nzonevn C ne Zo en hV i Si nh Vi en Zo ne C om Modern Methods in the Calculus p of Variations:L Spaces SinhVienZone.com https://fb.com/sinhvienzonevn Irene Fonseca Giovanni Leoni Si nh Vi en Zo ne C om Modern Methods in the Calculus p of Variations: L Spaces SinhVienZone.com https://fb.com/sinhvienzonevn Giovanni Leoni Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213 USA fonseca@andrew.cmu.edu Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213 USA giovanni@andrew.cmu.edu Zo ne C om Irene Fonseca ISBN: 978-0-387-35784-3 e-ISBN: 978-0-387-69006-3 en Library of Congress Control Number: 2007931775 Mathematics Subject Classification (2000): 49-00, 49-01, 49-02, 49J45, 28-01, 28-02, 28B20, 52A Si nh Vi © 2007 Springer Science+Business Media, LLC All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY, 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights Printed on acid-free paper springer.com SinhVienZone.com https://fb.com/sinhvienzonevn Si nh Vi en Zo ne C om To our families SinhVienZone.com https://fb.com/sinhvienzonevn Preface om In recent years there has been renewed interest in the calculus of variations, motivated in part by ongoing research in materials science and other disciplines Often, the study of certain material instabilities such as phase transitions, formation of defects, the onset of microstructures in ordered materials, fracture and damage, leads to the search for equilibria through a minimization problem of the type {I (v) : v ∈ V} , Si nh Vi en Zo ne C where the class V of admissible functions v is a subset of some Banach space V This is the essence of the calculus of variations: the identification of necessary and sufficient conditions on the functional I that guarantee the existence of minimizers These rest on certain growth, coercivity, and convexity conditions, which often fail to be satisfied in the context of interesting applications, thus requiring the relaxation of the energy New ideas were needed, and the introduction of innovative techniques has resulted in remarkable developments in the subject over the past twenty years, somewhat scattered in articles, preprints, books, or available only through oral communication, thus making it difficult to educate young researchers in this area This is the first of two books in the calculus of variations and measure theory in which many results, some now classical and others at the forefront of research in the subject, are gathered in a unified, consistent way A main concern has been to use contemporary arguments throughout the text to revisit and streamline well-known aspects of the theory, while providing novel contributions The core of this book is the analysis of necessary and sufficient conditions for sequential lower semicontinuity of functionals on Lp spaces, followed by relaxation techniques What sets this book apart from existing introductory texts in the calculus of variations is twofold: Instead of laying down the theory in the one-dimensional setting for integrands f = f (x, u, u ), we work in N dimensions and no derivatives are present In addition, it is self-contained in SinhVienZone.com https://fb.com/sinhvienzonevn VIII Preface Si nh Vi en Zo ne C om the sense that, with the exception of fundamentally basic results in measure theory that may be found in any textbook on the subject (e.g., Lebesgue dominated convergence theorem), all the statements are fully justified and proved This renders it accessible to beginning graduate students with basic knowledge of measure theory and functional analysis Moreover, we believe that this text is unique as a reference book for researchers, since it treats both necessary and