A picone Identity for variable exponent operators and applications

In this work, we establish a new Picone identity for anisotropic quasilinear operators, such as the $p(x)$-Laplacian defined as $\mbox{div}(|\nabla u|^{p(x)-2} \nabla u).$ Our extension provides a new version of the Diaz-Saa inequality and new uniqueness results to some quasilinear elliptic equations with variable exponents. This new Picone identity can be also used to prove some accretivity property to a class of fast diffusion equations involving variable exponents. Using this, we prove for this class of parabolic equations a new weak comparison principle.


Introduction and main results
The main aim of this paper is to prove a new version of the Picone identity involving quasilinear elliptic operators with variable exponent. The Picone identity is already known for homogeneous quasilinear elliptic as p-Laplacian with 1 < p < ∞. In [18], M. Picone considers the homogeneous second order linear differential system (a 1 (x)u ′ ) ′ + a 2 (x)u = 0 and proved for differentiable functions u, v = 0 the pointwise relation: and in [19], extended (1.1) to the Laplace operator, i.e. for differentiable functions u ≥ 0, v > 0 one has In [1], Allegretto and Huang extended (1.2) to the p-Laplacian operator with 1 < p < ∞. Precisely, for differentiable functions v > 0 and u ≥ 0 we have Picone identity plays an important role for proving qualitative properties of differential operators. In this regard, various attempts have been made to generalize Picone identity for different types of differential equations. At the same time, the study of differential equations and variational problems with variable exponents are getting more and more attention. Indeed, the mathematical problems related to nonstandard p(x)-growth conditions are connected to many different areas as the nonlinear elasticity theory and non-Newtonian fluids models (see [15,22]). In particular the importance of investigating these kinds of problems lies in modelling various anisotropic features that occur in electrorheological fluids models, image restoration [6], filtration process in complex media, stratigraphy problems [12] and heterogeneous biological interactions [4]. The mathematical framework to deal with these problems are the generalized Orlicz Space L p(x) (Ω) and the generalized Orlicz-Sobolev Space W 1,p(x) (Ω). We refer to [8,10,11,14,20,21] for the existence and regularity of minimizers in variational problems. In [3,7], several applications of Picone-type identity for p(·) = constant case have been obtained. This original identity is not further applicable for differential equations with p(x)-growth conditions. So, it is relevant to establish a new version of the Picone identity to include a large class of nonstandard p(x)-growth problems. In [14,16,22] convexity arguments to homogeneous functionals have been used to deal with quasilinear elliptic and parabolic problems with variable exponents. In the present paper, taking advantage of our new Picone pointwise identity, we give further applications in the context of elliptic and parabolic problems.
Before giving the statement of our main results, we first introduce notations and function spaces. Let Ω ⊂ R N , N ≥ 1. We recall some definitions of variable exponent Lebesgue and Sobolev spaces. Let P(Ω) be the set of all measurable function p : Ω → [1, ∞[ in N -dimensional Lebesgue measure. Define L p(x) (Ω) = {u : Ω → R | u is measurable and ρ p (u) < ∞} endowed with the norm The corresponding Sobolev space is defined as follows: (Ω) = W 1,1 0 (Ω) ∩ W 1,p(x) (Ω). In the sequel, we assume that Ω satisfies: (Ω) For N = 1, Ω is a bounded open interval and for N ≥ 2, Ω is a bounded domain whose the boundary ∂Ω is a compact manifold of class C 1,γ for some γ ∈ (0, 1) and satisfies the interior sphere condition at every point of ∂Ω.
(iii) L pc(x) (Ω) is the dual space of L p(x) (Ω) where we denote by p c the conjugate exponent of p defined as Moreover, we have also the generalized Hölder inequality: for p measurable function in Ω, there exists a constant C = C(p + , p − ) ≥ 1 such that for any f ∈ L p(x) (Ω) and g ∈ L pc(x) (Ω) In Section 2, we prove the Picone identity for a general class of nonlinear operator. More precisely, we consider a continuous operator A : Ω × R N → R such that (x, ξ) → A(x, ξ) is differentiable with respect to variable ξ and satisfies: (A1) ξ → A(x, ξ) is positively p(x)-homogeneous i.e. A(x, tξ) = t p(x) A(x, ξ), ∀ t ∈ R + , ξ ∈ R N and a.e. x ∈ Ω.
By using the convexity and the p(x)-homogeneity of the operator A, we prove the following extension of the Picone identity: (Ω)} for some r ≥ 1. Then where ., . is the inner scalar product and the above inequality is strict if r > 1 or v v0 ≡ Const > 0. From the above Picone identity, we can show an extension of the famous Diaz-Saa inequality to the class of operators with variable exponent. This inequality is strongly linked to the strict convexity of some associated homogeneous energy type functional. Theorem 1.2 (Diaz-Saa inequality). Let A : Ω × R N → R is a continuous and differentiable function satisfying (A1) and (A2) and define a(x, ξ) = (Ω), positive in Ω such that w 1 w 2 , w 2 w 1 ∈ L ∞ (Ω). Moreover, if the equality occurs in (1.6), then w 1 /w 2 is constant in Ω. If p(x) ≡ r in Ω then even w 1 = w 2 holds in Ω.
In sections 3, 4 and 5, we derive some applications of the new Picone identity. Precisely, we investigate the solvability of some boundary problems involving quasilinear elliptic operators with variable exponent.
In section 3, we consider the following nonlinear problem: The extended Picone identity can be reformulated as in Lemma 3.1 below. Together with the strong maximum principle and elliptic regularity, this identity can be used to prove the uniqueness of weak solutions to elliptic equations as (1.7). In particular, we establish the following result: Then, there exists a weak solution u to (1.7), i.e. u belongs to W 1,p(x) 0 (Ω) ∩ L s(x) (Ω) and satisfies for any φ ∈ W Furthermore u ∈ C 1,α (Ω) for some α ∈ (0, 1) and 0 ≤ u s−−q+ ≤ max{ h l L ∞ , 1} a.e. in Ω.

