Quasi uniformity for the abstract Neumann antimaximum principle and applications with a priori estimates

Quasi uniformity for the abstract Neumann antimaximum principle and applications with a

priori estimates

Genni Fragnelli Dipartimento di Matematica Universit` a degli Studi di Bari Via E. Orabona 4, 70125 Bari Italy

email: fragnelli@dm.uniba.it Dimitri Mugnai

Dipartimento di Matematica e Informatica Universit` a degli Studi di Perugia Via Vanvitelli 1, 06123 Perugia Italy

email: mugnai@dmi.unipg.it

Abstract

In this paper we prove a general result giving the maximum and the antimaximum principles in a unitary way for linear operators of the form L + λI, provided that 0 is an eigenvalue of L with associated constant eigenfunctions. To this purpose, we introduce a new notion of “quasi”–

uniform maximum principle, named k–uniform maximum principle: it holds for λ belonging to certain neighborhoods of 0 depending on the fixed positive multiplier k > 0 which selects the good class of right–hand–

sides. Our approach is based on a L

− L

estimate for some related problems. As an application, we prove some generalization and new re- sults for elliptic problems and for time periodic parabolic problems under Neumann boundary conditions.

Keywords: Maximum principle, antimaximum principle, uniform maximum principle with fixed multiplier, polyharmonic operators, periodic parabolic prob- lems

2000AMS Subject Classification: Primary 35B50; Secondary 35J25, 35J40,

35K10, 34C25

1 Introduction and abstract setting

Let us start considering the easy problem

( ∆u + λu = f (x) in Ω,

Bu = 0 on ∂Ω, (1.1)

where Ω is a bounded domain of R ^N , N ≥ 1, B denotes Dirichlet or Neumann boundary conditions and f ∈ L ^p (Ω). Denoting by λ 1 the first eigenvalue of −∆

under the corresponding homogeneous boundary conditions Bu = 0, a classical result is the following maximum principle:

(MP): if λ < λ 1 , then for any f ≥ 0 the associated solution u ≤ 0 in Ω.

On the other hand, a related stronger version, namely the strong maximum principle, holds:

(SMP): if in addition f 6= 0, then u < 0 in Ω.

We immediately observe that these results present two main features: they are uniform in λ and no special functional space of right hand sides f is involved.

Moreover, it is now well know that jumping after λ 1 changes the situation a lot: indeed, Cl´ement and Peletier in [8] proved the following antimaximum principle:

(AMP): for any f ≥ 0 in L ^p (Ω), p > N , there exists δ = δ(f ) > 0 such that if u solves (1.1) with λ ∈ (λ 1 , λ 1 + δ), then u ≥ 0 in Ω.

They also showed that under Neumann boundary conditions a uniform anti- maximum principle, i.e.

(UAMP): in (AMP) it is possible to take δ independent of f holds only when N = 1.

Therefore, it is clear that, in general, (AMP) is not uniform, while (MP) is.

Moreover, (MP) needs no special requirements on p, while (AMP) needs p > N , as showed in [30]: under Dirichlet boundary conditions, there exists f ∈ L ^N (Ω) with f > 0 such that for all λ > λ 1 different from an eigenvalue of −∆ in H 0 ¹ (Ω), the solution of (1.1) changes sign.

After [8] a large number of refinements of (UAMP) have been established.

We recall [6] for higher order ODE’s with periodic boundary conditions, while

for general second order PDE’s with Neumann or Robin boundary conditions

we mention [20] (where again it is proved that (UAMP) holds only if N = 1),

while polyharmonic operators in low dimensions (say for all those dimensions

for which the natural Sobolev space containing weak solutions is embedded in

C ⁰ (Ω)) are considered in [9], [10], [23]. In [30] it is showed that the condition

p > N in [8] is sharp for the validity of an antimaximum principle when L = ∆

under Dirichlet boundary conditions: in this context, taking a right–hand side

f ∈ L ² (Ω), forces to assume N = 1 also for the validity of (AMP) without any

uniformity. Finally, we recall a local extension in R ^N proved in [12] and some extensions in presence of the p–Laplace operator in [5] and [21].

Recently, Campos–Mawhin–Ortega in [7] show that it is possible to state maximum and antimaximum principles in a unitary way, at least in some cases.

Roughly speaking, having in mind (1.1) with Lu = u ^′′ and Neumann boundary conditions, so that λ 1 = 0, they start with the following definition of maximum principle:

Definition 1.1. Given λ ∈ R \ {0}, we say that the operator L + λI satisfies a maximum principle if for every f ∈ L ¹ (Ω) the equation

Lu + λu = f, u ∈ Dom(L) ⊂ C ⁰ (Ω) (1.2) has a unique solution with λu ≥ 0 for any f ≥ 0. Moreover, the maximum principle is said to be strong if λu(x) > 0 for any x ∈ Ω whenever f ≥ 0 and f (x) > 0 in a subset of Ω with positive measure.

It is clear that the definition above covers the case of a “classical” maximum principle when λ < λ 1 = 0 and a “classical” antimaximum principle when λ > 0;

more precisely, we remark that for λ > 0 Definition 1.1 includes a (UAMP) tout court.

The main result in [7] sounds as the following Theorem 1.2 ([7]). There exist λ − and λ + such that

−∞ ≤ λ ⁻ < 0 < λ + ≤ +∞

and L + λI satisfies a maximum principle if and only if λ ∈ [λ ⁻ , 0) ∪ (0, λ ⁺ ].

Moreover the maximum principle is strong if λ ∈ (λ − , 0) ∪ (0, λ + ).

