

n, x, y ∈ Rn,

t ∈ R, and also by P = (x1, . . . , xn, y1, . . . , yn, t) = (x1, . . . , xn, xn+1, . . . , x2n, t).If P = [z, t], Q = [ζ, τ ] ∈ H

n and r > 0, following the notations of [18], we definethe group operation

P ·Q :=

[z + ζ, t+ τ − 1

2ℑm(z · ζ)

], (1)

the family of non isotropic dilations δr(P ) := [rz, r2t], for r > 0 and the groupof left-translations τP (Q) = P ·Q. We denote as P−1 := [−z,−t] the inverse ofP and as 0 the origin of R

2n+1.Moreover H

n can be endowed with the homogeneous norm ‖P‖∞ := max|z|,|t|1/2 and the distance d∞ we shall deal with is defined as d∞(P,Q) :=‖P−1 ·Q‖∞. From now on, U∞(P, r) will be the open ball with centre P and ra-dius r with respect to the distance d∞. We notice that U∞(P, r) is an EuclideanLipschitz domain in R

n2n+ 1.(Hn, d∞) provides the simplest example of a metric space that is not Eu-

clidean, even locally, but is still endowed with a sufficiently rich compatibleunderlying structure, due to the existence of intrinsic families of left transla-tions and dilations respectively induced from the group law (1) and dilations.Indeed, the geometry of H

n is noneuclidean at every scale, since it was provedin [17] that there are no bilitschitz maps from H

n to any Euclidean space.H

n is a Carnot group of step 2. Indeed its Lie algebra hn is (linearly)generated by

Xj =∂

∂xj− yj

2

∂

∂t, Yj =

∂

∂yj+xj

2

∂

∂t, for j = 1, . . . , n; T =

∂

∂t, (2)

and the only non-trivial commutator relations are

[Xj , Yj ] = T, for j = 1, . . . , n. (3)

We could use the notation Wi = Xi if i ≤ n and Wi = Yi if n + 1 ≤ i ≤ 2n,W2n+1 = T .

We shall identify vector fields and associated first order differential oper-ators; thus the vector fields X1, . . . ,Xn, Y1, . . . , Yn generate a vector bundleon H

n, the so called horizontal vector bundle HHn according to the notation

of Gromov (see [13]), that is a vector subbundle of THn, the tangent vector

bundle of Hn. Since each fiber of HH

n can be canonically identified with a

512 F. Bigolin

vector subspace of R2n+1, each section ϕ of HH

n can be identified with a mapϕ : H

n → R2n+1. At each point P ∈ H the horizontal fiber is indicated as HH

nP

and each fiber can be endowed with the scalar product 〈·, ·〉P and the associatednorm | · |P that make the vector fields X1, . . . ,Xn, Y1, . . . , Yn orthonormal.

If Ω is an open subset of Hn and k ≥ 0 is a non negative integer, the symbols

Ck(Ω), C∞(Ω) indicate the usual (Euclidean) spaces of real valued continuouslydifferentiable functions. We denote by Ck(Ω;HH

n) the set of all Ck-sectionsof HH

n where the Ck regularity is understood as regularity between smoothmanifolds.

The similarity among some statements in Hn with others in R

2n+1 is clearusing intrinsic notions of gradient for functions f : H

n → R and of divergencefor sections of HH

n.

Definition 1. If Ω is an open subset of Hn, f ∈ C1(Ω) and ϕ =

(ϕ1, . . . , ϕ2n) ∈ C1(Ω;HHn), define ∇Hf := (X1f, . . . ,Xnf, Y1f, . . . , Ynf);

÷Hϕ :=∑n

j=1Xjϕj + Yjϕn+j .

We shall denote by CkH(Ω) the set of continuous real functions f in Ω such

that ∇Hf in the sense of distribution is of class Ck−1 in Ω. Moreover, weshall denote by Ck

H(Ω;HH

n) the set of all sections ϕ of HHn whose canonical

coordinates ϕj belong to CkH(Ω) for j = 1, . . . , 2n. We denote by Ck(Ω;HH

n)the set of all Ck-sections of HH

n where the Ck regularity is understood asregularity between smooth manifolds. The notions of Ck

c (Ω;HHn), C∞(Ω;HH

n)and C∞

c (Ω;HHn) are defined analogously.

