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Uniform Convexity and Variational Convergence

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Uniform Convexity and Variational Convergence Trudy Moskov. Matem. Obw. Trans. Moscow Math. Soc. Tom 75 (2014), vyp. 2 2014, Pages 205–231 S 0077-1554(2014)00232-6 Article electronically published on November 5, 2014 UNIFORM CONVEXITY AND VARIATIONAL CONVERGENCE V. V. ZHIKOV AND S. E. PASTUKHOVA Dedicated to the Centennial Anniversary of B. M. Levitan Abstract. Let Ω be a domain in Rd. We establish the uniform convexity of the Γ-limit of a sequence of Carath´eodory integrands f(x, ξ): Ω×Rd → R subjected to a two-sided power-law estimate of coercivity and growth with respect to ξ with expo- nents α and β,1<α≤ β<∞, and having a common modulus of convexity with respect to ξ. In particular, the Γ-limit of a sequence of power-law integrands of the form |ξ|p(x), where the variable exponent p:Ω→ [α, β] is a measurable function, is uniformly convex. We prove that one can assign a uniformly convex Orlicz space to the Γ-limit of a sequence of power-law integrands. A natural Γ-closed extension of the class of power-law integrands is found. Applications to the homogenization theory for functionals of the calculus of vari- ations and for monotone operators are given. 1. Introduction 1.1. The notion of uniformly convex Banach space was introduced by Clarkson in his famous paper [1]. Recall that a norm ·and the Banach space V equipped with this norm are said to be uniformly convex if for each ε ∈ (0, 2) there exists a δ = δ(ε)such that ξ + η ξ − η≤ε or ≤ 1 − δ 2 for all ξ,η ∈ V with ξ≤1andη≤1. In this case, one also says that the unit ball is uniformly convex. It is well known that the Euclidean norm | | 2 ··· 2 1/2 ξ =(ξ1 + + ξd) on Rd is uniformly convex; this follows from the parallelogram identity |ξ − η|2 + |ξ + η|2 =2(|ξ|2 + |η|2), which shows that one can set δ(ε)=1−(1−ε2)1/2. (This has a simple geometric meaning for the unit sphere in Rd: the midpoints of chords of length ≥ ε ∈ (0, 2) are at a distance of at least δ(ε) from the sphere.) The parallelogram identity also holds for the norm on a Hilbert space. As a consequence, every Hilbert space is uniformly convex. The uniform convexity of other norms of the form p p 1/p ξp =(|ξ1| + ···+ |ξd| ) 2010 Mathematics Subject Classification. Primary 35J20; Secondary 35J60, 46B10, 46B20, 49J45, 49J50. Key words and phrases. Uniform convexity, Γ-convergence, Orlicz spaces, power-law integrand, non- standard coercivity, and growth conditions. Supported by RFBR grant no. 14-01-00192a, grant no. NSh-3685.2014.1 of the President of the Russian Federation, and Russian Scientific Foundation grant no. 14-11-00398. c 2014 American Mathematical Society 205 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 206 V. V. ZHIKOV AND S. E. PASTUKHOVA d p on R ,where1<p<2orp>2, as well as the norms on the spaces lp and L (Ω), p>1, was proved in [1]. The proof is based on the Clarkson inequalities ξ − ηp + ξ + ηp ≤ 2p−1(ξp + ηp),p≥ 2, p ξ − ηq + ξ + ηq ≤ 2(ξp + ηp)q−1,q= , 1 <p≤ 2, p − 1 for these norms. d The norms ξ1 = |ξ1| + ···+ |ξd| and ξ∞ =max{|ξ1|,...,|ξd|} on R ,d≥ 2, are not uniformly convex. Neither are the Lebesgue spaces L1(Ω) and L∞(Ω)aswellastheir discrete counterparts, the spaces l1 and l∞. A uniformly convex Banach space possesses a number of interesting, useful proper- ties, of which we only note reflexivity (the Milman–Pettis theorem), the Banach–Saks property, and the uniform smoothness of the dual space; e.g., see [2, 3, 4, 5]. 1.2. It is quite natural to consider not only uniformly convex sets but also uniformly convex functions. Recall that a function f is said to be convex if (1.1) f(λξ +(1− λ)η) ≤ λf(ξ)+(1− λ)f(η) ∀ξ,η, 0 ≤ λ ≤ 1, and strictly convex if the inequality in (1.1) is strict whenever 0 <λ<1andξ = η.As to uniform convexity, the following definition is classical. Definition 1.1. A function f defined on a Banach space V is said to be uniformly convex if, for each ε ∈ (0, 1), there exists a δ = δ(ε) > 0 such that ξ + η (1.2) ξ − η≤ε max{ξ, η} or 2f ≤ (1 − δ)(f(ξ)+f(η)) 2 for any ξ,η ∈ V . Clearly, a uniformly convex function is strictly convex. For f(ξ)=ξ,weobtainan equivalent definition of uniformly convex norm. Uniformly convex