The basic ingredient of [7] is an L ^∞ − L ¹ estimate for solution–datum of the form

kuk L

(Ω) ≤ Mkfk L

(Ω) ,

which is common and natural for ODE’s and for the wave equation in 1D. On the other hand, for the classical theory of elliptic problems like (1.1), by the classical Agmon–Douglis–Nirenberg result ([1]), if Ω is smooth enough, say of class C ² , a more natural setting would be a W ^2,p − L ^p estimate of the form

kuk W

(Ω) ≤ Mkfk L

(Ω) , (1.3) since data belonging to L ¹ are not the good ones to perform a standard varia- tional approach (we refer to [3] for this case). For example, the easy problem

( ∆u + λu = f in Ω,

∂u

∂ν = 0 on ∂Ω, (1.4)

where Ω is a bounded domain of R ^N , N > 1, cannot be handled by Theorem 1.2

if f ∈ L ¹ (Ω), since L ¹ is not contained in the dual of the Sobolev space where

weak solutions are sought.

On the other hand, by Morrey’s Theorem, if N < 2p, then W ^2,p (Ω) ֒→

C ^0,α (Ω), for some α ∈ (0, 1), so that (1.3) implies kuk C

(Ω) ≤ Mkfk L

(Ω) , where, of course, kuk C

(Ω) = max _Ω |u| = kuk L

(Ω) .

In this paper we want to combine the spirit of [7] with the result stated in the other papers cited above showing two possible generalizations of Theorem 1.2. To this purpose, we first prove that, although a (UAMP) cannot hold (as for the Laplace operator with Neumann conditions in dimension greater than 1), a “quasi–(UAMP)” does (in the sense of k–(UMP) introduced in Definition 1.4 below), thus extending to any p ∈ (1, ∞) the sufficient condition proved for the case p = 2 in [16]. Moreover, with an additional hypothesis, we characterize completely the set of λ’s for which a strong k–(UMP) holds. To do this, we will also show some a priori estimates of independent interest, and some of which are in the spirit of bifurcation from 0. Finally, we also show the sharpness of the exponent p > N/2 for problem (1.4). In this context, we believe that our results can help understanding better all the situation.

We start introducing the abstract framework. Let Ω ⊂ R ^N be a bounded domain with C ² boundary ∂Ω; a positive and finite measure µ is given in Ω, and we write L ^q (Ω) := L ^q (Ω, µ) for any q ∈ [1, ∞].

From now on, we fix p ∈ (1, ∞) and for any f ∈ L ^p (Ω) we set f := 1

µ(Ω) Z

Ω

f dµ and f := f − f. ˜ We also set

L ^p := n

f ∈ L ^p (Ω) : f = 0 o

and C := C ˜ ⁰ (Ω) ∩ L ^p ; finally, fixed k > 0, we introduce the cone

F k ^p := n

f ∈ L ^p (Ω) : k ˜ f k L

(Ω) ≤ kkfk L

(Ω)

o and

P k ^p := n

f ∈ F k ^p : f ≥ 0 o , i.e. P k ^p is the sub–cone of positive functions in F k ^p .

Remark 1.3. It is clear that any f ∈ L ^p (Ω) belongs to a suitable F k ^p and to F ℓ ^p for any ℓ ≥ k. Moreover, ∪ ^k F k ^p ⊂ L ¹ (Ω) with strict inclusion.

We now consider a linear operator L : Dom(L) ⊂ C ⁰ (Ω) → L ^p (Ω) satisfying the following properties:

Ker(L) = n

constant functions o

, Im(L) = L ^p , (1.5)







the problem Lu = ˜ f has a unique solution ˜ u ∈ ˜ C and ∃ M = M(L) > 0 such that k˜uk C

(Ω) ≤ Mk ˜ f k L

(Ω) .

(1.6)

We remark that these requirements are the natural extensions to our setting of the assumptions made in [7]. Therefore, having in mind Definition 1.1, we give the following uniform maximum principle with fixed multiplier k > 0.

Definition 1.4. Fix k > 0. Given λ ∈ R \ {0}, we say that the operator L + λI satisfies a uniform maximum principle with multiplier k, k–(UMP) for short, if for every f ∈ F k ^p equation (1.2) has a unique solution u ∈ Dom(L) ⊂ C ⁰ (Ω)

(1.7) with λu ≥ 0 when f ≥ 0. We say that a strong k–(UMP) holds if λu(x) > 0 for any x ∈ Ω whenever f ≥ 0 and f(x) > 0 in a subset of Ω having positive measure.

Remark 1.5. Again as in Definition 1.1, the case λ < 0 corresponds to a clas- sical maximum principle, while the case λ > 0 corresponds to an antimaximum principle, which is “almost” uniform, since f belongs to F k ^p and not to the whole of L ^p (Ω).

Our first result concerns the existence of a neighborhood U k of 0 such that U ^k \ {0} ⊂ n

λ ∈ R : L + λI satisfies a k–(UMP) o . More precisely:

Theorem 1.6. Assume that (1.5) and (1.6) hold and fix k > 0. Then there exists Λ = Λ(k) > 0 such that L + λI satisfies a k–(UMP) if λ ∈ [−Λ, Λ] \ {0}.

Moreover a strong k–(UMP) holds if λ ∈ (−Λ, Λ) \ {0}.

Remark 1.7. As it will be clear from the proof, Λ = Λ(k) does not depend on p, so that the result is uniform also with respect to the integrability exponent.

As a corollary of the previous result, we obtain the validity of an (AMP), also in a quasi–uniform way, for problem (1.4) with f in L ^p (Ω), p > N/2, already established in [7] for f in L ² (Ω) and N = 1, 2, 3. On the other hand, we underline the fact that our somehow “standard” everywhere approach is quite natural for antimaximum principles, and so we cannot extend our results to an almost everywhere fashion, typical of a (MP) also for more general PDE’s and differential inequalities, possibly on Riemannian manifolds (see the recent [2], [27]) or in anisotropic settings ([13]).

Let us note that, in some sense, any function f defines a whole set of

“good” right–hand–sides, in the sense that a crucial rˆole is played by the ratio k ˜ f k L

(Ω) /kfk L

(Ω) .

Moreover, as a direct consequence of Theorem 1.6, in contrast to the Dirichlet

case considered in [30], we get the following result:

Corollary 1.8. Concerning problem (1.4), L ^N is not sharp for the validity of the antimaximum principle.

