Maximum principles for anisotropic elliptic inequalities Roberto Fortini

Maximum principles for anisotropic elliptic inequalities

Roberto Fortini ^a Dimitri Mugnai ^a Patrizia Pucci ^a

a

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

Abstract

We establish some maximum and comparison principles for weak distributional solutions of anisotropic elliptic inequalities in divergence form, both in the homogeneous and non homoge- neous case. The main prototypes we have in mind are inequalities involving the p(·)–Laplace operator and the generalized mean curvature operator.

Key words: Anisotropic elliptic inequalities in divergence form, maximum and comparison principles weak solutions.

2000 Mathematics Subject Classification. Primary: 35B40, 35L70, 35L80; Secondary: 35B99, 35Q35, 76A05.

1. Introduction

In the last years great attention was paid to (partial) differential equations having anisotropic nature:

this means that the associated operators possess non standard growth conditions.

In this paper we start by studying maximum principles for weak distributional solutions of anisotropic elliptic inequalities of the form

divA(x, u, Du) + B(x, u, Du) ≥ 0 in Ω, (1.1)

where Ω is a bounded domain contained in R ⁿ , and we consider both the homogeneous and the non homogeneous cases.

Anisotropic equations and systems of anisotropic equations appear in many situations, among them we recall some processes in elasticity theory, in modelling thermoconvective stationary fluxes for non New- tonian fluids (see [3,4,25]), and above all those processes concerning electrorheological fluids, which are extremely important in technology, especially in fast acting hydraulic valves and clutches. This type of non Newtonian fluids, also known as smart fluids, can change significantly their mechanical properties un- der the action of an external electromagnetic field, and exactly this physical characteristic produces the anisotropy in the equations which describe the phenomenon in which they are involved (for example, see [2–4,6,16,17,19,20,24,25]).

Email addresses: [email protected] (Roberto Fortini), [email protected] (Dimitri Mugnai), [email protected] (Patrizia Pucci).

1

This research was supported by the Project Metodi Variazionali ed Equazioni Differenziali Non Lineari.

The mathematical modelling of these fluids was made precise by many people, using different approaches, not excluding numerical ones (for instance, see [7]). Recently Ru˘ zi˘ cka in [20], improving the ideas of the previous paper [19] with Rajagopal, presented a very interesting model which reduces, in the stationary case, to the system

rot E = 0, div E = 0, div u = 0, divS(E, E (u)) − Dπ − [Du] u = f − X ^E [DE] E,

where E is the external electromagnetic field, u : Ω ⊂ R ³ → R ³ denotes the velocity of the fluid, S is the extra stress tensor, π is the pressure, X ^E is the constant of electric susceptibility and, according to the usual notation, E (u) denotes the symmetric part of the Jacobian matrix Du of u.

In [20,21] a particular form of S is considered, finally obtaining the following equation of generalized mean curvature

div 

(1 + |D(u)| ² ) ^[p(x)−2]/2 D(u) 

+ B(x, u, Du) = 0, 1 < p(x) ≤ 2, where p(·) is a variable exponent in Ω and now u : Ω → R.

In [23] the author presents a model for a thermistor describing the electric current which varies in a given conductor according to the temperature p = p(x); in other words the problem here is to find an electric potential u = u(x) satisfying an equation of the following type

div(|Du| ^p(x)−2 Du) + B(x, u) = 0 in Ω,

with p(x) > 1, x ∈ Ω. This is a generalization of equations involving the standard p–Laplace operator,

∆ p u ≡ div(|Du| ^p−2 Du), where p > 1 is a constant. Note that, differently from the p–Laplacian, i.e. when p(x) ≡ p is constant in Ω, the p(·)–Laplace operator

∆ _p(·) u ≡ div(|Du| ^p(x)−2 Du), p(x) > 1, is not homogeneous in general.

In order to face maximum principles in this framework, we make a short preamble to describe the tools of Lebesgue and Sobolev spaces with variable exponent p(·).

After that, in Section 3 we present a general maximum principle, which allows us to derive in the following sections some comparison principles for solutions of elliptic anisotropic homogeneous and also non homogeneous differential inequalities. Our results were inspired by some theorems of [18], in which the case p(x) ≡ p is considered.

Concerning comparison principles for elliptic operators A, covering the fundamental examples of the generalized mean curvature and of the p(·)–Laplace operator, a typical result that we prove has the following form, where we are intentionally vague for the regularity of the solutions, referring to the following sections for precise statements.

Prototype of Comparison Theorem. Let u, v be solutions of

divA(x, v, Dv) + B(x, v) ≤ 0 ≤ divA(x, u, Du) + B(x, u) in Ω, (1.2) where Ω is a bounded domain of R ⁿ and B(x, z) is non increasing in the variable z. If u ≤ v on ∂Ω, then u ≤ v in Ω.

On the other hand, by constructing suitable comparison functions, also general maximum principles are given. Again we present a na¨ıve version of our results in the following

Prototype of Maximum Principle. Assume that Ω is a bounded domain contained in B _R and that B(x, z) ≤ γ for a.e. x ∈ Ω and for all z ∈ R. Let u be a solution of

divA(x, Du) + B(x, u) ≥ 0 in Ω, u ≤ 0 on ∂Ω.

Then there exists a universal constant C > 0 such that

u(x) ≤ RC in Ω.

In Section 6 we give some final applications and consequences of the comparison principles proved in the earlier sections, providing uniqueness results for elliptic problems having non homogeneous boundary conditions.

2. Preliminaries

In this section we introduce some notation. Throughout the rest of the paper we assume that Ω is a bounded domain of R ⁿ . Put

C + (Ω) = {h ∈ C(Ω) : min

x∈Ω

h(x) > 1}, and for any h ∈ C + (Ω) define

h + = max

x∈Ω

h(x) and h ₋ = min

x∈Ω

h(x).

From now on we assume to work with a fixed p ∈ C + (Ω). The variable exponent Lebesgue space is the real vector space of all those measurable functions u : Ω → R such that R

Ω |u(x)| ^p(x) dx is finite. This space, endowed with the Luxemburg norm,

kuk p(·) = inf (

λ > 0 : Z

Ω

   

u(x) λ

   

p(x)

dx ≤ 1 )

,

is a separable and reflexive Banach space. For basic properties of the variable exponent Lebesgue spaces we refer to [15], where classical results for Lebesgue spaces are extended to variable exponent Lebesgue spaces.

