Hardy-type inequalities related to degenerate elliptic differential operators

Hardy-type inequalities related to degenerate elliptic differential operators

LORENZOD’AMBROSIO

Abstract. We prove some Hardy-type inequalities related to quasilinear second- order degenerate elliptic differential operatorsLpu:= −∇^∗_L(|∇Lu|^p−2∇Lu). If φis a positive weight such that−Lpφ≥0, then the Hardy-type inequality

c

|u|^p

φ^p |∇Lφ|^p dξ≤

|∇Lu|^pdξ

u∈

C

0¹()

holds. We find an explicit value of the constant involved, which, in most cases, results optimal. As particular case we derive Hardy inequalities for subelliptic operators on Carnot Groups.

Mathematics Subject Classification (2000):35H10 (primary); 22E30, 26D10, 46E35 (secondary).

1.

Introduction

AnN-dimensional generalization of the classical Hardy inequality is the following c

|u|^pw^−pd x ≤

|∇u|^pd x, u∈

C

0¹(), (1.1) where p > 1, ⊂

R

^N and the weightwis, for instance, w := |x| orw(x) :=

dist(x, ∂)(see for instance [5, 10, 23] and the references therein).

A lot of efforts have been made to give explicit values of the constant c, and even more, to find its best valuec_n,p(see e.g. [5, 10, 23, 24, 31, 40, 41, 42]).

The preeminent rˆole of the Hardy inequality in the study of linear and nonlinear partial differential equations is well-known. For instance, let us consider the linear initial value problem

u_t−u=λ_|x^u_|2, x ∈

R

ⁿ, n≥3, t ∈]0,T[, λ∈

R

,

u(x,0)=u₀(x), x ∈

R

ⁿ, u₀∈L²(

R

ⁿ), u₀>0. (1.2) Pervenuto alla Redazione il 9 febbraio 2005 e in forma definitiva il 3 agosto 2005.

The problem (1.2) has a solution if and only ifλ≤(ⁿ⁻₂²)²=c_n,2(see [3] for more details). In the last years this result has been extended in several directions, see e.g.

[9, 12, 28, 32, 45, 46, 49].

In the Heisenberg-group setting, Garofalo and Lanconelli in [29], Niu, Zhang and Wang in [47] and the author in [19] proved, among other results, the following Hardy-type inequality related to the sub-Laplacian H on the Heisenberg-group

H

^n:

c

Hⁿ

u²

ρ²ψ²_Hdξ ≤

Hⁿ|∇Hu|²dξ, u∈

C

0¹(

H

ⁿ\ {0}) (1.3) where∇H denotes the vector field associated to the real part of the Kohn Laplacian (H = ∇H· ∇H),ρ andψH are respectively a suitable distance from the origin and a weight function such that 0≤ψH ≤1.

Recently, in [32], it has been pointed out that the analogue problem of (1.2) involving the sub-LaplacianH, namely

u_t −Hu=λψ²_Hρ^u2 on

R

²ⁿ⁺¹×]0,T[, λ∈

R

,

u(·,0)=u₀(·) on

R

²ⁿ⁺¹, u₀∈L²(

R

²ⁿ⁺¹), u₀>0,

has a positive solution if and only ifλ≤ c_b,H, wherec_b,H is the best constant in (1.3).

Similar results have been established for equations involving the Baouendi- Grushin-type operators_γ :=x+ |x|^2γy = ∇_γ · ∇_γ (see [37]).

Recently, in [21] Mitidieri, Pohozaev and the author, among other results, found some conditions on the functionsu and f that assure the positivity of the solutions of the partial differential inequalities−Lu≥ f(ξ,u)on

R

^{N. HereLis a} quite general linear second-order differential operator, namely,Lu := −∇^∗_L· ∇Lu, where ∇L is a general vector field. This class of operators include all previously cited operators as well as the sub-Laplacian on Carnot groups.