It is well-know that ∇H acts as a gradient operator in Hn. In particular

Lemma 2. Let Ω ⊆ Hn be a connected open set and let f ∈ L1

loc(Ω) suchthat ∇Hf = 0 in the sense of distributions. Then f ≡ cost in Ω.

Let us now introduce the curlH operator. It was been explicitely given in[12] and [3] respectively for n = 1 and n = 2, using the theory and the languageof differential forms in H

n, for detailed calculations see [6], [16] too.

Definition 3. Let F = (F1, ..., F2n) be a smooth section of HHn, let us

define if n = 1

curlHF := (2W1W2F1−W2W1F1−W 21F2 , W1W2F2−2W2W1F2−W 2

2F1); (4)

if n ≥ 2 curlHF :=

(Fi,j ,

1√2(Fh,h+n − Fh+1,h+1+n)

), (5)

with 1 ≤ i < j ≤ 2n, j 6= i+ n and h = 1, ..., n − 1 and Fi,j := (WiFj −WjFi).

For a better comprehension of the problem of this paper and in view ofTheorem 11, let us study study the existence of a potential function f : Ω → R


for the problem of the horizontal gradient ∇Hf = F where F := (F1, . . . , F2n)is a vector-function defined by Fj : Ω → R. Like in the euclidean setting,where the problem of the existence of a potential function is related to the DeRham complex theory, in the setting of H

n this problem is related to the Rumincomplex theory, see [3], [16], [12]. The Rumin Theorem yields an exactitudes’result for 1-differential forms in H

n.

Theorem 4. Let Ω ⊆ Hn be a simply connected open set and let F =

(F1, ..., F2n) with Fj ∈ D′(Ω) j = 1, ..., n. Then the following conditions areequivalent

i there exists f ∈ D′(Ω) such that ∇Hf = F in Ω in the sense of distributions.

ii curlHF = 0 in Ω in the sense of distributions.

An alternative proof of this result is given in [6], [12] and does not use thedifferential forms’ language: it is obtained by the commutator relations of thevector fields Xj , Yj , T .

Let us now introduce the notion of H-regular hypersurfaces in Hn, recalling

the following definitions and preliminary results.

Definition 5. ([10]). We say that S ⊂ Hn is an H-regular hypersurface if,

for every p ∈ S, there exist a neighbourhood U of p and a function f ∈ C1H(U)

such that ∇Hf 6= 0 and S ∩U = q ∈ U : f(q) = 0 . The horizontal normal to

S at p is νS(p) := − ∇Hf(p)

|∇Hf(p)| .

In the following we will denote W := (x, y, t) ∈ Hn : x1 = 0 and we will

write (y1, x2, . . . , xn, y2, . . . , yn, t) instead of (0, x2, . . . , xn, y1, . . . , yn, t) if n ≥ 2and (y1, t) instead of (0, y1, t) if n = 1.

We will use the notation v = (v2, . . . , vn, vn+2, . . . , v2n) := (x2, . . . , xn,

y2, . . . , yn) ∈ R2n−2 too. We shall denote

Ir(A0) :=(y1, t) ∈ W : |y1 − y0

1 | < r, |t− t0| < r

for A0 = (y01, t

0) ∈ W ≡ R2 if n = 1 and

Ir(A0) :=(y1, x2, . . . , xn, y2, . . . , yn, t) ∈ W : |y1 − y0

1| < r,∑ni=2

[(xi − x0i )

2 + (yi − y0i )

2] < r2, |t− t0| < r

for A0 = (y01, x

02, . . . , x

0n, y

02 , . . . , y

0n, t

0) ∈ W ≡ R2n if n ≥ 2.

The following Implicit Function Theorem for H-regular hypersurfaces isproved in [10], [11].