Actually, this fact was already noticed in [20, Remark 4.5] and in [10, Propo- sition 2], were the sufficiency of the condition “p > N/2” was stated for the va- lidity of the (AMP), but not for any form of uniformity, as we give in Theorem 1.6. Moreover, the following result holds:

Proposition 1.9. In problem (1.4), L ^N/2 (Ω) is sharp for the validity of the antimaximum principle.

Proof. It follows the lines of [30] with obvious adaptations, so we will be sketchy, though we seize this opportunity to make precise a small step in the proof of the Dirichlet case which contained a misleading missprint. Assume Ω = B 1 , the unit ball, and take

f (x) = 1

|x| ² (2 − log |x|)



N − 2 + 1

2 − log |x|



∈ L ^N/2 (Ω).

Calculations show that v(x) = log(2 − log |x|) solves ∆v = −f. Now set w ^λ = u λ + χv, where χ is a smooth function with χ(x) = 1 if |x| ≤ 1/2, χ(x) = 0 if

|x| ≥ 7/8, and u λ solves problem (1.4). Thus w λ solves the problem ( ∆w = −λw + (1 − χ)f + v∆χ + 2Dχ · Dv + λχv in B 1

∂w/∂ν = 0 on ∂B 1 .

Since (1 − χ)f + v∆χ + 2Dχ · Dv + λχv ∈ L ^q (B 1 ) for any q ≥ 1, choosing λ different from an eigenvalue implies that w λ ∈ C( ¯ B 1 ). Hence

x→0 lim w λ (x)

v λ (x) = 0,

so that near 0 we have u λ (x) = − log(2 − log |x|) + w ^λ < 0.

Our final results, together with Theorem 1.6, give an almost complete char- acterization of the set of λ’s for which L + λI satisfies a k–(UMP), in the spirit of Theorem 1.2. Of course, due to the validity of a classical maximum principle for λ < 0, in the following we concentrate on λ > 0. As usual, we denote by

R λ = (L + λI) ⁻¹ : L ^p (Ω) → C ⁰ (Ω) the resolvent of L, whenever it exists. Finally, let us set

Λ + (k) := n

λ > 0 : L + λI satisfies a k–(UMP) o

and λ + = λ + (k) := sup Λ + (k). In the next result we show that the set of λ’s for which a k–(UMP) holds is an interval, in whose interior a strong k–(UMP) is satisfied; moreover, such a result characterizes completely such an interval.

We focus on λ + , an analogous result being valid for λ − .

Theorem 1.10. Assume (1.5) and (1.6), and suppose that for all λ < λ + the following condition holds:

if L + λI satisfies a k–(UMP), then R λ (P k ^p ) ⊆ P k ^p . (1.8) Then L + λI satisfies a k–(UMP) if and only if λ ∈ (0, λ + ]. Moreover, the k–(UMP) is strong if λ ∈ (0, λ + ).

In general, it is not clear if condition (1.8) can be verified, and without it, we cannot prove that L + λ + I satisfies a k–(UMP). For this, a useful tool is the following result on a priori estimates for R λ (P k ^p ):

Proposition 1.11 (A priori estimates). Take λ > 0 and suppose that L + λI satisfies a k–(UMP). Then, for any nonconstant f ∈ P k ^p , setting u = R λ f , we have

if k ˜ f k L

(Ω) < λk˜uk L

(Ω) , then 2M µ(Ω) ^1/p λ > 1. (1.9) Moreover, if λk˜uk L

(Ω) ≤ k ˜ f k L

(Ω) , then u ∈ P h ^p , where

h = min n

2M µ(Ω) ^1/p λk, k o

≤ k. (1.10)

If M µ(Ω) ^1/p λ < 1, then u ∈ P h ^p , where

h = max

 min n

2M µ(Ω) ^1/p λk, k o

, M µ(Ω) ^1/p λ 1 − Mµ(Ω) ^1/p λ k



. (1.11) As a consequence, we have the

Corollary 1.12. Suppose that L + λI satisfies a k–(UMP) and 2M µ(Ω) ^1/p λ ≤ 1. Then, for every f ∈ P k ^p , also R λ f ∈ P k ^p .

Proof. It is an immediate consequence of (1.10), since h ≤ k.

Remark 1.13. This corollary implies that condition (1.8) is satisfied if λ is small, in the spirit of bifurcation.

As a straightforward consequence, in addition to (1.10), we have the following sufficient condition for Theorem 1.10:

Proposition 1.14. Assume (1.5) and (1.6); moreover, suppose that λ + ∈ R and that 2M µ(Ω) ^1/p λ + ≤ 1. Then the conclusion of Theorem 1.10 holds.

2 Proof of Theorem 1.6

We introduce the operator ˜ R 0 : L ^p → ˜ C defined by

˜

u = ˜ R 0 f ⇐⇒ L˜u = ˜ ˜ f , which is well defined by assumption (1.6).

The first lemma we prove gives a condition that ensures the existence of R λ .

Lemma 2.1. There exists Λ 1 > 0 such that for all λ ∈ [−Λ 1 , Λ 1 ] \ {0} the resolvent R λ : L ^p (Ω) → C ⁰ (Ω) of L is well defined. Moreover, there exists C > 0 such that, if ˜ f ∈ L ^p and λ ∈ [−Λ ¹ , Λ 1 ] \ {0} then

kR λ f k ˜ C

(Ω) ≤ Ck ˜ f k L

(Ω) . More precisely, C := ^M

1−Λ

k ˜ R

k

, where M is the constant appearing in (1.6).

Here k ˜ R 0 k C→ ˜ ^˜ C denotes the norm of the restriction of the operator ˜ R 0 from C to ˜ ˜ C, which is well defined, since ˜ C ⊂ L ^p .