For example, since in this paper 0 < |Ω| < ∞, if q is a variable exponent in C ₊ (Ω), with p ≤ q in Ω, then the embedding L ^q(·) (Ω) ,→ L ^p(·) (Ω) is continuous by [15, Theorem 2.8].

Let L ^p

⁰

^(·) (Ω) be the conjugate space of L ^p(·) (Ω), obtained by conjugating the exponent pointwise that is, 1/p(x) + 1/p ⁰ (x) = 1, [15, Corollary 2.7]. Then for any u ∈ L ^p(·) (Ω) and v ∈ L ^p

⁰

^(·) (Ω) the following H¨ older type inequality

Z

Ω

|uv| dx ≤ r p kuk _p(·) kvk _p

⁰

_(·) , r p := 1 p ₋ + 1

p ⁰ ₊ , is valid by [15, Theorem 2.1]. Moreover, if u ∈ L ^p(·) (Ω), setting

ρ _p(·) (u) = Z

Ω

|u| ^p(x) dx, the following properties hold:

kuk _p(·) < 1 (= 1; > 1) ⇔ % _p(·) (u) < 1 (= 1; > 1) kuk p(·) > 1 ⇒ kuk ^p _p(·)

⁻

≤ % p(·) (u) ≤ kuk ^p _p(·)

⁺

kuk p(·) < 1 ⇒ kuk ^p _p(·)

⁺

≤ % p(·) (u) ≤ kuk ^p _p(·)

⁻

(2.1) ku n − uk p(·) → 0 ⇔ % p(·) (u n − u) → 0, (2.2) Analogously, the variable exponent Sobolev space W ^1,p(·) (Ω), consisting of functions u ∈ L ^p(·) (Ω) whose distributional gradient Du exists almost everywhere and belongs to L ^p(·) (Ω), endowed with the norm

kuk _1,p(·) = kuk _p(·) + kDuk _p(·) ,

is a separable and reflexive Banach space (see [15, Theorem 3.1]). As shown by Zhikov [22,23], in general smooth functions are not dense in W ^1,p(·) (Ω), but if the exponent p ∈ C + (Ω) is also logarithmic H¨ older continuous, i.e. there exists M > 0 such that

|p(x) − p(y)| ≤ − M

log |x − y| for all x, y ∈ Ω with |x − y| ≤ 1/2, (2.3)

and if ∂Ω is Lipschitz continuous, then smooth functions are dense in W ^1,p(·) (Ω) (see [10, Theorem 2.6]). Of

course a fundamental role in the applications is played by W ₀ ^1,p(·) (Ω), the Sobolev space with zero boundary

values, defined as the closure of C ₀ ^∞ (Ω) under the norm k · k _1,p(·) .

In addition, if p satisfies (2.3), the p(·)–Poincar´ e inequality

kuk _p(·) ≤ CkDuk _p(·) (2.4)

holds for all u ∈ W ₀ ^1,p(·) (Ω), where C depends on p, |Ω|, diam(Ω), [12, Theorem 4.3], and so kuk = kDuk _p(·)

is an equivalent norm in W ₀ ^1,p(·) (Ω). Hence W ₀ ^1,p(·) (Ω) is a separable and reflexive Banach space. Due to the intensive use of the Poincar´ e inequality, from now we assume that p ∈ C + (Ω) satisfies (2.3).

Note that if p ₊ < n, then the variable critical Sobolev exponent p ^∗ (x) = np(x)

n − p(x)

is well defined also in this setting, and the embedding W ₀ ^1,p(·) (Ω) ,→ L ^p

^∗

^(·) (Ω) is continuous (see [13, Proposition 4.2]), that is

kuk _p

^∗

_(·) ≤ SkDuk _p(·) , (2.5)

where S = S(p(·), n, |Ω|). Moreover, the embedding W ₀ ^1,p(·) (Ω) ,→ L ^h(·) (Ω) is compact and continuous for any h ∈ C(Ω), with 1 ≤ h(x) < p ^∗ (x) for all x ∈ Ω (see [10, Theorem 2.3]).

Details, extensions and further references about p(·)–spaces and embeddings can be found in [6] and [8]–[15].

3. Maximum Principle for homogeneous inequalities

Now consider inequality (1.1), where, without any regularity condition, we only assume A(x, z, ξ) : Ω × R × R ⁿ → R ⁿ , B(x, z, ξ) : Ω × R × R ⁿ → R.

Definition 3.1 Let u ∈ L ¹ _loc (Ω) be a function having weak gradient Du with the property that A(·, u, Du) and B(·, u, Du) ∈ L ¹ _loc (Ω). We say that u is a (weak distributional) solution of (1.1) if

Z

Ω

hA(x, u, Du), Dϕi dx ≤ Z

Ω

B(x, u, Du)ϕdx (3.1)

for all ϕ ∈ C ¹ (Ω) such that ϕ ≥ 0 in Ω and ϕ ≡ 0 near ∂Ω.

In particular we say that u is a p(·)–regular solution if (3.1) holds and in addition A(·, u, Du) ∈ L ^p

0

(·)

loc (Ω). (3.2)

Note that in Definition 3.1 condition (3.2) is automatic when p ⁰ ≡ 1, so that in this case the regularity is included in the notion of solution itself.

Furthermore by u ≤ M on ∂Ω for some M ∈ R we mean that for every δ > 0 there exists a neighborhood of ∂Ω in which u ≤ M + δ.

For simplicity, if s(·) ∈ C + (Ω), we write k·k _s(·),Γ in place of k·k _L

s(·)

(Γ) when Γ is a measurable subset of Ω, and k·k _s(·) for k·k _L

s(·)

(Ω) .

We recall that p ∈ C ₊ (Ω) is logarithmic H¨ older continuous in the sense of (2.3) throughout the paper.

Following the ideas of [1], we start proving some useful lemmas.

Lemma 3.2 Let ω ⊂ R ⁿ be a bounded domain with Lipschitz continuous boundary and let ψ : R → R be a piecewise smooth function, with ψ ⁰ ∈ L ^∞ (R). If u ∈ W ^1,p(·) (ω), then ψ ◦ u ∈ W ^1,p(·) (ω). Moreover, if S denotes the set where ψ is not differentiable, then

D(ψ ◦ u) =

( ψ ⁰ (u)Du, se u / ∈ S,

0, se u ∈ S.