Having in mind some extensions of the above results to the setting of second- order linear degenerate (or singular) partial differential operators, it appears that an important step towards this programme is to establish some fundamental inequali- ties of Hardy-type.

In this paper we shall prove some Hardy-type inequalities associated to the quasilinear operators

L_pu:= −∇_L^∗(|∇Lu|^p−2∇Lu) (p>1).

Our principal result can be roughly described as follows: if φ : →

R

^{is any} positive weight, for anyu∈

C

₀¹()we have

c

|u|^p

φ^p |∇Lφ|^p dξ ≤

|∇Lu|^pdξ, provided−L_pφ ≥0.

For this goal we shall mainly use a technique developed in [18, 19, 44]. An interesting outcome of this approach is that, in several cases, one can easily obtain the best constant. Furthermore, our main results represent a generalization of some results contained in [4, 5]. Indeed, in those papers the authors deal with a very spe- cial case, the usual Euclidean case whereφis a particular power of the Euclidean distance from a given surface, whereas, in our approach,∇L can be any quite gen- eral vector field and φ any positive weight, the generality of this approach being an important strength. It is, in fact, to remark that this unifying method allows, specializing the choice ofφ, to obtain almost all the fundamental Hardy inequali- ties known in Euclidean and subelliptic settings as well as to yield new Hardy-type inequalities. Moreover, let us to stress that our only hypothesis−L_pφ≥0 plays a relevant role in order to establish that the best constant is not achieved.

We pay particular attention to the following special cases of L_p: the Grushin type operators, the Heisenberg-Greiner operators and the sub-Laplacian on Carnot groups (see Section 3). Specializing the functionφ, we get more concrete Hardy- type inequalities for these operators with explicit values of the constants involved, which result the best possible in almost all the considered cases.

2.

Main results

The aim of this section is to present some preliminary results and derive some Hardy-type inequalities related to a general vector field.

In this paper∇ stands for the usual gradient in

R

^N. We indicate with I_k and with|·|respectively the identity matrix of orderkand the Euclidean norm.

Let µ := (µi j), i = 1, . . . ,l, j = 1, . . . ,N be a matrix with continuous entriesµi j ∈

C

(

R

^N). LetX_i (i =1, . . . ,l) be defined as

X_i :=^N

j=1

µi j(ξ) ∂

∂ξj

(2.1) and let∇Lbe the vector field defined by

∇L :=(X₁, . . . ,X_l)^T =µ∇.

Assuming that fori =1, . . . ,land j =1, . . . ,N the derivative _∂ξ^∂

jµi j ∈

C

(

R

^N), we set

X_i^∗:= − N

j=1

∂

∂ξjµi j(ξ)·

the formal adjoint of X_i and∇_L^∗ :=(X₁^∗, . . . ,X_l^∗)^T.

For any vector fieldh=(h₁, . . . ,h_l)^T ∈

C

¹(,

R

^l), we shall use the follow- ing notation

div_L(h):=div(µ^Th),

that is

div_L(h)= −^l

i=1

X_i^∗h_i = −∇_L^∗·h.

In what followsLstands for the linear second-order differential operator defined by L:=div_L(∇L)= −

l i=1

X^∗_i X_i = −∇_L^∗· ∇L

and for p>1, withL_pwe denote the quasilinear operator L_p(u):=div_L(|∇Lu|^p−2∇Lu)=−

l i=1

X_i^∗(|∇Lu|^p−2X_iu)=−∇_L^∗·(|∇Lu|^p−2∇Lu).

Example 2.1. Letl < N be a positive natural number and letµ^l be the matrix defined as

µ^l := I_l 0

.

The corresponding vector field∇^l results to be the usual gradient acting only on the firstl variables∇^l =(_∂ξ^∂₁,_∂ξ^∂₂, . . . ,_∂ξ^∂_l). It is clear that∇^N = ∇. The correspond- ing quasilinear operator L_pis the usual p−Laplacian acting on the firstlvariables of

R

^N.