Proof. Rewrite equation (1.2) as the system ( L˜ u + λ˜ u = ˜ f ,

λu = f . (2.1)

Applying ˜ R 0 , the first equation in (2.1) can be rewritten as

(I + λ ˜ R 0 )˜ u = ˜ R 0 f . ˜ (2.2) Now, observe that ˜ R 0 f ∈ ˜ ˜ C; take |λ| < 1/k ˜ R 0 k C→ ˜ ˜ C , so that I + λ ˜ R 0 is invertible from ˜ C to ˜ C, and equation (2.2) is solved by ˜u = (I + λ ˜ R 0 ) ⁻¹ R ˜ 0 f . ˜

Finally, take Λ 1 ∈ (0, 1/k ˜ R 0 k C→ ˜ ^˜ C ); therefore, from the triangle inequality, (2.2) and (1.6), for all λ with |λ| ≤ Λ 1 we have that

k˜uk L

(Ω) − Λ ¹ k ˜ R 0 k C→ ˜ ^˜ C k˜uk L

(Ω) ≤ k˜uk L

(Ω) − |λ|k ˜ R 0 k C→ ˜ ^˜ C k˜uk L

(Ω)

≤ k(I + λ ˜ R 0 )˜ uk L

(Ω) = k ˜ R 0 f k ˜ L

(Ω)

= k˜uk L

(Ω) ≤ Mk ˜ f k L

(Ω) . The thesis follows.

Note that the proof above provides the estimate Λ 1 < 1

k ˜ R 0 k C→ ˜ ^˜ C

.

After proving that R λ is well defined, we have our second

Lemma 2.2. For any k > 0 there exists Λ 2 := Λ 2 (k) ∈ (0, Λ 1 ] such that for all λ ∈ [−Λ ² , Λ 2 ] \ {0} the operator L + λI has a k–(UMP). Moreover, a strong k–(UMP) holds if λ ∈ (−Λ ² , Λ 2 ) \ {0}.

Proof. If f ∈ F k ^p , f ≥ 0, then f = µ(Ω) ¹ kfk L

(Ω) . Thus, using the second equation in (2.1), one has

λu = λR λ ( ˜ f + f ) = λR λ ( ˜ f ) + f = λR λ ( ˜ f ) + 1

µ(Ω) kfk L

(Ω)

≥ 1

µ(Ω) kfk L

(Ω) − |λ|kR λ f k ˜ L

(Ω) .

By the previous Lemma it results that if λ ∈ [−Λ 1 , Λ 1 ] \ {0}, then λu ≥ 1

µ(Ω) kfk L

(Ω) − |λ| M 1 − Λ 1 k ˜ R 0 k C→ ˜ ˜ C

k ˜ f k L

(Ω)

≥ 1

µ(Ω) − k|λ| M 1 − Λ 1 k ˜ R 0 k C→ ˜ ^˜ C

!

kfk L

(Ω) ,

since f ∈ F k ^p . The Lemma is thus completely proved, provided that

Λ 2 = min (

Λ 1 , 1 − Λ 1 k ˜ R 0 k C→ ˜ ˜ C

kM µ(Ω) )

.

Proof of Theorem 1.6. Fixed k > 0, the result follows by taking Λ = Λ 2 (k) as given in Lemma 2.2.

3 Proof of Theorem 1.10

In this section we prove Theorem 1.10, so that we assume all the conditions of the previous section. Let us fix k > 0.

Lemma 3.1. Assume that there exists λ 0 > 0 such that L+λ 0 I has a k–(UMP) and R λ

(P k ^p ) ⊆ P k ^p . Then R λ : P k ^p → C ⁰ (Ω) is defined for all λ ∈ (0, λ 0 ] and

1. the function λ 7→ R λ f is analytic in (0, λ 0 ] for all f ∈ P k ^p ; 2. L + λI satisfies a k–(UMP) for all λ ∈ (0, λ 0 ];

3. L + λI satisfies a strong k–(UMP) for all λ ∈ (0, λ ⁰ ).

Proof. First, by (1.7), we immediately obtain R λ

1 = _λ ¹

, and 1 ∈ P k ^p for any k > 0, since ˜1 = 0.

Now, take f ∈ P k ^p ∩C ⁰ ( ¯ Ω); using the k–(UMP), the inclusion R λ

(P k ^p ) ⊆ P k ^p , assumption (1.6) and system (2.1) for λ = λ 0 , we immediately get

0 ≤ R λ

f = ˜ u + u ≤ Mk ˜ f − λ 0 uk ˜ L

(Ω) + f

λ 0 ≤ Mk(kfk L

(Ω) + λ 0 kuk L

(Ω) ) + f λ 0

= 2M kkfk L

(Ω) + f λ 0 ≤



2M kµ(Ω) + 1 λ 0



kfk C

(Ω) .

(3.1) Thus

kR λ

k = sup

f ∈Pp k∩C0( ¯Ω)

kf k

≤1

kR λ

f k C

(Ω) ≤ η + 1 λ 0

,

where we have set η := 2M kµ(Ω)λ 0 .

Now, for any λ ∈ R we rewrite Lu + λu = f as Lu + λ 0 u = f + (λ 0 − λ)u, and applying R λ

, we get

u − (λ ⁰ − λ)R ^λ

u = R λ

f. (3.2) If (λ 0 −λ)kR ^λ

k < 1, then I −(λ ⁰ −λ)R ^λ

is invertible from the cone P k ^p ∩C ⁰ (Ω) to itself and (3.2) is solvable, since R λ

f ∈ P k ^p ∩ C ⁰ (Ω).

Now, if kR ^λ

k = 1/λ ⁰ , we find

(λ 0 − λ)kR ^λ

k < 1 ⇐⇒ λ 0 − λ

λ 0 < 1 ⇐⇒ λ > 0, which implies that (3.2) is solvable for any λ ∈ (0, λ 0 ).

On the other hand, if kR λ

k > 1/λ 0 , since kR λ

k ≤ ^η+1 _λ

, we have λ > η

η + 1 λ 0 ⇐⇒ (λ 0 − λ) η + 1

λ 0 < 1 =⇒ (λ 0 − λ)kR λ

k < 1.