514 F. Bigolin

Theorem 6. (Implicit Function Theorem) Let Ω be an open set in Hn,

0 ∈ Ω, and let f ∈ C1H(Ω) be such that X1f(0) > 0, f(0) = 0. Let S := [z, t] ∈

Ω : f([z, t]) = 0, then there exist a connected open neighbourhood U of 0 andthere exists a unique continuous function φ : Iδ(0) ⊂ W → [−h, h] such that

S ∩ U = Φ(Iδ(0)

), where δ, h > 0 and Φ is defined as

Φ(y1, v, t) =(φ(y1, v, t), v2, . . . , vn, y1, vn+2, . . . , v2n, t− y1

2φ(y1, v, t)

)

if n ≥ 2Φ(y1, t) =

(φ(y1, t), y1, t− y1

2φ(y1, t)

)if n = 1.

By Theorem 6 we can see an H-regular surfaces as an intrinsic graph. A setS ⊂ H

n is called X1-graph, induced by a function φ : ω ⊂ W → R, if

S = A · φ(A) e1 : A ∈ ω . (6)

Let us recall the following improvement of Theorem 6 contained in [2]:

Theorem 7. Under the same assumption of Theorem 6, let Bφ the dis-

tribution Bφ :=∂φ

∂y1

+1

2

∂φ2

∂ton Iδ(0), where φ and δ are given by Theorem 6.

Then if n = 1

Bφ = − Y1f

X1f Φ,

if n ≥ 2 Xjφ = −Xjf

X1f Φ, Yjφ = − Yjf

X1f Φ, Bφ = − Y1f

X1f Φ

where the equalities must be understood in the sense of distributions on Iδ(0).

In [2] it has been proved that each H- regular graph Φ(ω) admits an intrinsicgradient ∇φφ ∈ C0(ω; R2n), in the sense of distributions, which shares a lot ofproperties with the Euclidean gradient, and it is defined, in distributional sense,by

W φφ := Y1φ+1

2T (φ2),

∇φφ :=

(X2φ, . . . ,Xnφ,W

φφ, Y2φ, . . . , Ynφ) if n ≥ 2W φφ if n = 1

.(7)

We also denote by ∇φ := (∇φ2, . . . ,∇φ

2n) the family of vector fields on R2n,

∇φj := Xj for j 6= n + 1 and ∇φ

n+1= W φ := Y1 + φT . We use the notation

∇Hφ := (X2φ, . . . ,Xnφ, Y2φ, . . . , Ynφ) too. Let us notice that W φφ looks like

the classical Burgers’ operator∂φ

∂y1

+ φ∂φ

∂t. Therefore a correct notion of “weak

solution” becomes fundamental in the study of the intrinsic gradient. Given a


continuous vector function w = (w2, . . . , w2n) : ω ⊂ W → R2n−1, we introduce

the concept of broad* solution of the system

∇φφ = w in ω, (8)

i.e. a continuous function φ : ω ⊂ W → R such that for every A0 ∈ ω,∀ j = 2, ..., 2n there exists an exponential map,

γBj (s) = exp(s∇φ

j )(B) : [−δ2, δ2] × Iδ2(A0) → Iδ1(A0) ⊂ ω (9)

where 0 < δ2 < δ1, s ∈ [−δ2, δ2], such that ∀B ∈ Iδ2(A0)

(E.1) γBj ∈ C1([−δ2, δ2])

(E.2)

γB

j = ∇φj γB

j

γBj (0) = B

(E.3) φ(γB

j (s))− φ

(γB

j (0))

=

∫ s

0

wj

(γB

j (r))dr ∀s ∈ [−δ2, δ2]

Indeed let us recall the following characterizations of H-regular graphs Φ(ω),given respectively in [2] Theorem 1.3, [4] Theorem 1.2 and [5] Theorem 1.2 (seealso [9] for a characterization in general Carnot groups).

Theorem 8. Let ω ⊂ W ≡ R2n be an open set and let φ : ω → R be a

continuous function. The following conditions are equivalent:

(i) The set S := Φ(ω) is an H-regular hypersurface and ν1S(p) < 0 for all

p ∈ S, where νS(p) =(ν1

S(p), . . . , ν2nS (p)

)is the horizontal normal to S

at p.

(ii) There exist w = (w2, . . . , w2n) ∈ C0(ω; R2n−1) and a family (φǫ)ǫ>0 ⊂C1(ω) such that, as ǫ → 0+, φǫ → φ and ∇φǫφǫ → w in L∞

loc(ω) and (8)holds in the sense of distributions.