Example 2.2. (Baouendi-Grushin type operator) Let

R

^Nbe splitted inξ=(x,y)∈

R

ⁿ×

R

^{k. Let}γ ≥0 and letµbe the following matrix

I_n 0

0 |x|^γ I_k . (2.2)

The corresponding vector field is∇_γ=(∇x,|x|^γ ∇y)and the linear operatorLis the so-called Baouendi-Grushin operatorL=x+ |x|^2γy.

Notice that ifk = 0 orγ = 0, thenL andL_p coincide respectively with the usual Laplacian operator and p-Laplacian operator.

Example 2.3. (Heisenberg gradient) Letξ = (x,y,t) ∈

R

ⁿ ×

R

ⁿ×

R

=

H

ⁿ(=

R

^N)and letµbe defined as

I_n 0 2y 0 I_n −2x .

The corresponding vector field∇H is the Heisenberg gradient on the Heisenberg- group

H

^n.

This is the simplest case of a more general setting: the Carnot groups. More details are given in Section 3.3.

Example 2.4. (Heisenberg-Greiner operator) Letξ = (x,y,t) ∈

R

ⁿ×

R

ⁿ×

R

^, r := |(x,y)|,γ ≥1 and letµbe defined as

I_n 0 2γyr^2γ−2

0 I_n −2γxr^2γ−2 . (2.3)

The corresponding vector fields areX_i=_∂^∂_xi+2γy_ir^2γ−2_∂^∂_t,Y_i=_∂^∂_yi−2γx_ir^2γ−2_∂^∂_t fori =1, . . . ,n.

Forγ = 1L is the sub-LaplacianH on the Heisenberg-group

H

^{n. If}γ = 2,3, . . .,Lis a Greiner operator (see [33]).

Let A be an open subset of

R

^N with Lipschitz boundary ∂A and let hˆ ∈

C

¹(A,

R

^l)be a vector field. By the divergence theorem we have

A

div_Lhdˆ ξ =

A

div(µ^Thˆ)dξ =

∂A

hˆ·µνd=

∂A

hˆ·νLd,

whereνL := µν, andν denotes the exterior normal at pointξ ∈ ∂A. Ifhˆ has the formhˆ = f hwith f ∈

C

¹(A)andh∈

C

¹(A,

R

^l), then

A

fdiv_Lhdξ+

A∇Lf ·hdξ =

∂A

f h·νLd. (2.4) Moreover, ifh= ∇Luwithu∈

C

²(A), then (2.4) yields the Gauss–Green formula

A

f Ludξ+

A∇Lf · ∇Ludξ=

∂A

f∇Lu·νLd.

Letg∈

C

¹(

R

)be such thatg(0)=0 and let⊂

R

^N be open. For every vector fieldh∈

C

¹(A,

R

^l)and any compactly supported functionu∈

C

0¹(), choosing

f :=g(u)in (2.4), we obtain

g(u)div_Lhdξ = −

g(u)∇Lu·hdξ. (2.5) Leth ∈ L¹_loc(,

R

^l)be a vector field. As usual, we define the distribution div_Lh using the formula (2.5) withg(s)=s. If in (2.5) we choseg(t)= |t|^pwithp>1, then for everyu∈

C

₀¹()we have

|u|^pdiv_Lhdξ = −p

|u|^p−2u∇Lu·hdξ. (2.6) Leth ∈ L¹_loc(,

R

^l)be a vector field and let A ∈ L¹_loc()be a function. In what follows we write A ≤ div_Lh meaning that the inequality holds in distributional sense, that is for everyφ ∈

C

₀¹()such thatφ ≥0, we have

φA dξ ≤

φdiv_Lhdξ = −

∇Lφ·h dξ.

Identities (2.5) and (2.6) play an important role in the proof of the following Hardy- type inequalities and the Poincar´e inequality too.

