George Dinca

Pavel Matei

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 347 (2009) 885–889Partial Differential Equations/Functional AnalysisGeometry of Sobolev spaces with variable exponent:smoothness and uniform convexityGeorge Dinca a, Pavel Matei ba Faculty of Mathematics and Computer Science, 14, Academiei St, 010014 Bucharest, Romaniab Department of Mathematics, Technical University of Civil Engineering, 124, Lacul Tei Blvd., 020396 Bucharest, RomaniaReceived 17 February 2009; accepted 13 March 2009Available online 2 June 2009Presented by Philippe G. CiarletAbstractLet Ω ⊂ RN , N  2, be a smooth bounded domain. It is shown that: (a) if p ∈ L∞(Ω) and ess infx∈Ω p(x) > 1, then thegeneralized Lebesgue space (Lp(·)(Ω),‖ ‖p(·)) is smooth; (b) if p ∈ C(Ω̄) and p(x) > 1, for all x ∈ Ω̄ , then the generalizedSobolev space (W1,p(·)0 (Ω),‖ ‖1,p(·)) is smooth. In both situations, the formulae giving the Gâteaux derivative of the norm,corresponding to each of the above spaces, are given; (c) if p ∈ C(Ω̄) and p(x)  2, for all x ∈ Ω̄ , then (W1,p(·)0 (Ω),‖ ‖1,p(·)) isuniformly convex and smooth. To cite this article: G. Dinca, P. Matei, C. R. Acad. Sci. Paris, Ser. I 347 (2009).© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméGéométrie des espaces de Sobolev à coefficients variables : lissitude et convexité uniforme. Soit Ω ⊂ RN , N  2,un domain borné et régulier. On demontre que : (a) si p ∈ L∞(Ω) et ess infx∈Ω p(x) > 1, alors l’espace de Lebesgue gé-néralisé (Lp(·)(Ω),‖ ‖p(·)) est lisse ; (b) si p ∈ C(Ω̄) et p(x) > 1, pour tout x ∈ Ω̄ , alors l’espace de Sobolev généralisé(W1,p(·)0 (Ω),‖ ‖1,p(·)) est lisse. Dans les deux cas, les formules de la dérivée au sens de Gâteaux de chaque norme des espacesci-dessus sont données ; (c) si p ∈ C(Ω̄) et p(x)  2, pour tout x ∈ Ω̄ , alors (W1,p(·)0 (Ω),‖ ‖1,p(·)) est uniformément convexe etlisse. Pour citer cet article : G. Dinca, P. Matei, C. R. Acad. Sci. Paris, Ser. I 347 (2009).© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeLe résultat principal de cette Note est le théorème suivant :Théorème 1. Soit Ω ⊂ RN , N  2, un domaine borné et régulier.(a) Si p ∈ L∞+ (Ω) et p− > 1, alors (Lp(·)(Ω),‖ ‖p(·)) est lisse. En tout u ∈ Lp(·)(Ω) \ {0}, la dérivée au sens deGâteaux de la norme ‖ ‖p(·) est donnée par :E-mail addresses: dinca@fmi.unibuc.ro (G. Dinca), pavel.matei@gmail.com (P. Matei).1631-073X/$ – see front matter © 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2009.04.028886 G. Dinca, P. Matei / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 885–889〈‖ ‖′p(·)(u),h〉 =∫Ωp(x)|u(x)|p(x)−1 sgnu(x)‖u‖p(x)p(·)h(x)dx∫Ωp(x)|u(x)|p(x)‖u‖p(x)+1p(·)dx, pour tout h ∈ Lp(·)(Ω).(b) Si p ∈ C(Ω̄) et p(x) > 1, pour tout x ∈ Ω̄ , alors (W 1,p(·)0 (Ω),‖ ‖1,p(·)) est lisse. En tout u ∈ W 1,p(·)0 (Ω) \ {0},la dérivée au sens de Gâteaux de la norme ‖ ‖1,p(·) est donnée par :〈‖ ‖′1,p(·)(u),h〉 =∫Ωp(x)|∇u(x)|p(x)−2∇u∇h‖u‖p(x)1,p(·)dx∫Ωp(x)|∇u|p(x)‖u‖p(x)+11,p(·)dx, pour tout h ∈ W 1,p(·)0 (Ω).