sufficient conditions for well-posedness and lower semicontinuity of functionals, while usually only sufficient conditions are addressed The central part of this book is Part III, although Parts I and II contain original contributions Part I covers background material on measure theory, integration, and Lp spaces, and it combines basic results with new approaches to the subject In particular, in contrast to most texts in the subject, we not restrict the context to σ-finite measures, therefore laying the basis for the treatment of Hausdorff measures, which will be ubiquitous in the setting of the second volume, in which gradients will be present Moreover, we call attention to Section 1.1.4, on “comparison between measures”, which is completely novel: The Radon–Nikodym theorem and the Lebesgue decomposition theorem are proved for positive measures without our having first to introduce signed measures, as is usual in the literature The new arguments are based on an unpublished theorem due to De Giorgi treating the case in which the two measures in play are not σ-finite Here, as De Giorgi’s theorem states, a diffuse measure must be added to the absolutely continuous and singular parts of the decomposition Also, we give a detailed proof of the Morse covering theorem, which does not seem to be available in other books on the subject, and we derive as a corollary the Besicovitch covering theorem instead of proving it directly Part II streamlines the study of convex functions, and the treatment of the direct method of the calculus of variations introduces the reader to the close connection between sequential lower semicontinuity properties and existence of minimizers Again here we present an unpublished theorem of De Giorgi, the approximation theorem for real-valued convex functions, which provides an explicit formula for the affine functions approximating a given convex function f A major advantage of this characterization is that additional regularity hypotheses on f are reflected immediately on the approximating affine functions In Part III we treat sequential lower semicontinuity of functionals defined on Lp , and we separate the cases of inhomogeneous and homogeneous functionals The latter are studied in Chapter 5, where f (v(x)) dx I(u) := E with E a Lebesgue measurable subset of the Euclidean space RN , f : Rm → (−∞, ∞] and v ∈ Lp (E; Rm ) for ≤ p ≤ ∞ This material is intended for an introductory graduate course in the calculus of variations, since it requires only basic knowledge of measure theory and functional analysis We treat both SinhVienZone.com https://fb.com/sinhvienzonevn Preface IX Zo ne C om bounded and unbounded domains E, and we address most types of strong and weak convergence In particular, the setting in which the underlying conver gence is that of (Cb (E)) is new Chapter and Chapter are devoted to integrands f = f (x, v) and f = f (x, u, v), respectively, and are significantly more advanced, since the proofs of the necessity parts are heavily hinged on the concept of multifunctions An important tool here is selection criteria, and the reader will benefit from a comprehensive and detailed study of this subject Finally, Chapter describes basic properties of Young measures and how they may be used in relaxation theory The bibliography aims at giving the main references relevant to the contents of the book It is by no means exhaustive, and many important contributions to the subject may have failed to be listed here To conclude, this text is intended as a graduate textbook as well as a reference for more-experienced researchers working in the calculus of variations, and is written with the intention that