Theorem 2.5. Let p > 1. Let h ∈ L¹_loc(,

R

^l) be a vector field and let A_h ∈ L¹_loc()be a nonnegative function such that A_h ≤div_Lh and|h|^pA¹_h^−p∈L¹_loc(). Then for every u ∈

C

₀¹(), we have

|u|^pA_hdξ ≤ p^p

|h|^p

A⁽_h^p−1) |∇Lu|^pdξ. (2.7) Proof. We note that the right hand side of (2.7) is finite sinceu ∈

C

₀¹(). Using the identity (2.6) and H¨older inequality we obtain

|u|^p A_hdξ ≤

|u|^pdiv_Lhdξ≤ p

|u|^p−1|h| |∇Lu|dξ

= p

|u|^p−1A⁽_h^p−1)/p |h|

A⁽_h^p−1)/p|∇Lu|dξ

≤ p

|u|^pA_hdξ ^(p−1)/p

|h|^p

A_h^p−1|∇Lu|^pdξ 1/p

.

This completes the proof.

Specializing the vector fieldhand the functionA_h, we shall deduce from (2.7) some concrete inequalities of Hardy-type.

Remark 2.6. Setting A_h =div_Lhin (2.7), we have

|u|^pdiv_Lh dξ ≤ p^p

|h|^p

|div_Lh|^(p−1)|∇Lu|^pdξ. (2.8) Acting as Davies and Hinz in [24], the choiceh:= ∇LV withVsuch thatL V >0, yields

|u|^p|L V|dξ ≤ p^p

|∇LV|^p

|L V|^(p−1)|∇Lu|^pdξ. (2.9) In order to state a Hardy inequality, now the problem is to find a suitable function V. In the Euclidean setting for 1< p< N, choosingV(ξ)= |ξ|^2−pif 1< p<2, V(ξ)=ln|ξ|if p=2 andV(ξ)= − |ξ|^2−p if 2< p< N, we obtain the Hardy inequality (1.1) withw(ξ)= |ξ|.

Another strategy is to chose the vector fieldh ash = |∇LV|^p−2∇LV withV such thatL_pV >0. Thus, we have

|u|^pL_pVdξ ≤ p^p

|∇LV|^p(p−1)

L_pV^(p−1) |∇Lu|^pdξ. (2.10)

Hence, in the Euclidean setting for 1< p< N, choosingV(ξ)=ln|ξ|we reobtain the inequality (1.1) withw(ξ)= |ξ|.

In order to obtain the classical Hardy inequalities in Euclidean setting, these strategies are equivalent. This equivalence is basically due to the fact that|∇ |ξ|| = 1 forξ =0. The latter approach is slightly more simple: the choice ofVis indepen- dent of p. Moreover, it turned out to be more fruitful in the Heisenberg-group and in the Grushin plane settings (see [19, 18]) as well as in our more general framework.

Letd : →

R

be a nonnegative non constant measurable function. In order to state Hardy inequalities involving the weightd, the basic assumption we made ondis that, forα =0,d^αis a one side weak solution of−L_p(u)= 0, that isd^α is super-L_p-harmonic or sub-L_p-harmonic in weak sense. Namely, let α, β ∈

R

^, α =0, requiring

d^{(α−1)(p−1)}|∇Ld|^p−1∈L¹_loc(), (2.11) we assume that

−L_p(d^α)≥0 on [resp.≤0] (2.12) in weak sense, that is for every nonnegativeφ∈

C

₀¹()we have

∇Ld^αp−2∇Ld^α· ∇Lφ = α|α|^p−2

d^{(α−1)(p−1)}|∇Ld|^p−2∇Ld· ∇Lφ

≥ 0 [resp.≤0] (2.13)

and

α[(α−1)(p−1)−β−1]>0, [resp. <0]. (2.14) Gluing together the above conditions, we assume that

−L_p(cd^α)≥0 on (2.15)

in weak sense, wherec:=α[(α−1)(p−1)−β−1]

Theorem 2.7. Assume that(2.11)and(2.15)hold. Letβ∈

R

be such that d^β|∇Ld|^p ∈ L¹_loc(), (2.16)

d^β+p ∈ L¹_loc(). (2.17)

For every function u∈

C

₀¹(), we have (c_α,β,p)^p

|u|^pd^β|∇Ld|^pdξ ≤

d^β+p|∇Lu|^pdξ, (2.18) where c_α,β,p := |(α−1)(p−1)−β−1|/p.