Proof. Under our assumptions C ^∞ (ω) is dense in W ^1,p(·) (ω). Let us first suppose ψ ∈ C ¹ (R), with

|ψ ⁰ | ≤ λ. Set v = ψ ◦ u and take (u k ) _k∈N in C ^∞ (ω) such that u k → u in W ^1,p(·) (ω) and a.e. in ω. Then v k := ψ ◦ u k ∈ C ¹ (ω) and Dv k = ψ ⁰ (u k )Du k for any k ∈ N. Moreover, |v ^k | ≤ λ |u k | and |Dv k | ≤ λ |Du k |, so they are both p(·)−integrable on ω.

It remains to prove that v k → v and Dv k → Dv = ψ ⁰ (u)Du in L ^p(·) (ω). Of course, by the mean value theorem, we have |v k (x) − v(x)| ≤ λ |u k (x) − u(x)| and so

Z

ω

|v k (x) − v(x)| ^p(x) dx ≤ max {λ ^p

⁻

, λ ^p

⁺

} Z

ω

|u k (x) − u(x)| ^p(x) dx → 0.

Moreover,

|Dv k − Dv| ≤ |ψ ⁰ (u _k )Du _k − ψ ⁰ (u _k )Du| + |ψ ⁰ (u _k )Du − ψ ⁰ (u)Du|

≤ λ |Du k − Du| + |ψ ⁰ (u _k ) − ψ ⁰ (u)| · |Du| , hence, since ω is bounded and p ∈ C + (ω),

|Dv k (x) − Dv(x)| ^p(x) ≤ C 

|Du k (x) − Du(x)| ^p(x) + |ψ ⁰ (u k ) − ψ ⁰ (u)| ^p(x) |Du| ^p(x)  . Consequently, by the Lebesgue Theorem, as k → ∞

% _p(·) (Dv k − Dv) = Z

ω

|Dv k (x) − Dv(x)| ^p(x) dx → 0

by (2.2), since kDu k − Duk _p(·) → 0 in L ^p(·) (ω), while |ψ ⁰ (u k ) − ψ ⁰ (u)| · |Du| → 0 a.e. in ω and

|ψ ⁰ (u _k ) − ψ ⁰ (u)| ^p(·) |Du| ^p(·) ≤ 2 ^p

⁺

max {λ ^p

⁻

, λ ^p

⁺

} |Du| ^p(·) ∈ L ¹ (ω).

From (2.2) we obtain kDv k − Dvk _p(·) → 0. Finally we observe that Dv = ψ ⁰ (u)Du is the weak gradient of v. This completes the first case.

Now let ψ be a piecewise smooth function. By iterating the following argument, we can assume that ψ is not differentiable at only one point, say u 0 . Without loss of generality, we take u 0 = 0 and suppose ψ(0) = 0. Now let ψ ₁ , ψ ₂ ∈ C ¹ (R), with bounded derivatives, be such that ψ 1 (u) = ψ(u) for u ≥ 0 and ψ ₂ (u) = ψ(u) for u ≤ 0. Then ψ(u) = ψ ₁ (u ⁺ ) + ψ ₂ (u ⁻ ). Since ψ ₁ , ψ ₂ ∈ C ¹ (R), for the first step, it is enough to show that u ^± ∈ W ^1,p(·) (ω).

Take ε > 0 and define

ψ _ε (t) =

(p t ² + ε ² − ε, se t ≥ 0,

0, se t ≤ 0.

Of course ψ ε ∈ C ¹ (R) and |ψ ε ⁰ | ≤ 1. For the first step ψ ε ◦ u ∈ W ^1,p(·) (ω) and Z

ω

ψ _ε (u) ∂ϕ

∂x i

dx = − Z

ω

ϕ ψ ⁰ _ε (u) ∂u

∂x i

dx = − Z

ω

⁺

√ u

u ² + ε ² ϕ ∂u

∂x i

dx

for any ϕ ∈ C ₀ ^∞ (ω), i = 1, ..., n, where ω ⁺ := {x ∈ ω : u(x) > 0}. Moreover |ψ ε (u)| = ψ ε (u) ≤ u ⁺ a.e. in ω and ψ ε (u) → u ⁺ as ε → 0. Furthermore u/ √

u ² + ε ² → 1 a.e. in ω ⁺ and 

  

√ u

u ² + ε ² ϕ ∂u

∂x i

   

≤ |ϕ| ·    

∂u

∂x i

    .

Therefore, we can pass to limit for ε → 0 and we obtain D(u ⁺ ) = (Du) χ

_ω+

. The rest of the proof is

straightforward. 2

Lemma 3.3 Let ψ : R → R ⁺ 0 be a non decreasing continuous function such that ψ(t) = 0 for t ∈ (−∞, l], l > 0, and ψ ∈ C ¹ in (l, m) t (m, ∞), and with ψ ⁰ bounded in R. Let u ∈ W loc ^1,p(·) (Ω) be a p(·)–regular solution of (1.1) with u ≤ 0 on ∂Ω, then (3.1) is valid for ϕ = ψ ◦ u, in the sense that

Z

Ω

hA(x, u, Du), Dϕi dx ≤ Z

Ω

[B(x, u, Du)] ⁺ ϕ dx, (3.3)

where Dϕ = ϕ ⁰ (u)Du when u / ∈ {l, m}.

Proof. Clearly ϕ is compactly supported in Ω. Let ω ⊂⊂ Ω be an open domain with Lipschitz continuous boundary which contains the support of ϕ. Let ϕ k = ψ k ◦ u be the truncation of ψ ◦ u at the level k, that is, ϕ k (u) = ψ(u) when u < k and ϕ k (u) = ψ(k) when u ≥ k. By the properties of ψ and by Lemma 3.2, clearly ϕ k ∈ W ₀ ^1,p(·) (Ω). Moreover, ϕ k has compact support in ω and can be used as a test function in (3.1).

Hence

Z

Ω

hA(x, u, Du), Dϕ k i dx ≤ Z

Ω

[B(x, u, Du)] ⁺ ϕ k dx.