This implies that if λ ∈ 

η η+1 λ 0 , λ 0

 , then (3.2) is solvable.

If f ∈ P k ^p then R λ

f ∈ P k ^p ∩ C ⁰ (Ω) by assumption and (3.2) is solvable as well, provided that λ ∈ 

η η+1 λ 0 , λ 0

 , or λ ∈ (0, λ 0 ) if kR λ

k = 1/λ 0 , the easier case that we don’t discuss.

Now, since u = [I − (λ ⁰ − λ)R ^λ

] ⁻¹ R λ

f by (3.2), from Neumann’s formula for operators we get

R λ f = [I − (λ 0 − λ)R λ

] ⁻¹ R λ

f =

∞

X

n=0

(λ 0 − λ) ⁿ R ⁿ _λ

!

R λ

f, (3.3)

and thus the function λ 7→ R λ f is analytic.

Since f ≥ 0, by assumption we have R λ

f ≥ 0. Moreover, R λ

f be- longs to P k ^p by hypothesis; so we can iterate obtaining R λ f ≥ 0 whenever λ ∈ 

η η+1 λ 0 , λ 0

i and f ≥ 0.

In order to conclude the proof of claims 1 and 2, let us show that it is possible to extend the results above also for λ ∈ 

0, _η+1 ^η λ 0

i . Indeed, by assumption (1.6) and system (2.1) for λ 1 = _η+ ^η

2

λ 0 , operating as for (3.1), we find kR λ

f k C

(Ω) ≤ k˜uk C

(Ω) + kuk C

(Ω) ≤ Mk ˜ f k L

(Ω) + f

λ 1

≤ Mkkfk L

(Ω) + f

λ 1 ≤  η λ 0

+ 1 λ 1



kfk C

(Ω) ≤ η + 1

λ 1 kfk C

(Ω) . Moreover, since λ 1 ∈ 

η η+1 λ 0 , λ 0

 , by the previous part we know that R λ

(P k ^p ) ⊆

P k ^p , so that we can proceed as above, extending the results for λ belonging to the

interval 

η η+1 λ 1 , λ 0

 = 

η η+1

η η+

λ 0 , λ 0

 . Now, we repeat the procedure above starting from _η+1 ^η _η+ ^η

2

λ 0 , and by induction we can extend the results above for all those λ’s belonging to the whole interval 

0, ^2+η _1+η λ 0

 , so that claims 1 and 2 follow.

Now, take f ∈ P k ^p , f > 0 in a subset of Ω having positive measure, and let λ ∈ (0, λ 0 ]. First note that by (3.3) the following monotonicity property is immediate:

if 0 < λ 2 < λ 1 ≤ λ 0 then R λ

f ≥ R λ

f. (3.4) Then, fix x ∈ Ω and consider the function g : (0, λ 0 ] → C ⁰ (Ω) defined as g(λ)(x) = R λ f (x). Then, by (3.3) and (3.4), it is clear that g λ is nonnegative and non increasing.

By the first statement of this Lemma, we can assume that Λ 1 < ^2+η _1+η λ 0 . By Lemma 2.2 there exists Λ 2 = Λ 2 (k) ∈ (0, Λ ¹ ] such that for all λ ∈ (0, Λ ² ) the operator L + λI satisfies a strong k–(UMP), i.e g(λ) > 0 for all λ ∈ (0, Λ ² ), so that g is not identically trivial. On the other hand, g is analytic in (0, λ 0 ], so by analytic continuation we also get that g(λ) > 0 for all λ ∈ (0, λ ⁰ ).

Now we are able to give the

Proof of Theorem 1.10. From (1.8), we can apply Lemmas 2.2 and 3.1, and thus we infer that the set

Λ + (k) := n

λ > 0 : L + λI satisfies a k–(UMP) o

is a nonempty interval. By Lemma 3.1 we also know that a strong k–(UMP) holds if λ ∈ (0, λ + ).

Letting λ + ∈ R, we now show that λ + ∈ Λ + . Indeed, take a strictly in- creasing sequence λ j such that lim j→+∞ λ j = λ + and let j 0 be such that λ + < ^2+η _1+η λ j

. By assumption R λ

satisfies a k–(UMP); thus, from the first statement of Lemma 3.1 we know that R λ

is well defined, and by (3.3) applied with λ 0 = λ + we can conclude that u j = R λ

f → u = R λ

f uniformly in ¯ Ω as j → ∞. Since f ∈ P k ^p , then u j ≥ 0, and hence u ≥ 0 as well.

Proof of Proposition 1.11. By (2.1) we get λu = f and L˜ u = ˜ f − λ˜u, so that kfk L

(Ω) = λkuk L

(Ω) , since f ∈ P k ^p and u ≥ 0 by assumption. Moreover, f is nonconstant if and only if u is nonconstant as well.

Hence, from hypothesis (1.6), we have

k˜uk L

(Ω) ≤ µ(Ω) ^1/p k˜uk L

(Ω) ≤ Mµ(Ω) ^1/p k ˜ f − λ˜uk L

(Ω)

≤ Mµ(Ω) ^1/p [k ˜ f k L

(Ω) + λk˜uk L

(Ω) ]. (3.5) Now, if k ˜ f k L

(Ω) < λk˜uk L

(Ω) , we get

k˜uk L

(Ω) < 2M µ(Ω) ^1/p λk˜uk L

(Ω) ,

and if ˜ u 6= 0, i.e. u is not a constant, we find 2Mµ(Ω) ^1/p λ > 1, and (1.9) is proved.

