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Volume 2007, Article ID 48232, 16 pagesdoi:10.1155/2007/48232 Research Article Liouville Theorems for a Class of Linear Second-Order Operators with Nonnegative Characteristic Form Alessi

Volume 2007, Article ID 48232, 16 pages

doi:10.1155/2007/48232

Research Article

Liouville Theorems for a Class of Linear Second-Order Operators with Nonnegative Characteristic Form

Alessia Elisabetta Kogoj and Ermanno Lanconelli

Received 1 August 2006; Revised 28 November 2006; Accepted 29 November 2006 Recommended by Vincenzo Vespri

We report on some Liouville-type theorems for a class of linear second-order partial dif-ferential equation with nonnegative characteristic form The theorems we show improve our previous results

Copyright © 2007 A E Kogoj and E Lanconelli This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis-tribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

In this paper, we survey and improve some Liouville-type theorems for a class of hypoel-liptic second-order operators, appeared in the series of papers [1–4]

The operators considered in these papers can be written as follows:

ᏸ := N

i, j =1

∂ x i



a i j(x)∂ x j



+

N



i =1

b i(x)∂ x i − ∂ t, (1.1)

where the coefficients ai j, b i are t-independent and smooth in RN The matrix A =

(a i j)i, j =1, ,Nis supposed to be symmetric and nonnegative definite at any point ofRN

We will denote byz =(x,t), x ∈ R N,t ∈ R, the point ofRN+1, by Y the first-order

differential operator

Y : =N

i =1

and byᏸ0the stationary counterpart ofᏸ, that is,

ᏸ0:=

N



i, j =1

∂ x i



a i j(x)∂ x j



+

N



i =1

b i(x)∂ x i (1.3)

We always assume the operatorY to be divergence free, that is,N

i =1∂ x i b i(x) =0 at any pointx ∈ R N Moreover, as in [2], we assume the following hypotheses

(H1)ᏸ is homogeneous of degree two with respect to the group of dilations (d λ)λ>0 given by

d λ(x,t) =D λ(x),λ2t

,

D λ(x) = D λ



x1, ,x N



=λ σ1x1, ,λ σ N x N



whereσ =(σ1, ,σ N) is anN-tuple of natural numbers satisfying 1 = σ1≤ σ2≤

··· ≤ σ N When we say thatᏸ is d λ-homogeneous of degree two, we mean that

ᏸu

d λ(x,t)

= λ2(ᏸu)d λ(x,t)

∀ u ∈ C ∞

RN+1

(H2) For every (x,t),(y,τ) ∈ R N+1,t > τ, there exists an ᏸ-admissible path η : [0,T] →

RN+1such thatη(0) =(x,t), η(T) =(y,τ).

Anᏸ-admissible path is any continuous path η which is the sum of a finite number of

diffusion and drift trajectories

A di ffusion trajectory is a curve η satisfying, at any points of its domain, the inequality



η (s),ξ 2

≤ A

η(s)

ξ,ξ

Here·,· denotes the inner product inRN+1 andA(z) =  A(x,t) =  A(x) stands for the

(N + 1) ×(N + 1) matrix



A =

A 0

A drift trajectory is a positively oriented integral curve of Y.

Throughout the paper, we will denote byQ the homogeneous dimension ofRN+1with respect to the dilations (1.4), that is,

and assume

Then, theD λ-homogeneous dimension ofRNisQ −2≥3

We explicitly remark that the smoothness of the coefficients of ᏸ and the homo-geneity assumption in (H1) imply that thea i j’s and the b i’s are polynomial functions (see [5, Lemma 2]) Moreover, the “oriented” connectivity condition in (H1) implies the

hypoellipticity ofᏸ and of ᏸ0(see [1, Proposition 10.1]) For anyz =(x,t) ∈ R N+1, we define thed λ-homogeneous norm| z |by

| z | = (x,t) :=

| x |4+t2  1/4

where

| x | = x1, ,x N  = N

j =1



x2jσ/σ j 1/2σ

, σ =

N

j =1

σ j (1.11)

Hypotheses (H1) and (H2) imply the existence of a fundamental solutionΓ(z,ζ) of ᏸ

with the following properties (see [2, page 308]):