Since

kDϕ _k − Dϕk _p(·),Ω = kDϕk _{p(·),{u≥k}} → 0

as k → ∞, by the Beppo Levi theorem (ϕ _k ↑ ϕ) the result follows at once. 2 Now suppose that A and B in (1.1) satisfy the condition that there exist a 1 > 0 and a 2 ≥ 0 such that for all (x, z, ξ) ∈ Ω × R ⁺ × R ⁿ there holds

hA(x, z, ξ), ξi ≥ a 1 |ξ| ^p(x) − a 2 z ^p(x) , (3.4) or, more simply, it is enough that the inequality holds true for those z and ξ’s belonging to the range of the solution and of its gradient.

Our first result in the spirit of maximum principles is the following.

Theorem 3.4 (Maximum Principle). Assume that A satisfies (3.4) and B(x, z, ξ) ≤ 0. Let u ∈ W _loc ^1,p(·) (Ω) be a p(·)–regular solution of (1.1) in Ω. If u ≤ 0 on ∂Ω, then u ≤ 0 a.e. in Ω.

Proof. Assume by contradiction that V := ess sup

x∈Ω

u(x) > 0, possibly infinite, and let ε > 0 be such that ε < min {1, V }. Without loss of generality, we suppose p + > p ₋ , otherwise p(·) is constant and the proof proceeds as in [18, Theorem 3.2.2]. We define

ψ(t) =



 

 

 

 

0, for t ≤ ε,

1 −  ε t

 ^p

−

−1

, for ε < t ≤ 1,

1 − ε ^p

⁻

⁻¹ + ε ^p

⁺

⁻¹ −  ε t

 p

+

−1

, for t ≥ 1.

Lemma 3.3 applies with l = ε and m = 1, so that ϕ := ψ(u) can be used as a nonnegative test function in (3.1). That is, by (3.3),

0 ≥ Z

Γ

1

hA(x, u, Du), Dϕi dx + Z

Γ

2

hA(x, u, Du), Dϕi dx, (3.5)

where Γ ₁ := {x ∈ Ω : ε < u(x) ≤ 1}, Γ ₂ := {x ∈ Ω : u(x) > 1} and Γ := Γ ₁ ∪ Γ 2 . Since Dϕ = ψ ⁰ (u)Du and

ψ ⁰ (u) =







(p ₋ − 1)ε ^p

⁻

⁻¹ u ^−p

⁻

a.e. in Γ ₁ , (p + − 1)ε ^p

⁺

⁻¹ u ^−p

⁺

a.e. in Γ 2 , we obtain

0 ≥ Z

Γ

1

hA(x, u, Du), Dui(p ₋ − 1)ε ^p

⁻

⁻¹

u ^p

⁻

dx +

Z

Γ

2

hA(x, u, Du), Dui(p + − 1)ε ^p

⁺

⁻¹

u ^p

⁺

dx.

From (3.4) we get Z

Γ

1

hA(x, u, Du), Dui

u ^p

⁻

dx ≥ a 1

Z

Γ

1

|Du| ^p(x)

u ^p(x) dx − a 2

Z

Γ

1

u ^p(x) u ^p

⁻

dx and also

Z

Γ

2

hA(x, u, Du), Dui

u ^p

⁺

dx ≥ a 1

Z

Γ

2

|Du| ^p(x)

u ^p(x) dx − a 2

Z

Γ

2

u ^p(x)

u ^p

⁺

dx.

Since u ^p(x) /u ^p

⁻

≤ 1 in Γ 1 and R

Γ

1

{|Du| ^p(x) /|u| ^p(x) }dx = R

Γ

1

|D log u| ^p(x) dx, we have a ₂ |Γ ₁ | ≥ a ₁

Z

Γ

1

|D log u| ^p(x) dx + I _Γ

₂

ε ^p

⁺

^−p

⁻

, (3.6) where

I Γ

2

:= p ₊ − 1 p ₋ − 1

"

a 1

Z

Γ

₂

|Du| ^p(x)

|u| ^p(x) dx − a 2

Z

Γ

₂

u ^p(x) u ^p

⁺

dx

# .

Define ϕ 1 (x) = log(u(x)/ε) if u(x) > ε and ϕ 1 (x) = 0 if u(x) ≤ ε. Let also δ > 0 be such that ε < δ < min{1, V } and put

Σ := {x ∈ Ω : δ ≤ u(x) ≤ 1} .

Obviously Σ ⊂ Γ 1 has positive measure. Moreover, since ϕ 1 = 0 in Ω \ Γ, it then follows from the Sobolev inequality (2.5) that

kϕ 1 k _p

∗

(·),Γ ≤ S kDϕ 1 k _p(·),Γ = S D log u

ε p(·),Γ

= S kD log uk _p(·),Γ . (3.7) If kD log uk _p(·),Γ ≤ 1, we find from (2.1) 2 , (3.6) and (3.7) that

kϕ 1 k _p

∗

(·),Σ ≤ kϕ 1 k _p

∗

(·),Γ ≤ C

Z

Γ

₁

∪Γ

2

|D log u| ^p(x) dx

 1/p

+

≤ C J Γ

₂

+ (a 2 |Γ 1 | − I Γ

₂

ε ^p

⁺

^−p

⁻

)/a 1  ^1/p

+

,

(3.8)

where J _Γ

₂

= R

Γ

2

|D log u| ^p(x) dx. We observe that ϕ ₁ ≥ log (δ/ε) in Σ. If ε > 0 is so small that log (δ/ε) > 1, then from (2.1) we obtain

kϕ 1 k _p

∗

(·),Σ ≥

log δ ε p

^∗

(·),Σ

≥ min ( 

% _p

∗

(·),Σ

 log δ

ε

 1/p

^∗₊

,



% _p

∗

(·),Σ

 log δ

ε

 1/p

^∗₋

)

≥ min n

|Σ| ^1/p

^∗+

, |Σ| ^1/p

^∗−

o  log δ

ε

 p

^∗₋

/p

^∗₊

. By (3.8) we finally get

|Σ| ^1/p

^∗+

 log δ

ε

 ^p

^∗−

/p

^∗₊

≤ C J Γ

₂

+ (a 2 |Γ 1 | − I Γ

₂

ε ^p

⁺

^−p

⁻

)/a 1  ^1/p

+

, which gives a contradiction as ε → 0.