(iii) There exists w = (w2, . . . , w2n) ∈ C0(ω; R2n−1) such that φ is a broad*

solution of the system (8).

Moreover, for all p ∈ S we have

νS(p) =

(− 1√

1 + |∇φφ|2,

∇φφ√1 + |∇φφ|2

)(Φ−1(p)) .

516 F. Bigolin

(iv) There exists w = (w2, . . . , w2n) ∈ C0(ω; R2n−1) such that φ is a distribu-tional solution of the system ∇φφ = w

In view of or results, let us recall the following theorem, see [11], where wewrite

πP (x, y, t) :=

n∑

j=1

(xjXj(P ) + yjYj(P )

)for P, (x, y, t) ∈ H

n.

Theorem 9. (Whitney Extension Theorem) Let F ⊂ Hn be a closed set,

and let f : F → R, k : F → HHn be two continuous functions. We set

R(Q,P ) :=f(Q) − f(P ) − 〈k(P ), πP (P−1 ·Q)〉P

d(P,Q),

and, if K ⊂ F is a compact set, ρK(δ) := sup|R(Q,P )| : P,Q ∈ K, 0 <

d∞(P,Q) < δ. If ρK(δ) → 0 as δ → 0 for every compact set K ⊂ F , then thereexist f : H

n → R, f ∈ C1H(Hn) such that f|F ≡ f and ∇Hf|F ≡ k.

3. Main Results and Proofs

Let us consider the problem of the existence of a potential function of theintrinsic gradient ∇φφ in H

1. For this pourpose in the following we will writeφ ∈ h1/2(D) if f is continuous on D ⊂ R

m and

limr→0

sup

|f(ξ) − f(ζ)||ξ − ζ|1/2

: ξ, ζ ∈ D, 0 < |ξ − ζ| < r

= 0.

The notion of h1/2

loc (D) is defined analogously.

Theorem 10. Let A0 = (y01, t

0) ∈ R2 = Ry1

×Rt. Let φ0 ∈ h1

2 ([t0−r0, t0+

r0]), w0 ∈ C0([t0 − r0, t0 + r0]) be given. Then there exist φ, w ∈ C0

(Ir0

(A0))

such that φ is a broad* solution of the initial value problem

W φφ = w in Ir0

(A0)φ(y0

1 , t) = φ0(t) ∀t ∈ [t0 − r0, t0 + r0]

if n = 1 (10)

for r0 small enough and w ≡ w0 on [t0 − r0, t0 + r0].

Proof. First let us observe without loss of generality we can assume thatA0 = (0, 0). Otherwise let us consider φ∗(y1, t) = φ(y1 − y0

1, t − t0) and the


associated initial value problem.

W φ∗φ∗ = w∗ in Ir0

((0, 0))φ∗(0, t) = φ∗0(t) ∀t ∈ [−r0, r0]

(11)

where w∗(y1, t) = w(y1 − y01, t − t0), φ∗0(t) = φ0(t − t0), (y1, t) ∈ Ir0

((0, 0)),t ∈ [−r0, r0]. Then it is easy to see by definition that φ is a broad* solution of(10) if and only if φ∗ is a broad* solution of (11).

Using the notation of Theorem 9 let F := (φ0(t), 0, t) : t ∈ [−r0, r0],f ≡ 0, k : F → HH

1 ≡ R2,

k(ξ, η, τ) :=

(1,−w0

(η, τ +

ξη

2

))if (ξ, η, τ) ∈ F.

Let Q = (φ0(t′), 0, t′), P = (φ0(t), 0, t) with t 6= t′ ∈ [−r0, r0], then

|R(Q,P )| =|f(Q) − f(P ) − 〈k(P ), πp(P

−1 ·Q)〉P |d∞(P,Q)

= (12)

=| − (φ0(t

′) − φ0(t)) + w0(t) · 0|max

|φ0(t′) − φ0(t)|,

√|t′ − t|

≤ |φ0(t′) − φ0(t)|√|t′ − t|

Since φ0 ∈ h1

2 ([−r0,+r0]), for compact setK ⊆ F , by (12) we get limδ→0+

ρK(δ) = 0.