In particular, if−L_p(d^α)≥0, then for every function u∈

C

₀¹()we have |α|(p−1)

p

p

|u|^p

d^p |∇Ld|^pdξ ≤

|∇Lu|^pdξ (2.19) provided d^−p|∇Ld|^p∈L¹_loc().

Remark 2.8. In most examples we shall deal with, the constantc_α,β,^p _p, yielded by applying Theorem 2.7, results to be sharp. We shall now indicate an argument that can be used to prove the sharpness of the constantc_α,β,^p _pinvolved in the inequality of Theorem 2.7. Letc_b()be the best constant in (2.18). It is clear thatc_b() ≥ c_α,β,^p _p. We shall assume that the hypotheses of Theorem 2.7 are satisfied and that there existss > 0 such that^s := d⁻¹(]− ∞,s[)ands := d⁻¹(]s,+∞[)are not empty open subsets ofwith piecewise regular boundaries.

We assume that there exists0>0 such that for every∈]0, 0[ there hold 0<

d<s

d^c()p+β|∇Ld|^p<+∞, 0<

d>s

d^−c()p+β|∇Ld|^p<+∞, (2.20) where

c():= |(α−1)(p−1)−β−1| +

p =c_α,β,p+

p. (2.21)

By rescaling argument, we can assume thats=1. Let∈]0, 0[ and letv:→

R

be defined as

v(ξ):=

d^c()(ξ) ifd(ξ)≤1,

d^−c(/2)(ξ) ifd(ξ) >1. (2.22) By hypothesis,

v^pd^β|∇Ld|^pis finite. Thus, we have c()^p

v^pd^β|∇Ld|^p = c()^p

d<1

d^β+pd^(c()−1)p|∇Ld|^p +c()^p

d>1

d^β+pd^{(−c(/2)−1)p}|∇Ld|^p

=

d<1

d^β+p|∇Lv|^p+( c() c(/2))^p

d>1

d^β+p|∇Lv|^p

=

d^β+p|∇Lv|^p+( c()^p c(/2)^p −1)

d>1

d^β+p|∇Lv|^p. Observing thatc() >c(/2), we get

c()^p

v^pd^β|∇Ld|^p>

d^β+p|∇Lv|^p, (2.23) the converse of the Hardy inequality.

Now we assume that the Hardy inequality (2.18) holds for the functionvde- fined in (2.22). From (2.23) we deducec()^p > c_b(). Letting → 0, we get c_α,β,^p _p≥cb()and hence the claim.

The question of the existence of functions that realize the best constant arises.

In such a general framework a unique answer cannot be given. Indeed, even in the Euclidean setting several cases occur. Let p = 2, letd₁(·):= |·|be the Euclidean distance from the origin, and let d₂(·) := dist(·, ∂) be the distance from the boundary of a given domain. If⊂

R

^{N (N} ≥3) is a ball centered at the origin, then the best constants in the Hardy inequality (2.19) related tod₁andd₂are not achieved. On the other hand, there exist smooth bounded domainssuch that the best constant in the inequality related tod₁is not achieved and the best constant in the inequality related tod₂is achieved (see [40, 41]). Anyway, some steps in this direction can be done even in our general framework. For the sake of simplicity, we shall focus our attention on the inequality (2.19).