(i)Γ is smooth in{(z,ζ) ∈ R N+1 × R N+1 | z = ζ },

(ii)Γ(·,ζ) ∈ L1

loc(RN+1) andᏸΓ(·,ζ) = − δ ζ for everyζ ∈ R N+1, (iii)Γ(z, ·)∈ L1

loc(RN+1) andᏸ∗ Γ(z, ·)= − δ zfor everyz ∈ R N+1, (iv) lim supζ → z Γ(z,ζ) = ∞for everyz ∈ R N+1,

(v)Γ(0,ζ) →0 asζ → ∞,Γ(0,d λ(ζ)) = λ − Q+2 Γ(0,ζ),

(vi)Γ((x,t),(ξ,τ)) ≥0,> 0 if and only if t > τ,

(vii)Γ((x,t),(ξ,τ)) = Γ((x,0),(ξ,τ − t)).

In (iii)ᏸ∗denotes the formal adjoint ofᏸ

These properties of Γ allow to obtain a mean value formula at z =0 for the entire solutions toᏸu = 0 We then use this formula to prove a scaling invariant Harnack in-equality for the nonnegative solutions ᏸu = f inRN+1 Our first Liouville-type theorems will follow from this Harnack inequality All these results will be showed inSection 2

InSection 3, we show some asymptotic Liouville theorem for nonnegative solution to

ᏸu =0 in the halfspaceRN ×]− ∞, 0[ assuming thatᏸ, together with (H1) and (H2), is left invariant with respect to some Lie groups inRN+1

Finally, inSection 4some examples of operators to which our results apply are showed

2 Polynomial Liouville theorems

Throughout this section, we will assume thatᏸ in (1.1) satisfies hypotheses (H1) and (H2) LetΓ be the fundamental solution of ᏸ with pole at the origin With a standard procedure based on the Green identity forᏸ and by using the properties of Γ recalled in the introduction, one obtains a mean value formula atz =0 for the solution toᏸu =0 Precisely, for every point (0,T) ∈ R N+1andr > 0, define the ᏸ-ball centered at (0,T) and

with radiusr, as follows:

Ωr(0,T) : = ζ ∈ R N+1:Γ(0,T),ζ

>

1

r

Q −2 

Then, ifᏸu =0 inRN+1, one has

u(0,T) =

1

r

Q −2 

Ω (0,T) K(T,ζ)u(ζ)dζ, (2.2)

K(T,ζ) =



A(ξ) ∇ ξΓ,∇ ξΓ

andΓ stands for Γ((0,T),(ξ,τ)) Moreover, ·,· denotes the inner product inRNand∇ ξ

is the gradient operator (∂ ξ1, ,∂ ξ N)

Formula (2.2) is just one of the numerous extensions of the classical Gauss mean value theorem for harmonic functions For a proof of it, we directly refer to [6, Theorem 1.5]

We would like to stress that in this proof one uses our assumption divY =0

The kernelK(T, ·) is strictly positive in a dense open subset ofΩr(0,T) for every T,r >

0 (see [2, Lemma 2.3]) This property ofK(T, ·), together with thed λ-homogeneity ofᏸ, leads to the following Harnack-type inequality for entire solutions toᏸu =0

Theorem 2.1 Let u :RN+1 → R be a nonnegative solution to ᏸu = 0 inRN+1 Then, there exist two positive constants C = C(ᏸ) and θ = θ(ᏸ) such that

sup

C θr

u ≤ Cu(0,r2) ∀ r > 0, (2.4)

where, for ρ > 0, C ρ denotes the d λ -symmetric ball

C ρ:=z ∈ R N+1 | | z | < ρ

The proof of this theorem is contained in [2, page 310]

By using inequality (2.4) together with some basic properties of the fundamental solu-tionΓ, one easily gets the following a priori estimates for the positive solution to ᏸu = f

inRN+1

Corollary 2.2 Let f be a smooth function inRN+1 and let u be a nonnegative solution to

Then there exists a positive constant C independent of u and f such that

u(z) ≤ Cu

0,

|

z |

θ

 2

+| z |2 sup

| ζ |≤| z | /θ2

f (ζ) , (2.7)

where θ is the constant in Theorem 2.1

This result allows to use the Liouville-type theorem of Luo [5] to obtain our main result in this section

Theorem 2.3 Let u :RN+1 → R be a smooth function such that

ᏸu = p inRN+1,

where p and q are polynomial function Assume

u(0,t) = O

t m

Then, u is a polynomial function.