Now, let us prove (1.10): if λk˜uk L

(Ω) ≤ k ˜ f k L

(Ω) , we have two estimates:

first,

λk˜uk L

(Ω) ≤ k ˜ f k L

(Ω) ≤ kkfk L

(Ω) = λkkuk L

(Ω) , so that

k˜uk L

(Ω) ≤ kkuk L

(Ω) . Moreover, starting from (3.5), we have

k˜uk L

(Ω) ≤ 2Mµ(Ω) ^1/p k ˜ f k L

(Ω) ≤ 2Mµ(Ω) ^1/p kkfk L

(Ω) = 2M µ(Ω) ^1/p λkkuk L

(Ω) . Hence (1.10) holds.

Finally, suppose that M µ(Ω) ^1/p λ < 1; then from (3.5), we obtain

k˜uk L

(Ω) ≤ M µ(Ω) ^1/p

1 − Mµ(Ω) ^1/p λ k ˜ f k L

(Ω) ≤ M µ(Ω) ^1/p

1 − Mµ(Ω) ^1/p λ kkfk L

(Ω)

= M µ(Ω) ^1/p λ

1 − Mµ(Ω) ^1/p λ kkuk L

(Ω) ,

(3.6)

which, together with (1.10), concludes (1.11).

4 Applications

In this section we present some differential problems where Theorems 1.6 and 1.10 can be applied. The first two examples are almost straightforward, after the considerations made in Section 1.

The third application concerns time–periodic parabolic problems, which have raised a growing interest in the last years, especially in their nonlinear versions, mainly for the large number of biological applications they describe (see [14], [15], [18], [24], [29],[31] and also [17] and [28] for non periodic applications), and for which a general approach for the validity of an antimaximum principle seemed to miss so far, except for [11] and [22].

4.1 Laplace operator

Let us go back to the classical Neumann problem given in (1.4) ( ∆u + λu = f in Ω,

∂u

∂ν = 0 on ∂Ω,

where Ω is a bounded smooth domain of R ^N , N ≥ 1, and f ∈ L ^p (Ω), p ∈ (N/2, ∞). Then it is well known that problem (1.4) with λ = 0 has a solution if and only if R

Ω f = 0. On the other hand, setting L = ∆, it is clear that Ker(L) =

{constant functions} and that λ 1 = 0. In addition, by elliptic regularity, the problem

( L˜ u = ˜ f in Ω,

∂ ˜ u

∂ν = 0 on ∂Ω, has a unique solution

u ∈ W ˜ ^2,p (Ω) ֒→ C ⁰ (Ω) (since p > N/2) satisfying the additional condition R

Ω u = 0, that is ˜ ˜ u ∈ ˜ C, according to the nota- tions introduced in Section 1. Moreover, as already remarked at the beginning, by the classical estimates in [1], there exists ˜ M > 0 such that

k˜uk W

(Ω) ≤ ˜ M k ˜ f k L

(Ω) ;

note that, by uniqueness, the usual term k˜uk L

(Ω) that would appear on the right hand side of the previous inequality can be deleted. Hence, all the abstract requirements (1.5) and (1.6) for L are fulfilled, with the underlying measure µ equal to Lebesgue’s measure in Ω.

Applying Theorem 1.6 we immediately get the following

Proposition 4.1. Let p > N/2; for any k > 0 there exists Λ = Λ(k) > 0 such that ∆ + λI under homogeneous Neumann boundary conditions satisfies a a k–(UMP) if λ ∈ (0, Λ]. Moreover, a strong k–(UMP) holds if λ ∈ (0, Λ).

In [7] it was proved that this result was valid for N = p = 1, and that the analogous of Theorem 1.10 holds, while in general we cannot consider this char- acterization, unless Proposition 1.14 can be applied (see Section 4.4). However, the authors underlined the fact that they could not prove this result for N > 1, so that they were naturally turned to consider polyharmonic operators in low dimensions. We consider the same operators in the following section.

4.2 Polyharmonic operator

Let us now consider an elliptic Neumann problem in presence of the m–poly- harmonic operator ∆ ^m , m ∈ N,

( ∆ ^m u + λu = f in Ω,

∂u

∂ν = ^∂∆u _∂ν . . . = ^∂∆

_∂ν ^u = 0 on ∂Ω, (4.1) where Ω ⊂ R ^N is a bounded smooth domain and f ∈ L ^p (Ω) with p ∈ (N/2m, ∞).

As before, Ker(L) = {constant functions}, with L = ∆ ^m , and as already re- marked in [7], the assumption (1.5) for L is satisfied. Moreover, by [1], the weak solution u ∈ W ^m,p (Ω) of (4.1) actually belongs to W ^2m,p (Ω) and is such that the estimate

k˜uk W

(Ω) ≤ ˜ M k ˜ f k L

(Ω) (4.2)

holds for a suitable constant ˜ M . Indeed, ˜ u solves

( ∆ ^m u = ˜ ˜ f in Ω,

∂ ˜ u

∂ν = ^∂∆˜ _∂ν ^u . . . = ^∂∆

_∂ν ^{u ˜} = 0 on ∂Ω,

and it belongs to W ^2m,p (Ω); multiplying both sides of the equation above by

|∆ ^m u| ˜ ^p−2 ∆ ^m u, integrating and applying the H¨ ˜ older inequality, we obtain k∆ ^m uk ˜ L

(Ω) ≤ k ˜ f k L

(Ω) .

By the Open Mapping Theorem we can prove, in the spirit of [26, Remark 2.1], that k∆ ^m · k ^p is a norm (equivalent to the usual one) for functions in W ^2m,p (Ω) with zero mean. Thus, being p > N/2m, we have W ^2m,p (Ω) ֒→ C ⁰ (Ω), so that (4.2) holds, i.e. also assumption (1.6) is verified.

Now, Theorem 1.6 can be immediately applied, extending to higher dimen- sions the result showed in [7] when N ≤ 2m − 1 and p = 1. As in the previous case, we remark that in [7] there was a complete characterization of the “good”

λ’s, while here we give a sufficient condition for the validity of a k–(UMP):

Proposition 4.2. Let m ∈ N and p > N/2m; for any k > 0 there exists Λ = Λ(k) > 0 such that ∆ ^m + λI under homogeneous Neumann boundary conditions satisfies a k–(UMP) if λ ∈ (0, Λ]. Moreover, a strong k–(UMP) holds if λ ∈ (0, Λ).