Assume in addition that
≤ c 2 } and is the unique weak solution to (1.7).
Regarding the current literature, Theorem 1.3 does not require any subcritical growth condition for g to establish existence and uniqueness of the weak solution to (1.7).
In section 4, we study a nonlinear fast diffusion equation (F.D.E. for short) driven by p(x)-Laplacian. From the physical Fick's law, the diffusion coefficient of our problem is then proportional to |∇u(x, t)| p(x)−2 . It naturally leads to investigate the following F.D.E. type problem: × Ω and Γ = (0, T ) × ∂Ω for some T > 0. We suppose that h ∈ L ∞ (Q T ) and nonnegative. The assumptions on f are given by i.e. f has a strict subhomogeneous growth.
We set R the operator defined by Rv Note that D(R) contains for instance solutions to (4.18). One can also easily check that solutions to (4.19) belong to D(R) L 2 (Ω) . In the sequel, we denote the associated positive cone of a given real vector space X. In order to establish existence and properties of weak solutions to (1.8), we investigate the following related parabolic problem: The notion of weak solution for (1.9) is given as follows: (1.10) Concerning (1.9), we prove the following results: (Ω). In addition, there exists h 0 ∈ L ∞ (Ω), h 0 ≡ 0 and h(t, x) ≥ h 0 (x) ≥ 0 for a.e x ∈ Ω, for a.e. t ≥ 0. Assume in addition q ≤ min{ N 2 + 1, p − } and f satisfies (f 1 )-(f 3 ). Then there exists a weak solution to (1.9).
Based on the accretivity of R with domain D(R), we show the following result providing a contraction property for weak solutions to (1.9) under suitable conditions on initial data: Theorem 1.5. Let v 1 and v 2 are weak solutions of (1.9) with initial data (Ω) and such that u q 0 , v q 0 ∈ D(R) L 2 (Ω) and h, g ∈ L ∞ (Q T ), such that h ≥ h 0 , g ≥ g 0 with h 0 , g 0 as in Theorem 1.4. Then, for any 0 ≤ t ≤ T , Furthermore, using a similar approach as in [4], we consider for ǫ > 0 the we can prove (as in Proposition 2.6 in [4]) that Arguing as in Theorem 1.5 with the operator R ǫ instead of R and passing to the limit as ǫ → 0 + , we get: (Ω). Then Theorem 1.5 holds.
From Theorem 1.5, we derive the following comparison principle from which uniqueness of the weak solution to problem (1.9) follows: Corollary 1.2. Let u and v are the weak solutions of (1.9) with initial data u 0 , v 0 satisfying conditions in Theorem 1.5 or Corollary 1.
From the above remark, under assumptions given in Theorem 1.4, we obtain the existence of weak solutions to (1.8) satisfying the monotonicity properties in Theorem 1.5 and Corollaries 1.1, 1.2. We highlight that in our knowledge there is no result available in the current literature about F.D.E. with variable exponent. In this regard our results are completely new.
In the previous applications, the condition (A1) plays a crucial role to get suitable convexity property of energy functionals. In section 5, we study a quasilinear elliptic problem where this condition is not satisfied. Precisely, given ǫ > 0, we study the following nonhomogeneous quasilinear elliptic problem: where g satisfies (f 1 ) and (g) for some m ∈ [1, p − ]: (g) For any x ∈ Ω, s → g(x, s) s m−1 is decreasing in R + \{0} and a.e. in Ω.
To get the uniqueness result contained in Theorem 1.6, we exploit the hidden convexity property of the associated energy functional in the interior of positive cone of C 1 (Ω).