(c) Si p ∈ C(Ω̄) et p(x)  2, pour tout x ∈ Ω̄ , alors (W 1,p(·)0 (Ω),‖ ‖1,p(·)) est uniformément convexe et lisse.Les définitions des principales notions intervenant dans cet enoncé ainsi que les grandes lignes de la démonstrationsont données dans la version anglaise ci-dessous. Pour les détails, voir [2].1. Basic definitionsLet Ω ⊂ RN , N  2, be a bounded and smooth domain. We take Lebesgue measure in RN and denote byL∞+ (Ω) ={u ∈ L∞(Ω) ∣∣ 1  p− = ess infΩu  p+ = ess supΩu < ∞}.For p ∈ L∞+ (Ω), the generalized Lebesgue space Lp(·)(Ω) is defined byLp(·)(Ω) ={u : Ω → R∣∣∣ u measurable and ρp(·)(u) =∫Ω∣∣u(x)∣∣p(x) dx < ∞}and it is endowed with the norm‖u‖p(·) = inf{λ > 0∣∣∣∣ ρp(·)(uλ)=∫Ω∣∣∣∣u(x)λ∣∣∣∣p(x)dx  1}, for all u ∈ Lp(·)(Ω).The generalized Sobolev space W 1,p(·)(Ω) is defined byW 1,p(·)(Ω) = {u ∈ Lp(·)(Ω) ∣∣ |∇u| ∈ Lp(·)(Ω)}, |∇u|2 = N∑i=1(∂u∂xi)2and it is endowed with the norm‖u‖ = ‖u‖p(·) +∥∥|∇u|∥∥p(·), for all u ∈ W 1,p(·)(Ω).We define W 1,p(·)0 (Ω) as the closure of C∞0 (Ω) in (W 1,p(·)(Ω),‖ ‖). If p ∈ C(Ω̄) and p(x) > 1 for any x ∈ Ω̄ ,there is a constant c > 0 such that (Poincaré’s inequality)‖u‖p(·)  c∥∥|∇u|∥∥p(·), for all u ∈ W 1,p(·)0 (Ω)holds and, from this, we infer that ‖u‖ and ‖u‖1,p(·) = ‖|∇u|‖p(·) are equivalent norms on W 1,p(·)0 (Ω).For the basic properties of the above spaces, cf. Edmunds and Rákosník [3,4], Fan and Zhao [5], Kováčik andRákosník [7]. In what follows, W 1,p(·)0 (Ω) will be considered as endowed with the norm ‖ ‖1,p(·) and we will oftenwrite W 1,p(·)(Ω) instead of (W 1,p(·)(Ω),‖ ‖1,p(·)).0 0G. Dinca, P. Matei / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 885–889 8872. The main resultTheorem 1. Let Ω ⊂ RN , N  2, be a bounded and smooth domain.(a) If p ∈ L∞+ (Ω) and p− > 1, then (Lp(·)(Ω),‖ ‖p(·)) is smooth.At any u ∈ Lp(·)(Ω), u 	= 0, the Gâteaux derivative of the norm ‖ ‖p(·) is given by〈‖ ‖′p(·)(u),h〉 =∫Ωp(x)|u(x)|p(x)−1 sgnu(x)‖u‖p(x)p(·)h(x)dx∫Ωp(x)|u(x)|p(x)‖u‖p(x)+1p(·)dx, for all h ∈ Lp(·)(Ω). (1)(b) If p ∈ C(Ω̄) and p(x) > 1, for all x ∈ Ω̄ , then (W 1,p(·)0 (Ω),‖ ‖p(·)) is smooth.At any u ∈ W 1,p(·)0 (Ω), u 	= 0, the Gâteaux derivative of the norm ‖ ‖1,p(·) is given by〈‖ ‖′1,p(·)(u),h〉 =∫Ωp(x)|∇u(x)|p(x)−2∇u∇h‖u‖p(x)1,p(·)dx∫Ωp(x)|∇u|p(x)‖u‖p(x)+11,p(·)dx, for all h ∈ W 1,p(·)0 (Ω). (2)(c) If p ∈ C(Ω̄) and p(x)  2, for all x ∈ Ω̄ , then (W 1,p(·)0 (Ω),‖ ‖p(·)) is uniformly convex and smooth.Next, we give the idea of the proof (for more details, see [2]).(a) The key tools in proving (a) is the classical implicit function theorem and a basic property of the convex modularρp(·).Indeed, we have to prove that, for a given u0 ∈ Lp(·)(Ω), u0 	= 0, and any h ∈ Lp(·)(Ω), the function t ∈ R,t → ‖u0 + th‖p(·) is differentiable at t = 0. Let k > 1 be a fixed real number, D = (1,1) × ( 1k ‖u0‖p(·), k‖u0‖p(·))and Φ :D → R defined byΦ(t, λ) = ρp(·)(u0 + thλ)− 1 =∫Ω|u0(x) + th(x)|p(x)λp(x)dx − 1. (3)It may be shown that:(a) Φ ∈ C 1(D);(b) Φ(0,‖u0‖p(·)) = 0;(c) ∂Φ∂λ(0,‖u0‖p(·)) < 