If kD log uk _p(·),Γ

1

> 1 then from (2.1), (3.6) and (3.7) it follows analogously that kϕ 1 k _p

∗

(·),Σ ≤ C J Γ

₂

+ (a 2 |Γ 1 | − I Γ

₂

ε ^p

⁺

^−p

⁻

)/a 1  ^1/p

−

,

which gives again a contradiction when ε → 0. 2

4. The general elliptic case

In this section and from now on we consider (1.1) with B independent of ξ, and also the reverse inequality

divA(x, v, Dv) + B(x, v) ≤ 0 in Ω. (4.1)

We assume that B = B(x, z) is non increasing in the variable z, unless otherwise said, and on A that (i) A is continuous with respect to ξ and ∂ _z A is locally bounded in Ω × R × R ⁿ ,

(ii) There exists a non empty open subset P of R ⁿ (possibly P = R ⁿ ) such that A is continuously differentiable with respect to ξ in Ω × R × P ,

(iii) hA(x, z, ξ) − A(x, z, η), ξ − ηi ≥ 0 for all (x, z, ξ), (x, z, η) ∈ Ω × R × R ⁿ ,

hold.

P is called the regular set for A, while

Q = R ⁿ \ P

is the singular set. If Q = ∅ the problem is called regular, while otherwise it is singular. We say that the operator A is elliptic in a set K ⊂ Ω × R × P if the Jacobian matrix [∂ ^ξ A] is positive definite in K.

Note that also in the regular case it may happen that the operator A is elliptic only in a proper subset of P . A typical example is given by the standard p–Laplace operator, which is elliptic in R ⁿ \ {0} for any p > 1, but it is singular at 0 if p < 2 and regular if p ≥ 2. En passant we recall that the latter case is commonly quoted as a degenerate case.

Before giving the proof of the next result we establish Lemma 4.1 Let ξ, η be such that

ξ, η ∈ B _W , dist(ξ, Q) + dist(η, Q) ≥ 4d

for some positive constants W and d, with d ≤ W . Let Γ 1 be a compact subset of Ω, Γ 2 a compact subset of R, P ^d = {ζ ∈ R ⁿ : dist(ζ, Q) ≥ d}, K = Γ 1 × Γ 2 × {P ^d ∩ B W }. Assume that A is elliptic in K. Then for all x ∈ Γ 1 and u, v ∈ Γ 2

hA(x, u, ξ) − A(x, v, η), ξ − ηi ≥ a 1 |ξ − η| ² − a 2 |u − v| ² , (4.2) where

a 1 = d 4W inf

K {min eigenvalue of [∂ ξ A]} > 0, a 2 = c ² ₁ 4a ₁ ≥ 0, with c 1 = sup _K |∂ z A|.

Proof. For ξ 6= η we consider the line segment [ξ, η], that is

ζ(t) = (1 − t)ξ + tη, t ∈ [0, 1].

By hypothesis we may suppose without loss of generality that dist(η, Q) ≥ 2d, so that η ∈ P ^d . There are two cases:

Case I. [ξ, η] 6⊂ P ^d ; Case II. [ξ, η] ⊂ P ^d .

In Case I, let t 0 ∈ (0, 1) be such that ζ(t) ∈ P ^d for all t ∈ [t 0 , 1) while dist(ζ(t 0 ), Q) = d. We start evaluating a first auxiliary inequality, namely for all u ∈ Γ 2

I ≡ hA(x, u, ξ) − A(x, u, η), ξ − ηi

= hA(x, u, ξ) − A(x, u, ζ ₀ ), ξ − ηi + hA(x, u, ζ ₀ ) − A(x, u, η), ξ − ηi = I ₁ + I ₂ , where ζ ₀ = ζ(t 0 ). By (iii), since ξ − η = (ξ − ζ ₀ )|ξ − η|/|ξ − ζ ₀ |,

I ₁ = hA(x, u, ξ) − A(x, u, ζ ₀ ), ξ − ζ ₀ i |ξ − η|

|ξ − ζ ₀ | ≥ 0.

Moreover, since A is elliptic in K, noting that ξ − η = (ζ ₀ − η)|ξ − η|/|ζ ₀ − η|, we have I 2 = hA(x, u, ζ ₀ ) − A(x, u, η), ζ ₀ − ηi |ξ − η|

|ζ ₀ − η| ≥ a |ζ ₀ − η| ² |ξ − η|

|ζ ₀ − η| = a |ξ − η| ² |ζ ₀ − η|

|ξ − η| , where

a = inf

K {min eigenvalue of [∂ ξ A]} > 0,

and K, defined as in the statement, is a compact subset of Ω × R × P . Finally, |ζ 0 − η| ≥ d and

|ζ ₀ − η|/|ξ − η| ≥ d/2W , so that

I ≥ I ₂ ≥ ad

2W |ξ − η| ² ,

proving (4.2) for Case I. Case II is obvious by ellipticity, and I ≥ a|ξ − η| ² ≥ (ad/W )|ξ − η| ² . In conclusion we have shown that

I ≥ 2a 1 |ξ − η| ² .

Now take any u and v ∈ Γ 2 . The left hand side of (4.2) is equal to I + J , where

J = hA(x, u, η) − A(x, v, η), ξ − ηi = h∂ _z A(x, t, η), ξ − ηi(u − v) ≥ −c ₁ |ξ − η| · |u − v|, and t lies in the open interval between u and v. By Cauchy’s inequality this yields

J ≥ −a 1 |ξ − η| ² − c ² ₁ (u − v) ² /4a 1

and (4.2) follows at once. 2

We are now ready to give

Theorem 4.2 (Comparison Principle). Let P be an open set of R ⁿ and assume that A = A(x, z, ξ) is elliptic in Ω × R ⁺ × P . Let u, v ∈ W _loc ^1,∞ (Ω) be solutions of (1.2), with

ess inf

x∈Ω {dist(Du, Q) + dist(Dv, Q)} > 0, (4.3)

where Q = R ⁿ \ P . If u ≤ v on ∂Ω, then u ≤ v in Ω.