Proof We split the proof into two steps.

Step 1 There exists n > 0 such that

u(z) = O

| z | n

Indeed, lettingv : = u − q, we have

ᏸv = p − ᏸq inRN+1,

andv(0,t) = u(0,t) − q(0,t) = O(t n1) ast → ∞, for a suitablen1> 0 Moreover, since p

andᏸq are polynomial functions, (p − ᏸq)(z) = O( | z | m1) asz → ∞for a suitablem1> 0.

Then, by the previous corollary, there existsm2> 0 such that

v(z) = O

| z | m2 

From this estimate, sincev = u + q, and q is a polynomial function, the assertion (2.10) follows

Step 2 Since p is a polynomial function and ᏸ is d λ-homogeneous, there existsm ∈ N

such that

whereᏸ(m) =ᏸ◦ ··· ◦ ᏸ is the mth iterated of ᏸ It follows that

Moreover, since ᏸ is d λ -homogeneous and hypoelliptic, the same properties hold for

ᏸ(m+1) On the other hand, byStep 1,u(z) = O(z m) as z → ∞, so thatu is a tempered distribution Then, by Luo’s paper [5, Theorem 1],u is a polynomial function. 

Remark 2.4 It is well known that hypothesis (2.9) in the previous theorem cannot be removed Indeed, ifᏸ=Δ− ∂ tis the classical heat operator andu(x,t) =exp(x1+···+

x N+Nt), x =(x1, ,x N)∈ R Nandt ∈ R, we have

andu is not a polynomial function.

In the previous theorem, the degree of the polynomial functionu can be estimated in

terms of the ones ofp and q For this, we need some more notation If α =(α1, ,α N,α N+1)

is a multi-index with (N + 1) nonnegative integer components, we let

| α | d := σ1α1+···+σ N α N+ 2α N+1, (2.16)

and, ifz =(x,t) =(x1, ,x N,t) ∈ R N+1,

z α:= x α1

1 ··· x α N

As a consequence, we can write every polynomial functionp inRN+1, as follows:

p(z) = 

| α | dλ ≤ m

withm ∈ Z,m ≥0, andc α ∈ Rfor every multi-indexα If



| α | dλ = m

c α z α ≡0 inRN+1, (2.19) then we set

Ifp is independent of t, that is, if p is a polynomial function inRN, we denote by

the degree of p with respect to the dilations (D λ)λ>0 Obviously, in this case, degD λ p =

degd λ p.

Proposition 2.5 Let u, p :RN+1 → R be polynomial functions such that

Assume u ≥ 0 Thus, the following statements hold.

(i) If p ≡ 0, then u = constant.

(ii) If p ≡ 0, then

degd λ u =2 + degd λ p. (2.23) This proposition is a consequence of the following lemma

Lemma 2.6 Let u :RN+1 → R be a nonnegative polynomial function d λ -homogeneous of degree m > 0 Then ᏸu ≡ 0 inRN+1

Proof We argue by contradiction and assume ᏸu =0 Since u is nonnegative and d λ -homogeneous of strictly positive degree, we have

u(0,0) =0=min

Let us now denote byᏼ the ᏸ-propagation set of (0,0) inRN+1, that is, the set

ᏼ :=z ∈ R N+1: there exists anᏸ-admissible path η : [0,T] −→ R N+1,

s.t.η(0) =(0, 0),η(T) = z

From hypotheses (H2), we obtainᏼ= R N ×]− ∞, 0] so that, since (0, 0) is a minimum point ofu and the minimum spread all over ᏼ (see [7]), we have

u(z) = u(0,0) =0 ∀ z ∈ R N ×]− ∞, 0]. (2.26) Then, beingu a polynomial function, u ≡0 inRN+1 This contradicts the assumption