Then by Whitney’s extension Theorem 9 there exists f : H1 → R, f ∈

C1H(H1) such that

f = 0 and ∇Hf = k inF. (13)

Let P0 := (φ0(0), 0, 0) ∈ F , g(P ) := f(P0 · P ) for P ∈ H1, S = P ∈ H

1 :g(P ) = 0. Since g ∈ C1

H(H1), 0 ∈ S, X1g(0) = 1 by Implicit Function

Theorem 6 and Proposition 7 there exists an open neighborhood U ⊆ H1 of 0

such thatS ∩ U is H-regular. (14)

Moreover there exist δ > 0 and an unique continuous function φ : I = [−δ, δ] ×[−δ2, δ2] → R such that

Φ(I)

= G1

H,eφ

(I)

= S ∩ U (15)

if Φ(y1, t) = (0, y1, t) · φ(y1, t)e1 with (y1, t) ∈ I

Bφ = w in I (16)

518 F. Bigolin

in the sense of distributions, where

w(y1, t) =

(− Y1g

X1g Φ

)(y1, t) = − Y1f

X1f

(P0 · Φ(y1, t)

).

Let us perform now the change of variable ψ : I → R2

ψ(y1, t) =(y1, t+ φ0(0)y1

)= (y1, t)

and let I := ψ(I). Let us define φ(y1, t) := φ0(0)+φ(y1, t−φ0(0)y1), (y1, t) ∈ I.

Then by (15)

S0 := τP0

(S ∩ U

)= τP0

(G1

H,eφ(I))

= G1H,φ(I). (17)

Let r0 > 0 so small such that Ir0(0, 0) ⊂ I. By (13), (14) and (17) we get that

φ(0, t) = φ0(t) ∀t ∈ [−r0, r0], (18)

G1H,φ(Ir0

(0, 0)) is H-regular. (19)

On the other hand it is easy to see that by (16)

Bφ = w in Ir0(0, 0) (20)

in the sense of distributions, where w(y1, t) = w(ψ−1(y1, t)

), (y1, t) ∈ Ir0

(0, 0).

Thus by (19) and (20) and Theorem 8 we get

φ ∈ h1

2

loc (Ir0(0, 0)) (21)

W φφ = w in Ir0(0, 0). (22)

Finally by (18), (19), (22) and Theorem 8 we get that φ is a broad* solution ofW φφ = w in ω.

A local uniqueness result for broad* solutions of (10) uniformly boundedin ω is given in [4] (see Theorem 3.8) using the theory of entropy solutions, aclass of distributional solutions which are admissible by a physical viewpoint,see [15].

Let us now consider the problem in Hn with n ≥ 2. In this case there are

regular solutions of the system (8) provided compatibility’s conditions amongthe components wi’s.


Theorem 11. Let us denote ω = (y01 − r0, y

01 + r0) × ω where ω ⊆ R

2n−2

is an open set and P0 = (y01 , v

0, t0) and r > 0. Let w = (w2, . . . , w2n) ∈C2(ω; R2n−1), n ≥ 2. Let us define

ψ(y1, v, t) :=(X2w2+n − Y2w2

)(y1, v, t);

E(y1, v, t) := e

(−∫ y1

y01

ψ(y′1, v, t) dy′1

)

I(y1, v, t) :=

∫ y1

y01

wn+1(y′1, v, t)

E(y′1, v, t)

dy′1;

E1(y1, v, t) := E(y1, v, t)I(y1, v, t);

a = (a2, ..., an, an+2, ..., a2n) aj :=XjE

E;

b = (b2, ..., bn, bn+2, ..., b2n) bj :=wj − XjE1

E;

where y01 ∈ R is fixed and wn+1 := (w2, ..., wn, wn+2, ..., w2n). Then the follow-

ing statements are equivalent:

i There exists φ ∈ C2(ω) such that ∇φφ = w in ω, i.e. (8);

ii There exists C ∈ C2(ω) such that

∇HC(v, t) = wn+1(y01 , v, t) ∀ (v, t) ∈ ω, (23)

a(y1, v, t)C(v, t) = b(y1, v, t) − b(y01 , v, t). (24)