Picone identity and Diaz-Saa inequality 2.1 Picone identity
First we recall the notion of strict ray-convexity.
Definition 2.1. Let X be a real vector space. Let • V be a non empty cone in where the inequality is always strict unless v 1 = Cv 2 for some C > 0.
Then we have the following result: Proposition 2.1. Let A satisfying (A1) and (A2) and let r ≥ 1. Then, for any is positively r-homogeneous and ray-strictly convex. For r > 1, ξ → N r (x, ξ) is even strictly convex.
Proof. We begin by the case r = 1. For any t ∈ R + , we have N 1 (x, tξ) = tN 1 (x, ξ). Furthermore, for any x ∈ Ω, ξ 1 , ξ 2 ∈ R N and t ∈ [0, 1]. Therefore and this inequality is always strict unless ξ 1 = λξ 2 , for some λ > 0. Now we prove that N 1 is subadditive. Without loss of generality, we can assume that ξ 1 = 0 and ξ 2 = 0. Then we have N 1 (x, ξ 1 ) > 0 and N 1 (x, ξ 2 ) > 0. Therefore, from (2.1) and 1-homogeneity of N 1 (x, ξ) we obtain for any t ∈ (0, 1): We now fix t such that Then we get and by 1-homogeneity of N 1 , we obtain This proves that ξ → N 1 (x, ξ) is ray-strictly convex. Now consider the case r > 1. Since for any x ∈ Ω, ξ → N is ray-strictly convex and thanks to the strict convexity of t → t r on R + , we deduce that ξ → N r (x, ξ) = N r 1 (x, ξ) is strictly convex when r > 1. From Proposition 2.1 and from the r-homogeneity of N r , we easily deduce the following convexity property of the energy functional: is r-positively homogeneous and strictly convex if r > 1 and for r = 1 this function is ray-strictly convex.
By homogeneity, and equality holds if and only if v 1 = λv 2 for some λ > 0. Using the convexity of From Proposition 2.1, we deduce the proof of Picone identity. Proof of Theorem 1.1: Firstly, we deal with the case r > 1. Then from Proposition 2.1, for any Let v, v 0 > 0 and replacing ξ, ξ 0 by ξ/v and ξ 0 /v 0 respectively in the above expression, we get Taking ξ = ∇v and ξ 0 = ∇v 0 and using (r − 1)-homogeneity ofã(x, .), We haveã and by replacing in (2.2) we obtain Now we deal with the case r = 1. Let ξ, ξ 0 ∈ R N \{0} such that for any λ > 0, ξ = λξ 0 . Then, from Proposition 2.1, we have that Taking ξ = ∇v and ξ 0 = ∇v 0 , we deduce for any x ∈ Ω and the inequality is strict unless v = λv 0 for some λ > 0. The Picone identity also holds for anisotropic operators of the following type: Precisely we have: for all v, v 0 ∈V r + ∩ L ∞ (Ω) and i = 1, 2, . . . , N . Then by summing the expression over i = 1, 2, . . . , N , we obtain

An extension of the Diaz-Saa inequality
We prove the first application of Picone identity. Proof of Theorem 1.2: The Picone identity implies Using the Young inequality for r ∈ [1, p − ], we get Noting that for any ξ ∈ R N , A(x, ξ) = a(x, ξ).ξ, we deduce Commuting w 1 and w 2 , we have

Summing (2.3) and (2.4) and integrating over Ω yield
The rest of the proof is the consequence of Proposition 2.2.
Diaz-Saa inequality also holds for anisotropic operators. Here we require that ξ → B i (x, ξ) is p i (x)-homogeneous and strictly convex and b i (x, ξ) = 1 Corollary 2.2. Under the assumptions of Corollary 2.1 and in addition that Proof. We apply Theorem 1.2. For A = B i : Ω × R → R and by replacing ∇ by ∂ xi .

Application of Picone identity to quasilinear elliptic equations
The aim of this section is to establish Theorem 1.3.