Therefore, under the same hypotheses of Theorem 2.7 we assume that

−L_p(d^α) ≥ 0 on in weak sense, that (

|∇Lu|^pdξ)^1/p is a norm and that D¹_L^,p(), the closure of

C

₀^∞()in that norm, is well defined. We denote byc_b() the best constant in (2.19), namely

c_b():= inf

u∈D^1,p_L ,u =0

|∇Lu|^pdξ

|u|^pd^−p|∇Ld|^pdξ. (2.24) Theorem 2.9. Under the above hypotheses we have:

1. If d^αp−1p ∈D¹_L^,p(), then c_b()=(|α| ^p−_p¹)^pand d^αp−1p is a minimizer.

2. If d^αp−1p ∈ D¹_L^,p(), p ≥2,|∇Ld| = 0a.e. and c_b() =(|α| ^p−_p¹)^p then the best constant c_b()is not achieved.

Remark 2.10. In all the examples we shall deal with in the last section, it is possi- ble to apply Theorem 2.9 and, hence, for p≥2 the best constants mentioned in all the theorems of Section 3 are not achieved.

Remark 2.11. Let us to consider the special case of∇L = ∇, the usual Euclidean gradient,dis the Euclidean distance from a given regular surfaceK of codimension k (1 ≤ k ≤ N), α = ^p_p⁻₋^k₁ andβ = −p. In this case, replacing with\K, Theorem 2.7 assures that the inequality

|p−k| p

p

|u|^p d^p dξ ≤

|∇u|^pdξ (2.25)

holds for everyu∈

C

₀¹(\K)provided−p(d^α)≥0 on\K.

This particular case of Theorem 2.7 is contained in [4, 5], where the authors also study the remainder terms for inequality (2.25).

The reader interested in the study of Hardy inequalities with remainder terms can refer to [4, 5, 10, 11, 31] and the references therein for the Euclidean case and to [19] for the case∇L = ∇H, the Heisenberg gradient on the Heisenberg-group.

Proof of Theorem 2.7. We prove the thesis in the case−L_p(d^α) ≥ 0 andc := α[(α−1)(p−1)−β−1]>0. The alternative case is similar.

Letϕ∈

C

0¹()be a nonnegative function. Choosing in (2.13)φ:=d^{β+1−(α−1)(p−1)}ϕ, we have

0 ≤ α

d^β+1|∇Ld|^p−2∇Ld· ∇Lϕ

−α[(α−1)(p−1)−β−1]

d^β|∇Ld|^pϕ. (2.26) Using H¨older inequality and hypotheses (2.16) and (2.17), it is immediate to check that the above integrals are finite.

Lethbe the vector field defined byh := −αd^β+1|∇Ld|^p−2∇Ldand let A_h be the function defined as A_h := α[(α−1)(p−1)−β−1]d^β|∇Ld|^p. Thus, from (2.26) and the fact that c > 0, we obtain div_Lh ≥ A_h ≥ 0. Now we are in the position to apply Theorem 2.5 and this concludes the proof.

Proof of Theorem 2.9. 1) From (2.19), we havec_b() ≥(|α| ^p−_p¹)^p. It is imme- diate to check thatu:=d^αp−1p realizes the infimum in (2.24).

2) Letu∈

C

₀^∞(). We define the functionalI as I(u):=

|∇Lu|^pdξ−

|α|(p−1) p

p

|u|^p

d^p |∇Ld|^pdξ.

The functionalI is non negative, and the best constant will be achieved, if and only if,I(u)=0 for someu∈D¹_L^,p().

Letvbe the new variablev := d^−γu withγ := α^p−_p¹. By computation we have

|∇Lu|²= |γ|²v²d^2γ−2|∇Ld|²+d^2γ|∇Lv|²+2γ vd^2γ−1(∇Ld· ∇Lv). (2.27) (Ifdis not smooth enough, by standard argument one can considerd a regulariza- tion ofdand after the computation taking the limit as→0).