∀y1 ∈ (y01 − r, y0

1 + r), ∀(v, t) ∈ ω. Moreover φ and C are linked by the relation

φ(y1, v, t) = E1(y1, v, t) + E(y1, v, t)C(v, t). (25)

Proof. It is not restrictive to assume y01 = 0.

i ⇒ ii Let us assume that there exists φ ∈ C2(ω) such that W φφ = w in ω.Let us observe that

∂φ

∂t= Tφ = [X2Y2 − Y2X2]φ = X2w2+n − Y2w2 =: ψ. (26)

Thus we can linearize the system getting

∇Hφ = wn+1 inω (27)

520 F. Bigolin

∂φ

∂y1

+ φψ = wn+1 inω (28)

For fixed (v, t) ∈ ω, by the uniqueness of linear ODE (28), we can represent φas

φ(y1, v, t) = E1(y1, v, t) + E(y1, v, t)φ(0, v, t) (29)

Let us denoteC(v, t) := φ(0, v, t) (v, t) ∈ ω

and let us prove (24). By (27) and (29) we get that

wn+1 = ∇H(E1 + EC) = ∇HE1 + ∇HE · C + E∇HC

and then ∀ (y1, v, t) ∈ ω

∇HC(v, t) + a(y1, v, t)C(v, t) = b(y1, v, t). (30)

By choosing y1 = 0, since b(0, v, t) = wn+1(0, v, t) and a(0, v, t) ≡ 0 we get atonce (23) and (24).

ii ⇒ i Let us assume that there exists C ∈ C2(ω) such that (23) and (24)hold. Let us define φ as in (29) with C(y1, t) ≡ φ(0, v, t), then it is easy toverify that W φφ = w in ω.

Remark 12. We need the hypotesis φ ∈ C2(ω) for the existence of Tφ.In general we cannot use the proof’ strategy of Theorem 4. Indeed ”a priori” ifφ is only continuous there is not a good definition of commutator for the fieldsXj ,W

φ, Yj , because φ could be not Lipschitz continuous.

Remark 13. Let us explicitly point out the system (8) differs from system∇Hφ = V . For instance, let us assume that w ∈ C2(R2n,R2n−1) such that

wn+1(y1, v, t) ≡ 0 in ω := R2n and (31)

wn+1(y1, v, t) = wn+1(v, t) (32)

with∇Hwn+1 6≡ 0 in ω (33)

Then compatibility’s condition (23) is satisfied with C ≡ cost in ω := R2n−1 by

Lemma 2. On the other hand since ψ ≡ 0 we haveE ≡ 1, E1(y1, v, t) = I(y1, v, t) = (y1 − y0

1)wn+1(v, t), a ≡ 0,

b(y1, v, t) = −(y1 − y01)∇Hwn+1(v, t). Then by (33) b(y1, v, t) − b(y0

1 , v, t) =

−∇HE1(y1, v, t) = −(y1 − y01)∇Hwn+1(v, t) 6≡ 0.

Therefore compatibility’s condition (24) is not satisfied and by Theorem 11there are not C2 solutions of the system (8).


We are going now to give some explicit regular solutions of the system(8) in H

2 by means of Theorem 11. We will assume in the examples belowthat φ ∈ C2(ω) is a solution of system (8), ω = (y0

1 − r0, y01 + r0) × ω =

(y01 − r0, y

01 + r0)×U(v0, r0)× (t0 − r0, t

0 + r0), where U(v0, r0) is the euclideanopen ball in R

2n−2 with centre v0 and radius r0. We will use the same notationsof Theorem 11.

Remark 14. Let us assume that ∃φ ∈ C2(ω) solution of (8). If C(v, t) ≡ 0in ω then b(y1, v, t) ≡ 0 in ω.

Indeed let us notice that by (24) we have

b(y1, v, t) − b(0, v, t) = a(y1, v, t)C(v, t) = 0 ∀(y1, v, t) ∈ ω, (34)

then by (23)

∇HC(v, t) = w3(y01, v, t) ≡ 0 ∀(y1, v, t) ∈ U(v0, r0) × (t0 − r0, t

0 + r0)

and by (25) φ(y1, v, t) = E1(y1, v, t) in ω. Let us observe that by definition

E(y01 , v, t) ≡ 1 E1(y

01 , v, t) ≡ 0,

therefore b(y01 , v, t) ≡ 0 and by (34) we conclude b(y1, v, t) ≡ 0 in ω.