readers with varied backgrounds may access different parts of the text This book prepares the ground for a second volume, since it introduces and develops the basic tools in the calculus of variations and in measure theory needed to address fundamental questions in the treatment of functionals involving derivatives Finally, in a book of this length, typos and errors are almost inevitable The authors will be very grateful to those readers who will write to either fonseca@andrew.cmu.edu or giovanni@andrew.cmu.edu indicating those that they have found A list of errors and misprints will be maintained and updated at the web page http://www.math.cmu.edu/˜leoni/book1 Irene Fonseca Giovanni Leoni Si nh Vi en Pittsburgh, month 2007 SinhVienZone.com https://fb.com/sinhvienzonevn Contents Part I Measure Theory and Lp Spaces Measures 1.1 Measures and Integration 1.1.1 Measures and Outer Measures 1.1.2 Radon and Borel Measures and Outer Measures 22 1.1.3 Measurable Functions and Lebesgue Integration 37 1.1.4 Comparison Between Measures 55 1.1.5 Product Spaces 77 1.1.6 Projection of Measurable Sets 83 1.2 Covering Theorems and Differentiation of Measures in RN 90 1.2.1 Covering Theorems in RN 90 1.2.2 Differentiation Between Radon Measures in RN 103 1.3 Spaces of Measures 113 1.3.1 Signed Measures 113 1.3.2 Signed Finitely Additive Measures 119 1.3.3 Spaces of Measures as Dual Spaces 123 1.3.4 Weak Star Convergence of Measures 129 en Zo ne C om Lp Spaces 139 2.1 Abstract Setting 139 2.1.1 Definition and Main Properties 139 2.1.2 Strong Convergence in Lp 148 2.1.3 Dual Spaces 156 2.1.4 Weak Convergence in Lp 171 2.1.5 Biting Convergence 184 2.2 Euclidean Setting 190 2.2.1 Approximation by Regular Functions 190 2.2.2 Weak Convergence in Lp 198 2.2.3 Maximal Functions 208 2.3 Lp Spaces on Banach Spaces 218 Si nh Vi SinhVienZone.com https://fb.com/sinhvienzonevn XII Contents Part II The Direct Method and Lower Semicontinuity The Direct Method and Lower Semicontinuity 231 3.1 Lower Semicontinuity 231 3.2 The Direct Method 245 Convex Analysis 247 4.1 Convex Sets 247 4.2 Separating Theorems 254 4.3 Convex Functions 258 4.4 Lipschitz Continuity in Normed Spaces 262 4.5 Regularity of Convex Functions 266 4.6 Recession Function 288 4.7 Approximation of Convex Functions 293 4.8 Convex Envelopes 300 4.9 Star-Shaped Sets 318 om Part III Functionals Defined on Lp Integrands f = f (z) 325 5.1 Well-Posedness 326 5.2 Sequential Lower Semicontinuity 331 5.2.1 Strong Convergence in Lp 331 5.2.2 Weak Convergence and Weak Star Convergence in Lp 334 5.2.3 Weak Star Convergence in the Sense of Measures 340 5.2.4 Weak Star Convergence in Cb E; Rm 350 5.3 Integral Representation 354 5.4 Relaxation 364 5.4.1 Weak Convergence and Weak Star Convergence in Lp , ≤ p ≤ ∞ 365 5.4.2 Weak Star Convergence in the Sense of Measures 369 5.5 Minimization 373 Si nh Vi en Zo ne C Integrands f = f (x, z) 379 6.1 Multifunctions 380 6.1.1 Measurable Selections 380 6.1.2 Continuous Selections 395 6.2 Integrands 401 6.2.1 Equivalent Integrands 401 6.2.2 Normal and Carath´eodory Integrands 404 6.2.3 Convex Integrands 413 6.3 Well-Posedness 428 6.3.1 Well-Posedness, ≤ p < ∞ 428 6.3.2 Well-Posedness, p = ∞ 435 SinhVienZone.com https://fb.com/sinhvienzonevn = ∅ When V is only a locally convex topological vector space we can still prove a weaker form of the previous theorem: en Zo Proposition 4.58 Let V be a locally convex topological vector space and let f : V → (−∞, ∞] be a proper convex function If v0 ∈ V is such that f (v0 ) ∈ R and f is continuous at v0 , then ∂+f (v0 ) = max v , v V ,V ∂v v ∈∂f (v0 ) for all v ∈ V nh Vi Si v , v1 V ,V ≤ Proof By Theorem 4.51, f is