We remind that the inequality

(ξ−η)^s ≥ξ^s−sηξ^s−1 (2.28) holds for everyξ, η,s∈

R

^withξ >0, ξ > ηands≥1 (see [31]). Applying (2.28) and (2.27) with s = p/2, ξ = |γ|²v²d^2γ−2|∇Ld|² andη = −2γ vd^2γ−1(∇Ld ·

∇Lv)−d^2γ|∇Lv|², we have

|∇Lu|^p ≥ |γ|^pv^pd^γpd^−p|∇Ld|^p

+p|γ|^p−2γ|v|^p−2vd^{(α−1)(p−1)}|∇Ld|^p−2(∇Ld· ∇Lv) +p

2|γ|^p−2|v|^p−2d^{(α−1)(p−1)+1}|∇Ld|^p−2|∇Lv|².

Taking into account thatu:=d^γvwe have

I(u)≥ I₁(v)+I₂(v) where

I₁(v) :=

p|γ|^p−2γ|v|^p−2vd^{(α−1)(p−1)}|∇Ld|^p−2(∇Ld· ∇Lv)dξ, I₂(v) := p

2|γ|^p−2

|v|^p−2d^{(α−1)(p−1)+1}|∇Ld|^p−2|∇Lv|²dξ.

Re-arranging the expression inI₁and integrating by parts we obtain I₁(v) =

p−1 p

p−1

∇L|v|^p·∇Ld^αp−2∇Ld^α

dξ

=

p−1 p

p−1

∂|v|^p∇Ld^αp−2

∇Ld^α·νL

d

+

p−1 p

p−1

|v|^p(−L_p(d^α))dξ≥0,

where we have used the fact thatv ∈

C

₀^∞()and the hypothesis−L_p(d^α)≥ 0.

On the other hand we can rewriteI₂as I₂(v)= 2

p|γ|^p−2

d^{(α−1)(p−1)+1}|∇Ld|^p−2∇L|v|^2p2 dξ.

Thus, we conclude that for anyu∈ D¹_L^,p() I(u)≥ 2

p|γ|^p−2

d^{(α−1)(p−1)+1}|∇Ld|^p−2∇L|v|^p22 dξ, and this inequality implies the non existence of minimizers in D¹_L^,p().

Specializing the function d, we shall deduce from Theorem (2.7) some con- crete inequalities of Hardy-type. A first example is the following. We assume that there existsm ∈

N

^{, 1} ≤ m ≤ l such that the matrixµin (2.1) has the following form

µ:=

I_m µ1

0 µ2 (2.29)

where µ1 andµ2 denote matrixes with m × (N −m) and(l −m)×(N −m) continuous entries respectively and I_m stands for the identity matrix of order m.

Notice that this case occurs in all the examples cited above.

Set η := (ξ1, . . . , ξm), τ := (ξm+1, . . . , ξN)and let vp be defined for η ∈

R

^m\ {0}as

vp(η):=

|η|^p−mp−1 ifp =m,

−ln|η| ifp=m. (2.30)

The functionvpis p-harmonic on

R

^m\ {0} ×

R

^N−mfor the Euclideanp-Laplacian acting on theηvariablep,ηand hence also for the quasilinear operatorL_p. More- over, there exists a constantl_p =0 such that

−p,ηvp=l_pδ0 on

R

^m

in weak sense, whereδ0is the Dirac distribution at 0∈

R

^mandlp >0 if and only if 1< p≤m. These relations allow us to apply Theorem 2.7.

Theorem 2.12. Assume thatµhas the form(2.29)and letβ∈

R

^{be fixed.}

1. Let1< p <∞and let ⊂

R

^N be an open set. If m+β <0, then we also require that⊂(

R

^m\ {0})×

R

^N−m. Then for every u∈

C

0¹(), we have

b_β^p

|u(η, τ)|^p|η|^βdηdτ ≤

|∇Lu(η, τ)|^p|η|^p+βdηdτ, (2.31) where b_β:= ^|m+β|_p .