Remark 15. Let us assume that a(y1, v, t) ≡ 0 in ω and that ∃φ ∈ C2(ω)solution of (8), then it is of the type φ(y1, v, t) = ψ(y1)t+ k(y1, v).

Indeed by the definition of a

0 = ∇HE(y1, v, t) = −E(y1, v, t)

∫ y1

y01

∇Hψ(y1, v, t) dy′1

∀y1 ∈ (y01 − r0, y

01 + r0), ∀(v, t) ∈ ω. Since for fixed (v, t) ∈ ω ∇Hψ(·, v, t) ∈

C0((y01 − r0, y

01 + r0); R

2) we can conclude that

∇Hψ ≡ 0 in ω (35)

By (35) and Lemma 2 we get that ψ = ψ(y1) y1 ∈ (y01 − r0, y

01 + r0). Therefore

by Theorem 11 and Theorem 4 there exists φ ∈ C2(ω) solution of the system(8) of the type

φ(y1, v, t) = ψ(y1)t+ k(y1, v) ∀(y1, v, t) ∈ ω.

522 F. Bigolin

Example 16. Let us assume now w = w(y1, v). Let us observe that in thiscase ψ = ψ(y1, v), E = E(y1, v), E1 = E1(y1, v), a = a(y1, v) and b = b(y1, v).By Theorem 11 each solution φ ∈ C2(ω) of the system (8) is of the type

φ(y1, v, t) = E1(y1, v) + E(y1, v)C(v, t) (36)

and we have∇HC(v, t) = w3(y

01, v) (37)

a(y1, v)C(v, t) = b(y1, v) − b(y01, v) (38)

∀y1 ∈ (y01 − r0, y

01 + r0), ∀v ∈ U(v0, r0), ∀t ∈ (t0 − r0, t

0 + r0). Recalling thatv = (x2, y2), by Theorem 4 the condition (37) is equivalent to

0 =

(∂2w4

∂x22

− ∂2w2

∂y2∂x2

)(y0

1 , v) =

(∂2w4

∂x2∂y2

− ∂2w2

∂y22

)(y0

1, v) (39)

Let us assume now that a(y1, v) 6≡ 0 in ω. Then by (38) we get that C(v, t) =C(v). Thus by (36) φ(y1, v, t) = φ(y1, v) provided (39) holds.On the other hand let a(y1, v) ≡ 0 in ω. In this case φ could depend on t. Forinstance, it is immediate to see that

φ(y1, v, t) =t

y1 + 2(y1, v, t) ∈ (−1, 1) × U(0, 1) × (−1, 1)

is a solution of the system (8) with

w(y1, v, t) := w(y1, v) =

(− y2

2(y1 + 2), 0,

x2

2(y1 + 2)

)∀(y1, v, t) ∈ ω.

Example 17. In the same assumptions of example 16, if w = w(y1) then

a solution φ of ∇φφ = w is such that∂φ

∂t= 0. Indeed let us observe that, since

w = w(y1), ψ = X2w2+n − Y2w2 = 0. We conclude so∂φ

∂t= ψ = 0.

Example 18. In the case w = w(v, t) we can find φ(y1, v, t) solutions

of ∇φφ = w such that∂φ

∂y1

6= 0. Let us assume in H2 w =

(−y2

2, t,

x2

2

),

ω = (−1, 1)4. Then φ(y1, v, t) = t+ e−y1 is a solution of the problem ∇φφ = w

in ω.

Example 19. In the case w = w(y1, t) we can find φ(y1, v, t) solutions

of ∇φφ = w such that∂φ

∂vi6= 0 for some i ∈ 2, ..., 2n. Let us assume in H

2

w = (1, 2y1, 0) , ω = (−1, 1)4. Then φ(y1, v, t) = x2 + y21 is a solution of the

problem ∇φφ = w in ω.


Acknowledgments

The author thanks F. Serra Cassano for useful and fruitful discussions on thesubject.

The author is supported by PRIN 2008 and University of Trento, Italy.

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