subdifferentiable at v0 , and so by Remark 4.49, ∂+f (v0 ) ≥ sup v , v V ,V > −∞ ∂v v ∈∂f (v0 ) for all v ∈ V To prove the reverse inequality, as in Step of the previous theorem it suffices to show that the seminorm p (v) := SinhVienZone.com ∂+f (v0 ) , v ∈ V , ∂v https://fb.com/sinhvienzonevn 4.5 Regularity of Convex Functions is continuous Since p (v) = inf t>0 f (v0 + tv) − f (v0 ) ≤ f (v0 + v) − f (v0 ) t and f is continuous at v0 , it follows that p is bounded from above in a neighborhood of v0 , and so it is continuous by Theorem 4.43 Next we study the subdifferentiability of the sum of two convex functions Proposition 4.59 Let V be a locally convex topological vector space and let f1 , f2 : V → (−∞, ∞] be two proper convex, lower semicontinuous functions Assume that there exists v0 ∈ dome f1 ∩ dome f2 such that f1 is continuous at v0 Then for every v ∈ V , om ∂ (f1 + f2 ) (v) = ∂f1 (v) + ∂f2 (v) Proof Step 1: Let v ∈ V If v1 ∈ ∂f1 (v) and v2 ∈ ∂f2 (v), then C f1 (w) ≥ f1 (v) + v1 , w − v V ,V f2 (w) ≥ f2 (v) + v2 , w − v V ,V for all w ∈ V , for all w ∈ V , ne and so, adding the two inequalities, we conclude that v1 + v2 ∈ ∂ (f1 + f2 ) (v) Hence if ∂f1 (v) and ∂f2 (v) are nonempty, then ∂ (f1 + f2 ) (v) is nonempty and ∂ (f1 + f2 ) (v) ⊃ ∂f1 (v) + ∂f2 (v) Zo Step 2: Conversely, if ∂ (f1 + f2 ) (v) is nonempty, let v ∈ ∂ (f1 + f2 ) (v) Then en (f1 + f2 ) (w) ≥ (f1 + f2 ) (v) + v , w − v V ,V for all w ∈ V (4.28) In particular, f1 (v), f2 (v) ∈ R Define nh Vi Si 283 g (w) := f1 (w) − f1 (v) + v , w − v V ,V , w ∈V, and C := {(w, t) ∈ V × R : t ≤ f2 (v) − f2 (w)} Then g is a convex function and C a convex set We claim that g and C satisfy the hypotheses of Lemma 4.52 Indeed, since f1 is proper and continuous at v0 then so is g Moreover, in view of (4.28), if (w, t) ∈ epi g ∩ C, then g (w) = f1 (w) − f1 (v) + v , w − v V ,V ≤ t ≤ f2 (v) − f2 (w) ≤ f1 (w) − f1 (v) + v , w − v V ,V , SinhVienZone.com https://fb.com/sinhvienzonevn 284 Convex Analysis which implies that g (w) = t and f2 (v)−f2 (w) = t, that is, (w, t) ∈ ∂ (epi g)∩ ∂C Finally, (v0 , f2 (v) − f2 (v0 )) ∈ C Hence we may apply Lemma 4.52 to find w ∈ V and α ∈ R such that g (w) = f1 (w) − f1 (v) + v , w − v V ,V ≥ α + w , w V ,V ≥ f2 (v) − f2 (w) for all w ∈ V Taking w = v yields w ,v V ,V = −α, so that f1 (w) − f1 (v) + v , w − v V ,V ≥ w , w − v V ,V ≥ f2 (v) − f2 (w) for all w ∈ V , and so v − w ∈ ∂f1 (v) and w ∈ ∂f2 (v) Hence v = (v − w ) + w ∈ ∂f1 (v) + ∂f2 (v) , which proves that if ∂ (f1 + f2 ) (v) is nonempty, then om ∂ (f1 + f2 ) (v) ⊂ ∂f1 (v) + ∂f2 (v) C Step 3: To conclude the proof we observe that if either ∂f1 (v) or ∂f2 (v) is empty, then by Step so must be ∂ (f1 + f2 ) (v) If both ∂f1 (v) and ∂f2 (v) are nonempty, then by Steps and 2, ∂ (f1 + f2 ) (v) is also nonempty and ∂ (f1 + f2 ) (v) = ∂f1 (v) + ∂f2 (v) ne This completes the proof Zo We now turn to the relation between subdifferentiability and (Gˆ ateaux) differentiability en Definition 4.60 Let V be a locally convex topological vector space A function f : V → [−∞, ∞] is Gˆ ateaux differentiable at v0 ∈ V if f (v0 ) ∈ R and there exists v ∈ V such that for every v ∈ V , lim t→0+ f (v0 + tv) − f (v0 ) = v , v V ,V t Si nh Vi The element v is called the Gˆ ateaux differential of f at v0 and is denoted by f (v0 ) Theorem 4.61 Let V be a locally convex topological vector space and let f : V → [−∞, ∞] be a convex function If f is Gˆ ateaux differentiable at v0 ∈ V , then it is subdifferentiable at v0 and ∂f (v0 ) = {f (v0 )} Conversely, if f is continuous and finite at v0 ∈ V and the subdifferential of f at v0 is a singleton, then f is Gˆ ateaux differentiable at v0 SinhVienZone.com