Proof. We argue by contradiction, assuming that u > v somewhere. Then, since u and v are continuous, there exists ε > 0 such that ω = {x ∈ Ω : u > v + ε} is non empty and open. Let C be a component of ω and set w = u − v − ε. Hence w > 0 in C ⊂⊂ Ω, w = 0 on ∂C and w is a solution in C of

div ˜ A(x, w, Dw) + ˜ B(x, w) ≥ 0, (4.4)

where

A(x, w, Dw) = A(x, u, Du) − A(x, v, Dv) ˜ and ˜ B(x, w) = B(x, u) − B(x, v) ≤ 0 in C by the monotonicity of B.

Since u, v ∈ W _loc ^1,∞ (Ω) there exists W > 0 sufficiently large such that Du( C ), Dv(C ) ⊂ B W . Moreover Γ 1 = C and Γ 2 = u(Γ 1 ) ∪ v(Γ 1 ) are clearly compact, hence Lemma 4.1 applies, so that (4.2) becomes

h ˜ A(x, w, Dw), Dwi ≥ a ₁ |Dw| ² − a ₂ w ²

along the solution w of (4.4) in C . Finally, Theorem 3.4 implies that w ≤ 0 in C , which is the required

contradiction. 2

Of course 0 can be deleted in (1.2), provided that the inequality is understood in the weak sense and the definition of regularity needed in Theorem 3.4 is appropriately changed.

The first application concerns the generalized mean curvature operator.

Corollary 4.3 (Comparison Principle). Let p ₊ ≤ 2 and let u, v ∈ W _loc ^1,∞ (Ω) be solutions of div

 

1 + |Dv| ²  ^[p(x)−2]/2 Dv



+ B(x, v) ≤ 0 ≤ div

 

1 + |Du| ²  ^[p(x)−2]/2 Du



+ B(x, u) in Ω. (4.5) If u ≤ v on ∂Ω, then u ≤ v in Ω.

Proof. Clearly A(x, ξ) = 

1 + |ξ| ²  [p(x)−2]/2

ξ satisfies conditions (i) − (iii). In particular, (iii) holds since A(x, ξ) is the gradient of the convex function ξ 7→ [(1 + |ξ| ² ) ^p(x)/2 − 1]/p(x).

Now we show that ξ 7→ A(x, ξ) = 

1 + |ξ| ²  [p(x)−2]/2

ξ is elliptic in Ω × R ⁿ . Indeed,

∂ ξ A(x, ξ) = (1 + |ξ| ² ) ^[p(x)−2]/2

"

II n + (p(x) − 2) ξ ⊗ ξ 1 + |ξ| ²

# . and so the least eigenvalue of the Jacobian matrix [∂ ξ A(x, ξ)] is

(1 + |ξ| ² ) ^[p(x)−2]/2

"

1 + (p(x) − 2) |ξ| ² 1 + |ξ| ²

#

≥ (1 + |ξ| ² ) ^[p

⁻

^−4]/2 h

1 + (p ₋ − 1) |ξ| ² i

> 0.

Thus A(x, ξ) is elliptic in Ω × R ⁿ , and then Theorem 4.2 applies, since Q = ∅ and so condition (4.3) is

obviously verified. 2

A useful result for the p(·)–Laplace operator is given by

Corollary 4.4 Let u, v ∈ W _loc ^1,∞ (Ω) be solutions of

∆ _p(·) v + B(x, v) ≤ 0 ≤ ∆ _p(·) u + B(x, u) in Ω, (4.6) satisfying

ess inf

x∈Ω {|Du| + |Dv|} > 0. (4.7)

If u ≤ v on ∂Ω, then u ≤ v in Ω.

Proof. Clearly A(x, ξ) = |ξ| ^p(x)−2 ξ satisfies conditions (i) − (iii), since the function ξ 7→ |ξ| ^p(x) /p(x) is convex, so that its gradient is monotone. Moreover,

∂ _ξ A(x, ξ) = |ξ| ^p(x)−2

"

II _n + (p(x) − 2) ξ ⊗ ξ

|ξ| ²

#

, ξ 6= 0,

and so the least eigenvalue of the Jacobian matrix [∂ ξ A(x, ξ)] is (p(x) − 1) |ξ| ^p(x)−2 , if ξ 6= 0. Hence A is elliptic in Ω × P , where P = R ⁿ \ {0}, and so the assertion follows at once from Theorem 4.2, being (4.3)

expressed in this case by (4.7). 2

5. Non–homogeneous elliptic inequalities

In this section we assume that A = A(x, ξ) is independent of z and consider the pair of differential inequalities

divA(x, Dv) + B(x, v) ≤ 0 ≤ divA(x, Du) + B(x, u). (5.1) Put L[u] = divA(x, Du) + B(x, u), where A : Ω × R ⁿ → R ⁿ , B : Ω × R → R, and A satisfies hypotheses (i)–(iii) of Section 4.

Theorem 4.2 has one of its main consequences in the following maximum principle for non homogenous elliptic inequalities. It is interesting that for the next two results and their corollary the function B(x, z) need not be monotone in the variable z.

Theorem 5.1 (Maximum Principle) Assume that A = A(x, ξ) is elliptic in Ω × P . For any v ∈ C ¹ (Ω) define L [z, v] pointwise by

L [z, v](x) = divA(x, Dv(x)) + B(x, z) (5.2)

for all x ∈ Ω and z ∈ R ⁺ .

Let v = v(x) ∈ C ¹ (Ω) be a non negative comparison function for the operator L, in the sense that v(x) ≥ 0 and Dv(x) ∈ P for all x ∈ Ω, and L [z, v] ≤ 0 in Ω for all z > 0. If u ∈ W _loc ^1,∞ (Ω) is a solution of L[u] ≥ 0 in Ω and u ≤ v on ∂Ω, then u ≤ v in Ω.

Proof. Define

L [v] ≡ divA(x, Dv) + ˜ ˜ B(x) = L [z, v] z=u(x) , (5.3) where ˜ B(x) = B(x, u(x)). By hypothesis L [v] ≤ 0 whenever u(x) > 0, and clearly ˜ ˜ L [u] ≥ 0 in Ω. Of course both A and ˜ B are independent of z, and (i)–(iii) hold.

Set Ω ⁰ = {x ∈ Ω : u(x) > 0} and let C be a component of Ω ⁰ . It is easy to see that u ≤ v on ∂ C . Moreover u and v satisfy L [v] ≤ 0 ≤ ˜ ˜ L [u], with dist(Dv, Q) > 0 in C , being Dv(x) ∈ P for all x in C .