In particular, for every u ∈

C

₀¹(), we obtain |m−p|

p

p

|u(ξ)|^p

|ξ|^p dξ ≤

|m− p|

p

p

|u(η, τ)|^p

|η|^p dηdτ

≤

|∇Lu(ξ)|^pdξ. (2.32) 2. Let p=m >1. Let R>0and set:= {ξ =(η, τ)∈

R

^m×

R

^N−m, |η|< R}.

Ifβ <−1, then for every u∈

C

₀¹(), we have b˜_β^p

|u(η, τ)|^p

|η|^p (ln R

|η|)^βdηdτ ≤

|∇Lu(η, τ)|^p(ln R

|η|)^p+βdηdτ, (2.33) whereb˜_β:= ^|β+_p^1|.

In particular, for every u ∈

C

₀¹(), we obtain p−1

p

p

|u(η, τ)|^p

(|η|ln(R/|η|))^pdηdτ ≤

|∇Lu(ξ)|^pdξ. (2.34) Remark 2.13. It is easy to check that the inequality (2.33) holds also forβ >−1 provided the setis replaced by:= {(η, τ)∈

R

^m×

R

^N−m, 0<|η|< R}.

Proof. Let 1 < p < m. We claim that the function vp is super-L_p-harmonic on

R

^N. Indeed, let φ ∈

C

₀¹(

R

^N) be a non negative function. Observing that ∇Lvp=∇_ηvp, we have

R^N−L_pvpφdξ =

R^N−mdτ

R^mdη(−p,ηvp)φ

= l_p

R^N−mφ(0, τ)dτ ≥0 (2.35) Analogously, one can prove thatvp is super-L_p-harmonic when p = m and sub- L_p-harmonic when p>m.

First we consider the case p =m. We choosed^α=vpwithd(ξ)=d(η, τ)=

|η| andα = ^p_p⁻₋^m₁. Observing that|∇L|η|| =∇_η|η| = 1 a.e. and that the inte- grability conditions (2.11),(2.16), (2.17) are satisfied, applying Theorem 2.7 we get (2.31).

Let p=m>1. The choicesd(η, τ)=ln_|η|^R andα=1 in Theorem 2.7 yield the inequality (2.33).

Finally, we prove the missing inequality (2.31) when p=m. We consider the casem +β > 0. The casem +β < 0 is analogous and the casem +β = 0 is trivial. Letσ >0 be such thatm+β−σ >0. We chosed(ξ)= |η|,α= _m^σ₋₁. In this case it easy to check thatd^αis sub-L_m-harmonic on

R

^{N, that is}

−L_m(d^α)=−div_L σ

m−1

m−1

|η|^σ−m+1∇_η|η|

=− σ^m (m−1)^m−1

1

|η|^m−σ ≤0. The constantcin (2.15) isc = _m^σ₋₁ ^σ−m_m^−β < 0. Hence, we are in the position to apply Theorem 2.7; thus, we derive the inequality

m+β−σ p

p

|u|^p|η|^βdξ ≤

|∇Lu|^p|η|^p+βdξ.

Lettingσ →0, we get the claim.

Remark 2.14. In the caseµ= I_N, the vector field∇L is the usual gradient∇. For m < N, inequalities of type (2.31) are already present in [43] and in [44]. Secchi, Smets and Willem in [48] prove that the constantb_β^p is optimal whenm +β > 0 and=

R

^N (see next section for further generalization in this direction).

An immediate consequence of Theorem 2.12 is a Poincar´e inequality for the vector field∇L. The claim easily follows from inequality (2.31) withβ=0.

Theorem 2.15. Let be an open subset of

R

^{N bounded in}ξ1 direction, that is, there exists M >0such that for everyξ ∈it results|ξ1| ≤ M. Assume that the matrixµhas the form(2.29).