0.Consequently, there exist neighborhoods U of 0 and V of ‖u0‖p(·) such that U × V ⊂ D and a C 1-mappingλ :U → V which satisfies: λ(0) = ‖u0‖p(·), Φ(t, λ(t)) = 0 for any t ∈ U andλ′(t) = −∂Φ∂t(t, λ(t))∂Φ∂λ(t, λ(t)), for all t ∈ U. (4)Taking into account the definition of Φ (see (3)), Φ(t, λ(t)) = 0 for any t ∈ U rewrites asρp(·)(u0 + thλ(t))= 1, for all t ∈ U. (5)Since ‖u‖p(·) = a > 0 if and only if ρp(·)( ua ) = 1 (Fan and Zhao [5, Theorem 1.2]) we deduce from (5) thatλ(t) = ‖u0 + th‖p(·), for all t ∈ U. (6)By combining (4) and (6) we derive, in particular, that λ′(0) exists andλ′(0) = limt→0‖u0 + th‖p(·) − ‖u0‖p(·)t= −∂Φ∂t(0,‖u0‖p(·))∂Φ (0,‖u ‖ ) , (7)∂λ 0 p(·)888 G. Dinca, P. Matei / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 885–889i.e. the norm ‖ ‖p(·) is Gâteaux differentiable at u0 and the Gâteaux derivative of this norm at u0 is given by〈‖ ‖′p(·)(u0), h〉 = − ∂Φ∂t (0,‖u0‖p(·))∂Φ∂λ(0,‖u0‖p(·)), for all h ∈ W 1,p(·)0 (Ω), (8)with Φ given by (3).Since a Banach space is smooth if and only if its norm is Gâteaux differentiable (Diestel [1, chapter 2, §1,Theorem 1]) we conclude on the smoothness of (Lp(·)(Ω),‖ ‖p(·)).In order to obtain formula (1) it is enough to compute explicitly the derivatives ∂Φ∂t(0,‖u0‖p(·)) and ∂Φ∂λ (0,‖u0‖p(·))in (8).The key point in so doing is to show that these derivatives may be computed by “differentiating under the integralsign”, i.e. at any (t, λ) ∈ D, ∂Φ∂t(t, λ) = ∫Ω[ ∂∂t(|u0(x)+th(x)|p(x)λp(x))]dx and a similar formula for ∂Φ∂λ(t, λ). In thesecomputations the condition p− > 1 is essentially involved (see [2] for necessary details).(b) Since at any u ∈ W 1,p(·)0 (Ω), u 	= 0W 1,p(·)0 (Ω)(⇔ |∇u| 	= 0Lp(·)(Ω)) the functional Ψ (u) = ‖u‖1,p(·) = ‖|∇u|‖p(·)may be written as a composed function Ψ (u) = Q(Pu) withP :W 1,p(·)0 (Ω) → Lp(·)(Ω), Pu = |∇u|,Q :Lp(·)(Ω) → R, Qv = ‖v‖p(·),the differentiation chain rule gives us〈‖ ‖′1,p(·)(u),h〉 = 〈Q′(Pu),P ′(u)h〉 = 〈‖ ‖′p(·)(|∇u|),P ′(u)h〉, for all h ∈ W 1,p(·)0 (Ω).Thus, by simple computation and using (1), formula (2) follows.The following auxiliary results are needed for the proof of point (c):Proposition 1. Let A(t) = ∫ |t |0 a(s)ds, t ∈ R be an N function (cf. [8], p. 6) with a(s)s non-decreasing on (0,∞). LetX be a real vector space and T :X ×X → R a bilinear symmetric and positive form. Then, for any f,g ∈ X, one has:A(√T (f,f )) + A(√T (g,g) )  2[A(√T(f + g2,f + g2) )+ A(√T(f − g2,f − g2))]. (9)This result is due to Gröger [6] (see also Langenbach [9, p. 153] and [2] for a simple proof).Proposition 2. Let p ∈ L∞+ (Ω). Then one has:(a) For every ε > 0 there exists a number η(ε) > 0 such that if u ∈ Lp(·)(Ω) satisfies ‖u‖p(·)  ε, then ρp(·)(u)  η;(b) For every η ∈ (0,1) there exists a number δ ∈ (0,1) such that if u ∈ Lp(·)(Ω) \ {0} satisfies ρp(·)(u)  1 − η, then‖u‖p(·)  1 − δ.Proof of point (b). (Point (a) may be proven by using the same type of arguments.)Assuming the contrary, a number η ∈ (0,1) and a sequence (un) ⊂ Lp(·)(Ω) \ {0} such that ρp(·)(un)  