Theorem 4.2 gives u ≤ v in C , completing the proof. 2

Of course the previous result is quite abstract, since it is not easy to know a priori the existence of a comparison function. However, if further conditions on the operators are in force such a comparison function can be constructed and the theorem can be applied. For the next application the following property is useful.

( E ) For all a, b ∈ R ⁺ , with a < b, there exists α = α(a, b) > 0 such that Trace [∂ x A(x, ξ)] ≤ α |ξ| E(x, ξ) in Ω × B b \ B a , where

E(x, ξ) = hξ, ∂ ξ A(x, ξ) ξi

|ξ| ² , ξ 6= 0.

Theorem 5.2 (Maximum Principle). Suppose that A = A(x, ξ) is elliptic in Ω × P , Q = R ⁿ \ P ⊂ B %

for some % ≥ 0, and B(x, z) ≤ γ for some γ > 0. Assume also that ( E ) holds and that

|ξ| E(x, ξ) ≥ Ψ(|ξ|) in Ω × P , (5.4)

for some function Ψ = Ψ(s) which is strictly increasing in (%, ∞), % ≥ 0.

Let R > 0 be such that

s→∞ lim Ψ(s) > R, and set

C = Ψ ⁻¹ (R), K = R max{%, C} and m = 1 + 1/ R. (5.5) Take a = mK, b = mKe ^mR and denote by α = α(mK, mKe ^mR ) the corresponding number given in ( E ).

Finally assume

(m − α)R ≥ γ. (5.6)

Fix a subdomain Ω R of Ω such that

Ω R ⊂ {x ∈ Ω : 0 < x i < R}

for some i ∈ {1, . . . , n}. Then every solution u ∈ W _loc ^1,∞ (Ω R ) of

divA(x, Du) + B(x, u) ≥ 0 in Ω _R , u ≤ 0 on ∂Ω _R , satisfies

u(x) ≤ R max {%, C} (e ^mR − 1) in Ω R . (5.7)

Proof. It is now enough to construct a suitable comparison function v = v(x) such that v(x) ≥ 0 in Ω _R and L [z, v] ≤ 0 in Ω R for all z > 0, so that the assertion of Theorem 5.1 holds. For this purpose define

v(x) := K(e ^mR − e ^mx

ⁱ

), x ∈ Ω R . Then

∂ _x

_i

v(x) = −mKe ^mx

ⁱ

.

Therefore |Dv| ≥ mK and Dv(x) ∈ P for any x ∈ Ω _R , since m > 1/R. Furthermore

∂ ² _x

_i

v(x) = −m ² Ke ^mx

ⁱ

= −m |Dv| . Of course

L [z, v] = Trace [∂ x A(x, Dv)] − m |Dv| ∂ _ξ

_i

A _i (x, Dv) + B(x, z).

But ∂ ξ

_i

A i (x, Dv) = E(x, Dv) so that

L [z, v] = Trace [∂ x A(x, Dv)] − m |Dv| E(x, Dv) + B(x, z).

Since mK ≤ |Dv| ≤ mKe ^mR , by ( E ), applied with a = mK and b = mKe ^mR , it follows that L [z, v] ≤ (α − m) |Dv| E(x, Dv) + γ.

Hence, if

(m − α) |Dv| E(x, Dv) ≥ γ, (5.8)

then L [z, v] ≤ 0 in Ω R . By (5.4) and the fact that Ψ is strictly increasing, we have

|Dv| E(x, Dv) ≥ Ψ(|Dv|) ≥ Ψ(mK) > Ψ(C) = γR,

since mK > max {%, C} ≥ C. Therefore (5.8) holds by (5.6) for C and K given in (5.5). Consequently L [z, v] ≤ 0 in Ω R , and by Theorem 5.1

u(x) ≤ v(x) ≤ K(e ^mR − 1) in Ω _R ,

that is (5.7). 2

We now give a useful application of the previous result to the p(·)–Laplace operator. Assume that p is differentiable in Ω. In this case direct calculations show that

Trace [∂ x A(x, ξ)] = |ξ| ^p(x)−2 log |ξ| hDp(x), ξi , E(x, ξ) = [p(x) − 1] · |ξ| ^p(x)−2 , ξ 6= 0.

By the Cauchy–Schwartz inequality we get

Trace [∂ _x A(x, ξ)] = hDp(x), ξi |ξ| ^p(x)−2 log |ξ| ≤ |Dp(x)| · |ξ| ^p(x)−1 |log |ξ||

for all x ∈ Ω. Hence, if p ∈ C + (Ω) ∩ C ¹ (Ω) with sup _x∈Ω |Dp(x)| < ∞, and 0 < a ≤ |ξ| ≤ b, we can put α = max {|log a| , |log b|}

p ₋ − 1 sup

x∈Ω

|Dp(x)| , (5.9)

so that

Trace [∂ x A(x, ξ)] ≤ α |ξ| E(x, ξ) (5.10)

for all x ∈ Ω and a ≤ |ξ| ≤ b. Put Λ(x, s) = (p(x) − 1)s ^p(x)−1 with s = |ξ| and define Ψ(s) = Ψ _p(·) (s) = inf

x∈Ω Λ(x, s) = (p ₋ − 1)

( s ^p

⁺

⁻¹ , 0 < s < 1, s ^p

⁻

⁻¹ , s ≥ 1.

Then Ψ : R ⁺ 0 → R ⁺ 0 is a strictly increasing function, such that lim

s→∞ Ψ(s) = ∞ and

|ξ| E(x, ξ) = Λ(x, |ξ|) ≥ Ψ(|ξ|) in Ω × P , (5.11) where, we recall, here P = R ⁿ \ {0}. Hence (5.4) in Theorem 5.2 is satisfied. Therefore we can state the following

Corollary 5.3 (Maximum Principle). Let p ∈ C + (Ω) ∩ C ¹ (Ω) be such that sup _x∈Ω |Dp(x)| < ∞.

Suppose that B(x, z) ≤ γ for some γ > 0. For all a, b ∈ R ⁺ , with a < b, take α > 0 as in (5.9). Consider R > 0 and set

C = Ψ ⁻¹ _p(·) (R) and K = RC.