Proof of Proposition 2.5 Obviously, if u =constant, we have nothing to prove If we as-sumem : =degd λ u > 0 and prove that

m ≥2, p ≡0, degd λ p = m −2, (2.27) then it would complete the proof Let us writeu as follows:

u = u0+u1+···+u m, (2.28) whereu jis a polynomial functiond λ-homogeneous of degree j, j =0, ,m, and u m ≡0

inRN+1

Then

p = ᏸu = ᏸu0+ᏸu1+···+ᏸu m, (2.29) and, sinceᏸ is d λ-homogeneous of degree two,



ᏸu j

d λ(x)

= λ j −2ᏸu j(x) (2.30)

so thatᏸu0= ᏸu1≡0 and degd λ ᏸu j = j −2 if and only ifᏸu j ≡0

On the other hand, the hypothesisu ≥0 impliesu m ≥0 so that, beingu m ≡0 andd λ -homogeneous of degreem > 0, byLemma 2.6, we getᏸu m ≡0 Hencem ≥2 Moreover,

by (2.29),p = ᏸu ≡0 and

degd λ p =degd λ ᏸu m = m −2. (2.31)

 This proposition allows us to make more precise the conclusion ofTheorem 2.3 In-deed, we have the following

Proposition 2.7 Let u, p,q :RN+1 → R be polynomial functions such that

ᏸu = p inRN+1,

Then

degd λ u ≤max

2 + degd λ p,deg d λ q

In particular, and more precisely, if q = 0, that is, if u ≥ 0, then

degd λ u =2 + degd λ p if p ≡0,

Proof If q ≡0, the assertion is the one ofProposition 2.5 Supposeq ≡0 By lettingv : =

u − q, we have

ByProposition 2.5, we have

degd λ v ≤2 + degd λ(p − ᏸq) ≤2 + max

degd λ p,deg d λ q −2

=max

2 + degd λ p,deg d λ q

(2.36)

Proposition 2.7, together withTheorem 2.3, extends and improves the Liouville-type theorems contained in [2,4] (precisely [2, Theorem 1.1] and [4, Theorem 1.2])

FromTheorem 2.3andProposition 2.7, we straightforwardly get the following poly-nomial Liouville theorem for the stationary operatorᏸ0in (1.3)

Theorem 2.8 Let P,Q :RN → R be polynomial functions and let U :RN → R be a smooth function such that

ᏸ0U = P, U ≥ Q, inRN (2.37)

Then, U is a polynomial function and

degD λ U ≤max

2 + degD λ P,deg D λ Q

In particular, and more precisely, if Q ≡ 0, that is, if U ≥ 0, then

degD λ U =2 + degD λ P if P ≡0,

Proof Let us define

u(x,t) = U(x), p(x,t) = P(x), q(x,t) = Q(x). (2.40) Thenp, q are polynomial functions inRN+1andu is a smooth solution to the equation

such thatu ≥ q Moreover,

u(0,t) = U(0) = O(1) as t −→ ∞ (2.42) Then, byTheorem 2.3,u is a polynomial function inRN+1 This obviously implies that

U is a polynomial inRN The second part of the theorem immediately follows from

Remark 2.9 The class of our stationary operatorsᏸ0also contains “parabolic”-type op-erators like, for example, the following “forward-backward” heat operator

ᏸ0:= ∂2x1+x1∂ x2 inR 2

Nevertheless, in Theorem 2.8, we do not require any a priori behavior at infinity, like condition (2.9) inTheorem 2.3

3 Asymptotic Liouville theorems in halfspaces

The operatorᏸ in our class do not satisfy the usual Liouville property Precisely, if u is a

nonnegative solution to

then we cannot conclude that u ≡constant without asking an extra condition on the solutionu (seeTheorem 2.3andRemark 2.4)

However, if we also assume thatᏸ is left translation invariant with respect to the com-position law of some Lie group inRN+1 , then we can show that every nonnegative solution

of ( 3.1 ) is constant at t = −∞

To be precise, let us fix the new hypothesis onᏸ and give the definition of ᏸ-parabolic trajectory.