https://fb.com/sinhvienzonevn 4.5 Regularity of Convex Functions Proof Assume that f is Gˆateaux differentiable at v0 ∈ V , let v ∈ V , and define g (t) := f (v0 + tv) By Proposition 4.34 the difference quotient t → g (t) − g (0) t−0 is nondecreasing in R \ {t0 } Hence g (t) − g (0) g (1) − g (0) ≥ lim+ 1−0 t−0 t→0 f (v0 + tv) − f (v0 ) = lim = f (v0 ) , v V ,V , t t→0+ f (v0 + v) − f (v0 ) = which implies that f (v0 ) ∈ ∂f (v0 ) We claim that ∂f (v0 ) = {f (v0 )} Indeed, if v ∈ ∂f (v0 ), then for any v ∈ V and t > 0, By letting t → 0+ we obtain that om f (v0 + tv) − f (v0 ) ≥ v , v V ,V t f (v0 ) , v V ,V ≥ v , v V ,V for all v ∈ V , Zo ne C which implies that v = f (v0 ) Conversely, assume that f is continuous and finite at v0 ∈ V and the subdifferential of f at v0 is a singleton {v } Then f > −∞ by Remark 4.23, and so by Proposition 4.58 and the fact that ∂f (v0 ) = {v } we deduce that ∂+f (v0 ) = v , v V ,V ∂v for all v ∈ V , which shows that f is Gˆ ateaux differentiable at v0 en The next result shows that for smooth functions convexity is equivalent to the monotonicity of the Gˆ ateaux differential Theorem 4.62 Let E be a convex subset of a locally convex topological vector space V and let f : V → [−∞, ∞] be Gˆ ateaux differentiable in E Then the following three conditions are equivalent: nh Vi Si 285 (i) f : E → R is convex; (ii) for all v, w ∈ E, f (v) ≥ f (w) + f (w) , v − w V ,V ; (iii) for all v, w ∈ E, f (v) − f (w) , v − w V ,V ≥ SinhVienZone.com https://fb.com/sinhvienzonevn 286 Convex Analysis Proof Assume that (i) holds Since f is Gˆateaux differentiable in E, by the previous theorem f is subdifferentiable at every w ∈ E and ∂f (w) = {f (w)} Hence (ii) holds Assume next that (ii) holds Then for all v, w ∈ E, f (v) ≥ f (w) + f (w) , v − w V ,V , f (w) ≥ f (v) − f (v) , v − w V ,V , and by adding these inequalities we obtain that ≥ f (w) − f (v) , v − w V ,V , which gives (iii) Finally, assume that (iii) holds and fix v, w ∈ E Since f is Gˆ ateaux differentiable in E, the function g : [0, 1] → R, defined by g (t) := f (tv + (1 − t) w) , t ∈ [0, 1] , is differentiable and If s > t, then ne C g (s) − g (t) = f (sv + (1 − s) w) − f (tv + (1 − t) w) , v − w V ,V f (sv + (1 − s) w) − f (tv + (1 − t) w) , = s−t (sv + (1 − s) w) − (tv + (1 − t) w) V ,V ≥ Hence g is nondecreasing, and so g is convex In particular, Zo f (tv + (1 − t) w) = g (t) ≤ (1 − t) g (0) + tg (1) = (1 − t) f (w) + tf (v) , which implies the convexity of f en Remark 4.63 A similar result holds for strictly convex functions provided we require the inequalities (i) and (ii) to be strict when v = w As a consequence of the previous theorem we now specialize the result of Theorem 4.36 to obtain a p-Lipschitz condition for separately convex functions with algebraic growth nh Vi Si om g (t) = f (tv + (1 − t) w) , v − w V ,V Proposition 4.64 Let f : Rm → R be a separately convex function such that p |f (z)| ≤ C (1 + |z| ) for some C > 0, p ≥ 1, and all z ∈ Rm Then p−1 p−1 |z − w| |f (z) − f (w)| ≤ C + |z| + |w| for all z, w ∈ Rm SinhVienZone.com https://fb.com/sinhvienzonevn 4.5 Regularity of Convex Functions Proof Step 1: We first assume that f ∈ C ∞ (Rm ) Let z := (z1 , , zm ) ∈ Rm be fixed and consider g (t) := f (z1 , , zi−1 , t, zi+1 , , zm ) , t ∈ R Since g is convex and smooth by Theorem 4.62 (see also Proposition 4.34), for all s and t ∈ R we have g (t + s) − g (t) ≥ g (t) s Thus, with s := + |z|, t := zi , g (t) = (1 + |z| ) ∂f g (t + s) − g (t) p−1 ≤C ≤ C + |z| (z) ≤ ∂zi s + |z| p Also, g (t − s) − g (t) ≥ −g (t) s, and so (1 + |z| ) g (t − s) − g (t) p−1 ≤C ≤ C + |z| s + |z| p om −g (t) ≤ ( ( ( ∂f ( ( ( ≤ C + |z|p−1 (z) ( ∂zi ( C Hence for every