Assume also that (5.6) holds. Fix a subdomain Ω R of Ω such that Ω R ⊂ {x ∈ Ω : 0 < x i < R}

for some i ∈ {1, . . . , n}. Then every solution u ∈ W _loc ^1,∞ (Ω R ) of

∆ _p(x) u + B(x, u) ≥ 0 in Ω R , u ≤ 0 on ∂Ω R , satisfies

u(x) ≤ RC(e ^mR − 1) in Ω _R .

Clearly ( E ) and (5.4) are verified by (5.10) and (5.11). The conclusion comes directly from the application of Theorem 5.2, with % = 0.

Let us note that an explicit value for C is C = max

(  R p − − 1

 ^1/(p

−

−1)

,

 R

p − − 1

 ^1/(p

+

−1) ) . Moreover, condition (5.6) is easily obtained if p is close to a constant in norm C ¹ .

We conclude with a list of results for which we need the following theorem, whose proof is an obvious adaptation to W ₀ ^1,p(·) (Ω) of that of [18, Theorem 3.4.1] for W ₀ ^1,p (Ω), with the standard Poincar´ e inequality replaced now by (2.4). From now on the function B(x, z) will be considered again non increasing in the variable z.

Theorem 5.4 Let u and v be p(·)–regular solutions of (5.1) of class W _loc ^1,p(·) (Ω). Suppose that A is mono- tone in ξ, i.e.

hA(x, ξ) − A(x, η), ξ − ηi > 0, when ξ 6= η. (5.12)

If u ≤ v on ∂Ω, then u ≤ v a.e. in Ω.

As a first application let us go back to the generalized mean curvature operator.

Corollary 5.5 (Comparison Principle). Let u, v ∈ W _loc ^1,p(·) (Ω) be p(·)–regular solutions of (4.5). If u ≤ v on ∂Ω, then u ≤ v a.e. in Ω.

Proof. It is enough to recall that the map ξ 7→ (1 + |ξ| ² ) ^[p(x)−2]/2 ξ is monotone, as seen in the proof of

Corollary 4.3, and apply Theorem 5.4. 2

A useful result for the p(·)–Laplacian operator for p(·)–regular solutions is given by

Corollary 5.6 If u, v ∈ W _loc ^1,p(·) (Ω) are p(·)–regular solutions of (4.6), with u ≤ v on ∂Ω, then u ≤ v a.e.

in Ω.

Proof. It is enough to recall that the map ξ 7→ |ξ| ^p(x)−2 ξ is monotone, as seen in the proof of Corollary 4.4, and apply Theorem 5.4.

Note that in Corollary 5.6 solutions are p(·)–regular, while in Theorem 4.2 and in the corresponding Corollary 4.4 they were assumed to be locally Lipschitz continuous. This allows us to delete the main condition (4.7), peculiar of the non regular case.

6. Uniqueness results

We conclude this paper with obvious consequences of all the comparison results stated above, which imply at once uniqueness theorems for solutions of the related boundary value problem

( divA(x, u, Du) + B(x, u) = 0 in Ω,

u = u ₀ on ∂Ω, (6.1)

where, from now on, u 0 ∈ C(∂Ω). In the following we assume that A satisfies the assumptions (i)–(iii) of Section 4 and that B = B(x, z) is non increasing in the variable z.

Theorem 6.1 Suppose that A = A(x, z, ξ) is elliptic in Ω × R ⁺ × P with P ⊂ R ⁿ open. If u, v ∈ W _loc ^1,∞ (Ω) are solutions of (6.1) such that (4.3) holds. Then u = v a.e. in Ω.

The assertion follows from Theorem 4.2. Hence, the following uniqueness results hold true.

Corollary 6.2 Let p ₊ ≤ 2 and let u, v ∈ W _loc ^1,∞ (Ω) be solutions of





 div

 

1 + |Du| ²  ^[p(x)−2]/2 Du



+ B(x, u) = 0 in Ω,

u = u 0 on ∂Ω.

(6.2)

Then u = v in Ω.

It is just a direct consequence of Corollary 4.3.

Corollary 6.3 Let u, v ∈ W _loc ^1,∞ (Ω) be solutions of

( ∆ _p(·) u + B(x, u) = 0 in Ω,

u = u ₀ on ∂Ω, (6.3)

satisfying (4.7). Then u = v in Ω.

This follows from Corollary 4.4.

Concerning uniqueness of p(·)–regular solutions we have these results.

Theorem 6.4 Suppose that A = A(x, ξ) satisfies (5.12) and let u, v be p(·)–regular solutions in W _loc ^1,p(·) (Ω) of

( divA(x, Du) + B(x, u) = 0 in Ω,

u = u 0 on ∂Ω.

Then u = v a.e. in Ω.

It follows from Theorem 5.4.

In the same way, as Theorem 5.4 had obvious consequences in Corollaries 5.5 and 5.6, also Theorem 6.4 implies applications to the prototype operators, which, however, can be derived directly from those corol- laries.

Corollary 6.5 Suppose that p ₊ ≤ 2. Let u, v ∈ W _loc ^1,p(·) (Ω) be p(·)–regular solutions of (6.2). Then u = v a.e. in Ω.

Corollary 6.6 Let u, v ∈ W _loc ^1,p(·) (Ω) be p(·)–regular solutions of (6.3). Then u = v a.e. in Ω.

Since the sum of elliptic (resp. monotone) operators is still elliptic (resp. monotone), Theorems 6.1 and 6.4 admit interesting consequences when the operator A is given by combinations of the prototypes above. For example, when p + ≤ 2,

A(x, ξ) = |ξ| ^p(x)−2 ξ + (1 + |ξ| ² ) ^[p(x)−2]/2 ξ, or, if q ∈ C + (Ω) with q(x) < p(x) for all x ∈ Ω,

A(x, ξ) = |ξ| ^p(x)−2 ξ + |ξ| ^q(x)−2 .

The latter operator is interesting for the study of solitons (see [5]). For completeness we give here an example of such applications.

Corollary 6.7 Let p + ≤ 2 and let u, v ∈ W _loc ^1,p(·) (Ω) be p(·)–regular solutions of ( div 

1 + |Du| ²
[p(x)−2]/2

Du 

+ ∆ _p(·) u + B(x, u) = 0 in Ω,

u = u ₀ on ∂Ω.

Then u = v a.e. in Ω.

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