Preliminary results
The first lemma is the Picone identity in the context of the p(x)-Laplacian operator.
Following the proof of Theorem 1.1 in [24], we first prove the following comparison principle: (Ω) and in Ω.
Using lemma 3.2, we show the following strong maximum principle: (Ω) be nonnegative and a nontrivial solution to Assume in addition either Then u is positive in Ω.
Proof. We follow the idea of the proof of Theorem 1.1 in [24]. For the reader's convenience we have included the detailed proof. We rewrite our equation (3.1) under condition (c1) as follows: We choose λ small enough such that for any u( Suppose that there exists x 1 such that u(x 1 ) = 0 then using the fact that u is nontrivial, we can find x 2 ∈ Ω and a ball B( We have a C e −l 1 C We choose C < 1 and using ∇u(x 1 ) = 0, a C < 1 small enough such that for any Without loss of generality we can take x 2 = 0 and we set Since j(t) < a < λ, we deduce On ∂P , w(C) = j(C) = a ≤ u(x) and w(2C) = j(0) = 0 ≤ u(x). Then by Lemma 3.2, we obtain w ≤ u on P . Finally,  and Ω satisfies the interior ball condition at x 1 , then ∂u ∂ n (x 1 ) < 0 where n is the outward unit normal vector at x 1 .

Proof of Theorem 1.3
Proof of Theorem 1.3: We perform the proof along five steps. First we introduce notations. Define F, G : Ω × R → R + as follows: We also extend the domain of f and g to all Ω × R by setting Step 1 : Existence of a global minimizer Since We argue similarly when u L s(x) → ∞ and we deduce E is coercive. The continuity of E on W 1,p(x) 0 (Ω) ∩ L s(x) (Ω) is given by Theorem 3.2.8 and 3.2.9 of [8]. Hence we get the existence of at least one global minimizer, say u 0 , to (3.4).
Step 3: u 0 satisfies the equation in (1.7) Since u 0 is a global minimizer and E is Step 4: Regularity and positivity of weak solutions First we prove that all nonnegative weak solutions of (1.7) belongs to L ∞ (Ω) which yields C 1,α (Ω) regularity.
Step 5: Uniqueness of the positive solution of (1.7) Let u, v be two positive solutions of (1.7). Thus for any φ,φ ∈ W 1,p(x) 0 By the previous steps, u and v belong to C 1 (Ω) and Lemma 3.4 implies u, v ∈ C 0 d (Ω) + . Hence taking the testing functions as φ = (Ω) (with the following notation t − def = max{0, −t}) and from Lemma 3.1 we obtain Since q + ≤ p − ≤ s − , the both terms in right-hand side are nonpositive. This implies v(x) ≥ u(x) a.e in Ω. Finally reversing the role of u and v, we get u = v.
Remark 3.2. Theorem 1.3 still holds when the condition l h ∈ L ∞ (Ω) is replaced by p + < s − and using strong maximum principle in [24].