Supposeᏸ satisfies (H2) of the introduction and, instead of (H1), the following con-dition

(H1)∗There exists a homogeneous Lie group inRN+1,

L =RN+1,◦,d λ

(3.2)

such thatᏸ is left translation invariant onLandd λ-homogeneous of degree two

We assume the composition law◦is Euclidean in the time variable, that is,

(x,t) ◦(x ,t )=c(x,t,x ,t ),t + t 

wherec(x,t,x ,t ) denotes a suitable function of (x,t) and (x ,t )

It is a standard matter to prove the existence of a positive constantC such that

| z ◦ ζ | ≤ C

| z |+| ζ | ∀ z,ζ ∈ R N+1 (3.4) Letγ : [0, ∞[→ R Nbe a continuous function such that

lim sup

s →∞

γ(s) 2

(here| · |denotes theD λ-homogeneous norm (1.11))

Then, the path

s −→ η(s) =γ(s),T − s

will be called anᏸ-parabolic trajectory.

Obviously, the curve

s −→ η(s) =(α,T − s), α ∈ R N,T ∈ R (3.7)

is anᏸ-parabolic trajectory It can be proved that every integral curve of the vector fields

Y in (1.2) also is anᏸ-parabolic trajectory (see [3, Lemma 3])

Our first asymptotic Liouville theorem is the following one

Theorem 3.1 Let ᏸ satisfy hypotheses (H1) ∗ and (H2), and let u be a nonnegative solution

to the equation

in the halfspace

Then, for every ᏸ-parabolic trajectory η,

lim

s →∞ u

η(s)

=inf

In particular

lim

t →−∞ u(x,t) =inf

The proof of this theorem relies on a left translation and scaling invariant Harnack

inequality for nonnegative solutions toᏸu =0

For everyz0∈ R N+1andM > 0, let us put

P z0(M) : = z0◦ P(M), (3.12) where

P(M) : =(x,t) ∈ R N+1:| x |2≤ − Mt

Then, the following theorem holds

Theorem 3.2 (left and scaling invariant Harnack inequality) Let u be a nonnegative so-lution to

ᏸu =0 inRN ×]− ∞, 0[. (3.14)

Then, for every z0∈ R N ×]− ∞ , 0[ and M > 0, there exists a positive constant C = C(M), independent of z0and u, such that

sup

P z0(M)

u ≤ Cu

z0 

Proof It follows fromTheorem 2.1and the left translation invariance ofᏸ The details

From this theorem we obtain the proof ofTheorem 3.1

Proof of Theorem 3.1 We may assume inf S u =0 Letη(s) =(γ(s),s0− s), s0≤0,s ≥ s0be

anᏸ-parabolic trajectory Then, there exists M0> 0 such that

γ(s) 2

wheres ∗ > 0 is big enough Let us put M =2C(M2+ 1)1/4whereC is the positive constant

in the triangular inequality (3.4) Letε > 0 be arbitrarily fixed and choose z ε =(x ε,t ε)∈ S

such that

u

z ε



Now, for everys ≥ s ∗, we have

z −1

ε ◦ η(s) ≤ C z −1

ε + η(s) 

≤ C z −1

ε +

M2+ 1 1/4 √

s

= C

s − s0+t ε z −1

ε

√

s − s0+t ε+



M2+ 1 1/4



s

s − s0+t ε .

(3.18)

Then, there existsT = T(ε) > 0 such that

z −1

ε ◦ η(s) ≤ M

s − s0+t ε ∀ s > T. (3.19) This implies that

η(s) ∈ z ε ◦ P(M) ≡ P z ε(M) ∀ s > T. (3.20)

On the other hand, by the Harnack inequality ofTheorem 3.2, there existsC ∗ = C ∗(M) >

0 independent ofz εandε such that

sup

P zε(M)

u ≤ C ∗ u

z ε

Therefore,

u

η(s)

SinceC ∗is independent ofε, this proves the theorem. 

where p and q are polynomial function Assume

u(0,t)... data-page ="9 ">

Remark 2.9 The class of our stationary operators< /i>ᏸ0also contains “parabolic”-type op-erators like, for example, the following “forward-backward” heat operator...