1 − η and‖un‖ > 1 − 1n would exist.From ρp(·)(un)  1 − η we derive (see [5, Theorem 1.3]) that ‖un‖p(·) < 1.Thus, ‖un‖ → 1 as n → ∞.Since ([5, Theorem 1.3]) ‖un‖p+p(·)  ρp(·)(un)  ‖un‖p−p(·), it follows that ρp(·)(un) → 1, in contradiction withρp(·)(un)  1 − η. G. Dinca, P. Matei / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 885–889 889Proof of point (c) in Theorem 1. Let x ∈ Ω . Consider the N -function A(t) = |t |p(x). Since a(s) = p(x)sp(x)−1 andp(x)  2, it follows that a(s)sis non-decreasing in (0,∞). Let T :W 1,p(·)0 (Ω) × W 1,p(·)0 (Ω) → R be the bilinear formdefined by T (f,g) = (∇f )(x) · (∇g)(x). Inequality (9) applies and we obtain:∣∣∇f (x)∣∣p(x) + ∣∣∇g(x)∣∣p(x)  2[∣∣∣∣∇(f + g2)(x)∣∣∣∣p(x)+∣∣∣∣∇(f − g2)(x)∣∣∣∣p(x)].By integrating over Ω we get:ρp(·)(|∇f |) + ρp(·)(|∇g|)  2[ρp(·)(∣∣∣∣∇(f + g2)∣∣∣∣)+ ρp(·)(∣∣∣∣∇(f − g2)∣∣∣∣)]. (10)Now, let ε ∈ (0,2] be given and let f,g ∈ W 1,p(·)0 (Ω) be such that‖f ‖1,p(·) =∥∥|∇f |∥∥p(·) = 1, (11)‖g‖1,p(·) =∥∥|∇g|∥∥p(·) = 1, (12)‖f − g‖1,p(·) =∥∥∣∣∇(f − g)∣∣∥∥p(·)  ε ⇔∥∥∥∥∣∣∣∣∇(f − g2)∣∣∣∣∥∥∥∥  ε2 . (13)From (11) and (12) we deduce (cf. [5, Theorem 1.2]) that ρp(·)(|∇f |) = 1, ρp(·)(|∇g|) = 1 while from (13) andProposition 2(a) it follows that there exists η > 0 such that ρp(·)(|∇( f −g2 )|)  η (clearly, we may assume η < 1).Consequently, from (10) it follows that ρp(·)(|∇( f +g2 )|)  1 − η and from Proposition 2(b) we deduce the existenceof a number δ ∈ (0,1) such that ‖|∇( f +g2 )|‖p(·)  1 − δ which rewrites as ‖f −g2 ‖1,p(·)  1 − δ. References[1] J. Diestel, Geometry of Banach Spaces – Selected Topics, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1975.[2] G. Dinca, P. Matei, Geometry of Sobolev spaces with variable exponent and a generalization of the p-Laplacian, Analysis and Applications, inpress.[3] D.E. Edmunds, J. Rákosník, Sobolev embeddings with variable exponent, I, Studia Mathematica 143 (2000) 267–292.[4] D.E. Edmunds, J. Rákosník, Sobolev embeddings with variable exponent, II, Math. Nachr. 246–247 (2002) 53–67.[5] X.L. Fan, D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2001) 424–446.[6] K. Gröger, Eine Verallgemeinerung der Sobolewschen Räume in unbeschränkten Gebieten, Math. Nachr. 32 (1966) 115–130.[7] O. Kováčik, J. Rákosník, On spaces Lp(x) and Wk,p(x) , Czechoslovak Math. J. 41 (1991) 592–618.[8] M.A. Krasnosel’skij, Ya.B. Rutickij, Convex Functions and Orlicz Spaces, Gröningen, Noordhoff, 1961.[9] A. Langenbach, Monotone potential operatoren in theorie und anwendung, Springer Verlag der Wissenschaften, Berlin, 1976.

Geometry of Sobolev spaces with variable exponent: smoothness and uniform convexity

Dinca, George

Matei, Pavel

Numérisation de Documents Anciens Mathématiques

http://www.numdam.org/item/10.1016/j.crma.2009.04.028.pdf

Geometry of Sobolev spaces with variable exponent: smoothness and uniform convexity

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Comptes Rendus Mathématique

Numérisation de Documents Anciens Mathématiques