4.3 Periodic parabolic problems

Here we consider the periodic parabolic problem



 

 

u t − αu xx + λu = f in Ω × (0, ∞),

u x = 0 on ∂Ω × (0, ∞),

u(0) = u(T ),

(4.3)

where Ω is a bounded interval of R, α > 0, T > 0, λ ∈ R and f ∈ L ^p (Q T ), with Q T = Ω × (0, T ) and p > 1. Using the notation of Section 1, we set Lu := u t − αu xx with

D(L) = n

u ∈ H ¹ (0, T ; W ^2,p (Ω)) : u x = 0 on ∂Ω for a.e. t ∈ (0, T ) o . First, by parabolic regularity (see [25]), any weak solution of (4.3) belongs to C ⁰ ([0, T ]; W ^1,p (Ω)). This fact, with an obvious adaptation of the proof in [16], lets us prove that Ker(L) = {constant functions} and that f ∈ Im(L) if and only if

f ∈ L ^p =



f ∈ L ^p (Q T ) : Z

Q

f dx dt = 0

 . Moreover, ˜ u ∈ C ⁰ ([0, T ]; W ^1,p (Ω)) and

k˜uk L

(0,T ;W

(Ω)) ≤ C k˜u(0)k W

(Ω) + kfk L

(Q

)

(4.4)

for a universal constant C = C(α, T, Ω).

We are not able to apply Theorem 1.6 to every problem of the form (4.3).

Thus, at this point we assume to deal with data (α, T, Ω) such that the related constant C(α, T, Ω) is such that

C = C(α, T, Ω) < 1, (4.5)

which holds, for example, if α is large enough.

Thus from (4.4) we easily get

k˜uk L

(0,T ;W

(Ω)) ≤ C

1 − C kfk L

(Q

) . (4.6) By Poincar´e–Wirtinger inequality (see, for example, [4, Chapter VIII]) we know that

k˜u(t)k L

(Ω) ≤ p|Ω|k˜u(t)k W

(Ω) ∀ t, so that (4.6) implies

k˜uk L

(Q

) ≤ C p|Ω|

1 − C kfk L

(Q

) ; thus (1.6) is satisfied with M = ^C

√ |Ω|

1−C .

Without other assumptions we can now apply Theorem 1.6 to problem (4.3) to get:

Theorem 4.3. Assume (4.5), fix k > 0 and set Lu := u t − αu xx . Then there exists Λ = Λ(k) > 0 such that L + λI satisfies a k−(UMP) if λ ∈ (0, Λ].

Moreover a strong k−(UMP) holds if λ ∈ (0, Λ).

To our best knowledge, there are not many results concerning (AMP) or (UAMP) for parabolic problems like (4.3). We quote [11], where an eventual antimaximum principle is proved; more precisely, the authors prove that solu- tions of Cauchy–Dirichlet problems are positive for large times, also when the data is negative.

In [22], a periodic parabolic problem under both homogeneous Dirichlet or Neumann conditions is considered, and an (AMP) also in presence of a weight is proved. But if N = 1 they assume that the right–hand–side of the parabolic equation belongs to L ^p (Ω) with p > 3, and in addition their result is not uniform.

On the contrary, we can handle the case f ∈ L ^p (Ω), p > 1, and we can prove a k−(UMP), so that a certain uniformity is guaranteed for the (AMP), at least if α is large enough.

Of course, a result analogous to Theorem 4.3 can be proved if −∆ in dimen- sion 1 is replaced by a polyharmonic operator in a higher dimension N with the natural Neumann boundary conditions:



 

 

u t + α(−∆) ^m u + λu = f in Ω × (0, ∞),

∂u

∂ν = ^∂∆u _∂ν . . . = ^∂∆

_∂ν ^u = 0 on ∂Ω × (0, ∞),

u(0) = u(T ),

with N ≤ 2m − 1, so that C ⁰ ([0, T ]; H ^m (Ω)) ⊂ C ⁰ (Q T ) and α is large. The details are left to the reader.

4.4 The limit case: validity of Proposition 1.14

We give an example for the validity of Proposition 1.14, taking p = 2 and considering the problem

( u ^′′ + λu = f in (0, π),

u ^′ (0) = u ^′ (π) = 0. (4.7)

Let ˜ u ∈ W ^2,2 (0, π) solve

( ˜ u ^′′ = ˜ f in (0, π),

˜

u ^′ (0) = ˜ u ^′ (π) = 0.

Then we claim that

k˜uk L

(0,π) ≤  π 3

 1/2

k ˜ f k L

(0,π) . (4.8) Indeed, ˜ u ^′ ∈ H 0 ¹ (0, π), so that, being 1 the first eigenvalue of −d ² /dx ² , by the Poincar´e inequality, we have

k˜u ^′ k L

(0,π) ≤ k˜u ^′′ k L

(0,π) = k ˜ f k L

(0,π) .

Moreover, after an even reflection of ˜ u around x = 0, we have that ˜ u is a 2π-periodic function such that R π

−π u = 0. Hence, by [19, Proposition 2.5.35 ˜ (b)], we have

k˜uk ² L

(0,π) = k˜uk ² L

(−π,π) ≤ π

6 k˜u ^′ k ² L

(−π,π) = π

3 k˜u ^′ k ² L

(0,π) , since k˜u ^′ k ² L

(−π,π) = 2k˜u ^′ k ² L

(0,π) , and then (4.8) easily follows.

In conclusion, the condition in (1.6) holds with M =  π

3

 1/2

, so that the requirement in Proposition 1.14 holds if

2 π

√ 3 λ + ≤ 1.