Application to Fast diffusion equations
In this section, we establish Theorems 1.4 and 1.5. To this aim, we use a time semi-discretization method associated to (1.9). With the help of accurate energy estimates about the related quasilinear elliptic equation and passing to the limit as the discretization parameter goes to 0, we prove the existence and the properties of weak solutions to (1.8). In the subsection below, we study the associated elliptic problem.
Step 1: Existence of a weak solution Consider the energy functional J defined on W equipped with .
We also extend the domain of f to all of Ω × R by setting f (x, t) = ∂F ∂t (x, t) = 0 for (x, t) ∈ Ω×(−∞, 0). From (4.2), Hölder inequality (1.5) and since W Then by choosing ǫ small enough we conclude the coercivity of J on W and J is also continuous on W therefore we deduce the existence of a global minimizer v 0 to J . Furthermore we note which implies v 0 ≥ 0. Now we claim that v 0 ≡ 0 in Ω. Since J (0) = 0, it is sufficient to prove the existence ofṽ ∈ W such that J (ṽ) < 0. For that takeṽ = tφ where φ ∈ C 1 c (Ω) is nonnegative function such that φ ≡ 0 and t > 0 small enough.
Step 2: Contraction property (4.4) Let v 1 and v 2 two positive weak solutions of (4.1) such that v 1 , v 2 ∈ C 0 d (Ω) + . For any φ, Ψ ∈ W: are welldefined and belong to W. Subtracting the two above expressions and using (f 3 ) together with Lemma 3.1 we obtain Finally, applying the Hölder inequality we get (4.4).
From Theorem 4.1, we deduce the accretivity of R: and h 0 ∈ L ∞ (Ω) + . Consider the following problem Then there exists a unique distributional solution u ∈ D(R) ∩ C 1 (Ω) of (4.6) i.e. ∀φ ∈ C 1 c (Ω) Moreover, if u 1 and u 2 are two distributional solutions of (4.6) in D(R)∩C 1 (Ω) associated to h 1 and h 2 respectively, then the operator R satisfies Proof. Define the energy functional E onV q + ∩ L 2 (Ω) as E(u) = J (u 1/q ) where J is defined in (4.5). Let φ ∈ C 1 c (Ω) and v 0 is the global minimizer of (4.5) which is also the weak solution of (4.1) and u 0 = v q 0 then there exists t 0 = t 0 (φ) > 0 such that for t ∈ (−t 0 , t 0 ), u 0 + tφ > 0. Hence we have Then divide by t and passing to the limits t → 0 we obtain u 0 = v q 0 is the distributional solution of (4.6). Finally (4.7) and uniqueness follow from (4.4).
We now generalize some above results for a larger class of potentials h 0 :
Proof. Let h n ∈ C 1 c (Ω) such that h n ≥ 0 and h n → h in L 2 (Ω). Define (v n ) ⊂ C 1,α (Ω) ∩ C 0 d (Ω) + as for a fixed n, v n is the unique positive weak solution of (4.1) with h 0 = h n i.e. v n satisfies: for φ ∈ W (4.8) thus we deduce that (v n ) converges to v ∈ L 2q (Ω).
We infer that the limit v does not depend on the choice of the sequence (h n ). Indeed, considerh n = h n such thath n → h 0 in L 2 (Ω) andṽ n the positive solution of (4.1) corresponding toh n which converges toṽ. Then, for any n ∈ N, (4.4) implies and passing to the limit we getṽ ≥ v and then by reversing the role of v andṽ we obtain v =ṽ. So define, for any n ∈ N * , h n = min{h, n}. Thus (v n ) is nondecreasing and for any n ∈ N * , v n ≤ v a.e. in Ω which implies v ≥ v 1 > 0 in Ω. We choose φ = v n in (4.8). Applying the Hölder inequality and (4.2), we obtain (Ω) ֒→ L p− (Ω) and by (1.3) we deduce for some positive constant C > 0: Choosing ǫ small enough and gathering with the case ∇v n L p(x) ≤ 1, we conclude (v n ) is uniformly bounded in W 1,p(x) 0 (Ω) and L p− (Ω). Hence v n converges weakly to v in W 1,p(x) 0 (Ω) and by monotonicity of (v n ) strongly in L p− (Ω) and in L 2q (Ω). Taking now φ = v n − v in (4.8), from (4.2) with ǫ = 1 and by Hölder inequality Finally (4.8) becomes Lemma A.2 and Remark A.3 of [16] give the strong convergence of v n to v in W 1,p(x) 0 (Ω). Since (v 2q−1 n ) and (h n v q−1 n ) are uniformly bounded in L 2q/(2q−1) (Ω) and by in Ω. Then by Lebesgue dominated convergence theorem we have (up to a subsequence), for φ ∈ W Finally we pass to the limit in (4.8) and we obtain v is a weak solution of (4.1).

Existence of a weak solution to (1.8)
In this section, in light of Remark 1.3, we consider the problem (1.9) and establish the existence of weak solution (Ω). Proof of Theorem 1.4: Let n * ∈ N * and set ∆ t = T /n * . For 0 ≤ n ≤ n * , we define t n = n∆ t .
Step 2: Time discretization of (1.9) Define the following implicit Euler scheme and for n ≥ 1, v n is the weak solution of (4.16) Note that the sequence (v n ) n=1,2,...,n * is well-defined. Indeed for n = 1 the existence and the uniqueness of v 1 ∈ C 1,α (Ω) ∩ C 0 d (Ω) + follows from Theorems 4.1 and 4.3 with h = ∆ t h 1 + v q 0 ∈ L ∞ (Ω) + . Hence by induction we obtain in the same way the existence and the uniqueness of the solution v n for any n = 2, 3, . . . , n * where v n ∈ C 1,α (Ω) ∩ C 0 d (Ω) + .
Step 3: Existence of a subsolution and supersolution Now we construct a subsolution and a supersolution w and w of (4.16) such that for each n ∈ {0, 1, 2, . . . , n * }, v n satisfies 0 < w ≤ v n ≤ w.
As above, there exists a unique weak solution to (4.19), w K ∈ C 1 (Ω) ∩ C 0 d (Ω) + . Let w K be the unique weak solution of (4.20) where ν ∈ (0, 1) and w K L ∞ (Ω) → ∞ as K → ∞. Then by weak comparison principle we can choose K large enough such that there exists such that v 0 ≤ w K < w def = w K . We easily check that w is the supersolution of (4.17) for n = 1 i.e.