functions have the property, important in applications, of strong convergence of a minimizing sequence. Proposition 1.2. Let f : V → (−∞, +∞] be a uniformly convex lower semibounded function such that limξ→∞ f(ξ)=+∞.LetA =infξ∈V f(ξ),andletξn be a minimizing sequence; i.e., limn→∞ f(ξn)=A.Thenξn is a Cauchy sequence. Proof. It follows from the properties of f that the sequence ξn is bounded (i.e., ξ≤M) and that for each ε ∈ (0, 1) there exists a δ = δ(ε) > 0 such that ξ + ξ (1.3) ξ − ξ ≤εM or 2f n n+m ≤ (1 − δ)(f(ξ )+f(ξ )) n n+m 2 n n+m for any positive integers, n and m.Sinceξn is a minimizing sequence, we see that f(ξn) <A/(1 − δ) for sufficiently large n, n ≥ N(ε). Then the second inequality in (1.3) cannot hold for these n, because ξ + ξ f n n+m ≥ A 2 by the definition of A, while, on the other hand, (1 − δ)2A (1 − δ)(f(ξ )+f(ξ )) < =2A. n n+m 1 − δ Thus, the first inequality in (1.3) holds for n ≥ N(ε), which means that ξn is a Cauchy sequence. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use UNIFORM CONVEXITY AND VARIATIONAL CONVERGENCE 207 The definition of uniform convexity given above uses the norm. One can introduce uniform convexity for functions in a somewhat different way. Definition 1.3. We say that a function f : V → (−∞, +∞] defined on a linear space V is uniformly convex if for each ε ∈ (0, 1) there exists a δ = δ(ε) > 0 such that ξ − η 1 ξ + η 1 (1.4) f ≤ ε (f(ξ)+f(η)) or f ≤ (1 − δ) (f(ξ)+f(η)) 2 2 2 2 for any ξ,η ∈ V . The function δ(ε)iscalledthemodulus of convexity. With this approach, a uniformly convex function does not even have to be convex. In what follows, we touch upon the comparison of various types of convexity. As to uniform convexity, we usually stick to Definition 1.3, often without mentioning this explicitly. 1.3. We are interested in problems related to the uniform convexity of the vector Orlicz– Lebesgue space (see Section 8). This is the space of measurable vector functions v(x) such that the power |v(x)|p(x) with a given variable exponent p(x) is integrable. The Orlicz–Lebesgue space is a natural generalization of the classical Lebesgue spaces. Being equipped with the Luxembourg norm, it becomes a uniformly convex Banach space. The Luxembourg norm is essentially the Minkowski functional of a convex set defined via an integral functional. This motivates our interest in the uniform convexity of integral functionals of the formˆ (1.5) f(x, v(x)) dx, where Ω is a domain in Rd, Ω and in the uniform convexity of the integrands f(x, ξ), ξ ∈ Rd (see Section 5). The uniform convexity of functionals is of interest in that this property, as well as the usual convexity, is preserved under Γ-convergence. The definition of Γ-convergence and a discussion of its properties will be given in Sections 3 and 4. Γ-convergence is often referred to as variational convergence. The point is that this is a type of convergence Fε → F of functionals which, under certain growth conditions on the functionals, implies the convergence of minima and minimizers in the corresponding variational problems (1.6) min Fε(v), where V is the space of admissible functions. v∈V Similar passages to the limit in variational problems arise in various fields of nonlinear analysis, e.g., in homogenization theory and optimal control theory. The limit problem for the family (1.6) is (1.7) min F (v),F= Γ-lim Fε. v∈V It is important to find out what properties of Fε are inherited by the Γ-limit functional F . Uniform convexity is one of such properties (see Section 5). Power-law functionals and their variational closure are of special interest owing to the search for a natural Γ-closed extension of this class (see Section 8). 2. Uniform convexity of functions on Rd We start from the power function f(ξ)=|ξ|α, ξ ∈ Rd. Proposition 2.1. The power function (2.1) f(ξ)=|ξ|α,α>1, is uniformly convex. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 208 V. V. ZHIKOV AND S. E. PASTUKHOVA Proof. 1◦. The power function ϕ(t)=tα, t ∈ [0, ∞), α>1, on the real line is strictly convex (because ϕ(t) > 0on(0, ∞)). This readily implies that the function |ξ|α on Rd is strictly convex. (See 3◦ below for details.) 2◦. Let us verify that the function |ξ|α is uniformly convex in the sense of Defini- tion 1.3.
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