But in [8] it was shown that for d ² /dx ² +λI with Neumann boundary conditions on (0, π), the associated Green function is nonnegative only if λ ≤ 1/4 = λ + , of course independently of the Lebesgue space containing f . Thus

2M µ(Ω) ^1/2 λ + < 1,

as required in Proposition 1.14.

References

[1] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for so- lutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 6230–727.

[2] P. Antonini, D. Mugnai, P. Pucci, Quasilinear elliptic inequalities on com- plete Riemannian manifolds, J. Math. Pures Appl. 87 (2007), no. 6, 582–600.

[3] P. Benilan, H. Brezis, M.G. Crandall, A semilinear equation in L ¹ (R ^N ), Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 4, 523–555.

[4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equa- tions, Springer New York Dordrecht Heidelberg London, 2011.

[5] A. Cabada, J.A. Cid, M. Tvrd´ y, A generalized anti-maximum principle for the periodic one-dimensional p–Laplacian with sign-changing potential, Non- linear Anal. 72 (2010), no. 7-8, 3436–3446.

[6] A. Cabada, S. Lois, Maximum principles for fourth and sixth order periodic boundary value problems, Nonlinear Anal. 29 (1997), no. 10, 1161–1171.

[7] J. Campos, J. Mawhin, R. Ortega, Maximum principles around an eigen- value with constant eigenfunctions, Commun. Contemp. Math. 10 (2008), no. 6, 1243–1259.

[8] P. Cl´ement, L. A. Peletier, An anti-maximum principle for second-order elliptic operators, J. Differential Equations 34 (1979), no. 2, 218–229.

[9] P. Cl´ement, G. Sweers Uniform anti-maximum principle for polyharmonic boundary value problems, Proc. Amer. Math. Soc. 129 (2001), no. 2, 467–

474.

[10] P. Cl´ement, G. Sweers Uniform anti-maximum principles, J. Differential Equations 164 (2000), no. 1, 118–154.

[11] J.I. D´ıaz, J. Fleckinger–Pell´e, Positivity for large time of solutions of the heat equation: the parabolic antimaximum principle, Discrete Contin. Dyn.

Syst. 10 (2004), no. 1-2, 193–200.

[12] J. Fleckinger–Pell´e, J.-P. Gossez, F. de Th´elin, Antimaximum principle in R ^N : local versus global, J. Differential Equations 196 (2004), no. 1, 119–133.

[13] R. Fortini, D. Mugnai, P. Pucci, Maximum Principles for anisotropic el- liptic inequalities, Nonlinear Anal. 70 (2009), 2917–2929.

[14] G. Fragnelli, Positive periodic solutions for a system of anisotropic parabolic equations, J. Math. Anal. Appl. 367 (2010), 204–228.

[15] G. Fragnelli, A. Andreini, Non–negative periodic solution of a system of

(p(x); q(x))–Laplacian parabolic equations with delayed nonlocal terms, Dif-

fer. Equ. Dyn. Syst. 15 (2007), 231–265.

[16] G. Fragnelli, D. Mugnai, A k–uniform maximum principle when 0 is an eigenvalue, “Modern Aspects of the Theory of Partial Differential Equa- tions” M. Ruzhansky, J. Wirth (Eds.), ”Operator Theory: Advances and Applications” 216, Birkh¨ auser/Springer, Basel, 2011, 139–151.

[17] G. Fragnelli, D. Mugnai, Nonlinear delay equations with nonautonomous past, Discrete Contin. Dyn. Syst. 21 (2008), no 4, 1159–1183.

[18] G. Fragnelli, P. Nistri, D. Papini, Positive periodic solutions and optimal control for a distributed biological model of two interacting species, Nonlinear Anal., Real World Appl. 12 (2011), no. 3, 1410–1428.

[19] L. Gas´ınski, N.S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications Edited by Ravi P. Agarwal and Donal ORegan Vol. 9, Chapman & Hall/CRC Boca Raton 2005.

[20] T. Godoy, J.P. Gossez, S. Paczka, A minimax formula for principal eigen- values and application to an antimaximum principle, Calc. Var. Partial Dif- ferential Equations 21 (2004), no. 1, 85–111.

[21] T. Godoy, J.P. Gossez, S. Paczka, On the antimaximum principle for the p–Laplacian with indefinite weight, Nonlinear Anal. 51 (2002), no. 3, 449–

467.

[22] T. Godoy, E. Lami Dozo, S. Paczka, On the antimaximum principle for parabolic periodic problems with weight, Rend. Semin. Mat. Univ. Politec.

Torino 60 (2002), no. 1, 33–44.

[23] H.-C. Grunau, G. Sweers, Optimal conditions for anti-maximum principles, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), no. 3-4, 499–513.

[24] P. Hess, Periodic-parabolic boundary value problems and positivity Pitman Research Notes 247, John Wiley, New York, 1991.

[25] J. L. Lions, E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I Die Grundlehren der mathematischen Wissenschaften 181, Berlin-Heidelberg-New York: Springer-Verlag, XVI, 1972.

[26] D. Mugnai, On a “reversed” variational inequality, Topol. Methods Non- linear Anal. (2001) 17, no. 2, 321–358.

[27] D. Mugnai, P. Pucci, Maximum Principles for inhomogeneous elliptic inequalities on complete Riemannian manifolds, Adv. Nonlinear Stud. 9 (2009), 429–452.

[28] P. Quittner, P. Souplet, Superlinear parabolic problems. Blow-up, global existence and steady states, Birkh¨ auser Advanced Texts, Birkh¨ auser Verlag, Basel, 2007.

[29] T. I. Seidman, Periodic solutions of a non-linear parabolic equation, J.

Differential Equations 19 (1975), no. 2, 242–257.

[30] G. Sweers, L ⁿ is sharp for the anti-maximum principle, J. Differential Equations 134 (1997), no. 1, 148–153.

[31] Y.Wang, J. Yin, Z.Wu, Periodic solutions of evolution p-Laplacian equa-

tions with nonlinear sources, J. Math. Anal. Appl. 219 (1998), no. 1, 76–96.