i = 1, , m Let z, w ∈ Rm By the mean value theorem there is θ ∈ (0, 1) such that Zo ne |f (z) − f (w)| = |∇f (θz + (1 − θ) w) · (z − w)| p−1 ≤C + |θz + (1 − θ) w| |(z − w)| p−1 p−1 ≤C + |z| |z − w| , + |w| en where we have used the fact that if < q < ∞ and a, b ≥ 0, then q (a + b) ≤ max 1, 2q−1 (aq + bq ) Step 2: To remove the smoothness hypothesis consider a standard mollifier nh Vi Si 287 ϕε and let fε (z) := Rm ϕε (w) f (z − w) dw, ε > The function fε is still separately convex and, in addition, p |fε (z)| ≤ ϕε (w) |f (z − w)| dw ≤ C ϕε (w) (1 + |z − w| ) dw m R B(0,ε) p p ≤ C (1 + |z| ) ϕε (w) dw = C (1 + |z| ) , Rm SinhVienZone.com https://fb.com/sinhvienzonevn 288 Convex Analysis where the constants are independent of ε By the previous step, p−1 p−1 |z − w| , + |w| |fε (z) − fε (w)| ≤ C + |z| and since fε (z) → f (z) pointwise as ε → 0+ (see Theorem 2.75) we obtain the desired result for f We conclude this subsection by proving that differentiable separately convex functions are of class C Theorem 4.65 Let B ⊂ Rm be an open ball If f : B → R is separately convex and E is the set of points in B at which f is differentiable, then ∇f : E → Rm is continuous Proof Let z0 ∈ E and define h (z) := f (z) − f (z0 ) − ∇f (z0 ) · (z − z0 ) , z ∈ B om Then the function h is separately convex and differentiable in E By Theorem 4.36, C |∇f (z) − ∇f (z0 )| = |∇h (z)| ≤ Lip (h; B (z0 , r)) ≤ lim sup ne for any z ∈ E with |z − z0 | < r and with B (z0 , 2r) ⊂ B Since f is differentiable at z0 we have that r→0+ z∈E, |z−z0 | such that B (w + z, r) ∩ C = ∅ However, w + zn ∈ C for every n and thus om < r ≤ dist (w + z, C) ≤ |(w + z) − (w + zn )| → 0, ne C and this is a contradiction Similarly, in part (iii), if z = limn→∞ θn wn , with wn ∈ C and θn → 0+ , and z were not in C ∞ , then there would exist w ∈ C and r > such that B (w + z, r) ∩ C = ∅ If n is sufficiently large we may assume that θn ∈ [0, 1], and so (1 − θn ) w + θn wn ∈ C by convexity of C, with Zo w + z = lim {(1 − θn ) w + θn wn + θn w} n→∞ en Once again we would reach a contradiction because < r ≤ dist (w + z, C) ≤ |θn w| → Conversely, if z ∈ C ∞ and w0 ∈ C, then nz + w0 = wn ∈ C for all n ∈ N, and so θn wn → z with θn := n1 nh Vi Si 289 Remark 4.68 It can be shown that the notion of recession cone of an arbitrary set E ⊂ Rm introduced as E ∞ := z ∈ Rm : there exist wn ∈ E, θn → 0+ such that θn wn → z inherits several properties of the recession cone of a convex set (see [RocWe98]) Note, however, that E ∞ defined in this way is always closed, which is in contrast to Definition 4.66 (see Theorem 4.67 (iii)) We now introduce the concept of recession function SinhVienZone.com https://fb.com/sinhvienzonevn 290 Convex Analysis Definition 4.69 Let f : Rm → (−∞, ∞] be a proper convex function The recession function of f is the function f ∞ : Rm → [−∞, ∞] defined by f ∞ (z) := sup {f (w + z) − f (w) : w ∈ dome f } , The next result relates the recession function of a convex function to the recession cone of its epigraph Theorem 4.70 Let f : Rm → (−∞, ∞] be a proper convex function The recession function f ∞ of f is a positively homogeneous proper convex function and ∞ (4.29) epi f ∞ = (epi f ) Moreover, if f is lower semicontinuous, then so is f ∞ , and for every w ∈ dome f we have t>0 f (w + tz) − f (w) f (w + tz) − f (w) = lim t→∞ t t Proof We begin by showing that om f ∞ (z) = sup epi f ∞ = (epi f ) ∞ (4.30) (4.31) ∞ ne (epi f ) C Note that in view of Propositions 4.22 and 4.67(i) this entails that epi f ∞ is convex, and so f ∞ is convex Convexity of f ∞ also follows from the fact that f ∞ is the supremum of a family of convex functions By Proposition 4.67(i) we have = { (z, t) ∈ Rm × R : (w, s) + (z, t) ∈ epi f Zo for all (w, s) ∈ epi f } = { (z, t) ∈ Rm × R : f (w + z) ≤ s + t for all w ∈ dome f and all s ∈ R such that s ≥ f (w)} en = { (z, t) ∈ Rm × R : f (w + z) ≤ f (w) + t for all w ∈ dome f } = { (z, t) ∈ Rm × R : f ∞ (z) ≤ t} = epi f ∞ Since for a fixed w ∈ dome f , nh Vi Si z ∈ Rm f ∞ (z) ≥ f (w + z) − f (w) for all z ∈ Rm , we have that f ∞ never takes the value −∞ On the other hand, f ∞ is not identically ∞ because f ∞ (0) = Hence f ∞ is proper To prove that f ∞ is positively homogeneous let z ∈ Rm and t > We first claim that (4.32) tf ∞ (z) ≥ f ∞ (tz) If f ∞ (z) = ∞ there is nothing to prove Thus assume that f ∞ (z) < ∞ Then (z, f ∞ (z)) ∈ epi f ∞ , and so by (4.29) and the definition of recession cone, SinhVienZone.com https://fb.com/sinhvienzonevn 4.6 Recession Function t (z, f ∞ (z)) + epi f ⊂ epi f In particular, t (z, f ∞ (z)) + (w, f (w)) ∈ epi f for all w ∈ dome f , and thus f (w + tz) ≤ f (w) + tf ∞ (z), or equivalently, (4.33) f (w + tz) − f (w) ≤ tf ∞ (z) for all w ∈ dome f Taking the supremum on the left-hand side over all w ∈ dome f , it follows that f ∞ (tz) ≤ tf ∞ (z) om for all z ∈ Rm and t > To prove the reverse inequality it suffices to replace z and t in the previous inequality with tz and 1t Assume next that f is lower semicontinuous Note that by (4.31) and Proposition 4.67(ii), epi f ∞ is closed, and thus by Proposition 3.10, f ∞ is lower semicontinuous To prove (4.30), in view of Proposition 4.34 it suffices to establish the equality f (w + tz) − f (w) f ∞ (z) = sup t t>0 ne C for all z ∈ Rm and for any fixed w ∈ dome f Fix w0 ∈ dome f We show first that f (w0 + tz) − f (w0 ) f ∞ (z) ≥ t for all z ∈ Rm Reasoning as in the proof of (4.32) we have that (4.33) holds for all w ∈ dome f and for every t ≥ In particular, Zo f (w0 + tz) ≤ f (w0 ) + tf ∞ (z) en for all t ≥ and z ∈ Rm , which gives sup t>0 f (w0 + tz) − f (w0 ) ≤ f ∞ (z) t (4.34) for all z ∈ Rm Conversely, if for z ∈ Rm , nh Vi Si 291 sup t>0 f (w0 + tz) − f (w0 ) = ∞, t then there is nothing to prove; otherwise, let s≥ f (w0 + tz) − f (w0 ) t for all t > Then st + f (w0 ) ≥ f (w0 + tz) for all t > 0, and so (w0 + tz, st + f (w0 )) ∈ epi f SinhVienZone.com https://fb.com/sinhvienzonevn (4.35) 292 Convex Analysis for all t > Write (z, s) = lim n→∞ ((w0 , f (w0 )) + n (z, s)) n ∞ By (4.35) and Proposition 4.67(iii), it follows that (z, s) ∈ (epi f ) , and now invoking (4.31), we conclude that f ∞ (z) ≤ s Given the arbitrariness of s, we deduce that f (w0 + tz) − f (w0 ) f ∞ (z) ≤ sup t t>0 This, together with (4.34), yields (4.30) Remark 4.71 For nonconvex functions it is also possible to define the recession ∞ function using formula (4.29), and where (epi f ) is given as in Remark 4.68 (see [RocWe98]) We call attention, however, to the fact that the recession function introduced in this way is always lower semicontinuous, in contrast to the case of a convex function (see Theorem 4.70) Exercise 4.72 Prove that: p om (i) If ≤ p < ∞, then the recession function of f (z) := |z| is ∞ if z = 0, f ∞ (z) = if z = 0, C for p > 1, while f ∞ (z) = f (z) = |z| if p = 1; |z| + is f ∞ (z) = |z|; ne (ii) the recession function of f (z) := (iii) the recession function of f (z) := Az · z, en Zo where A is a symmetric positive semidefinite matrix in Rm×m , is ∞ if Az = 0, f ∞ (z) = if Az = The next result shows that the recession function is of interest only for convex functions that are not superlinear at infinity nh Vi Theorem 4.73 Let f : Rm → (−∞, ∞] be a proper convex lower semicontinuous function Then lim inf |z|→∞ f (z) = inf f ∞ (z) |z| |z|=1 Si Proof By (4.29), inf f ∞ (z) = inf inf {t ∈ R : (z, t) ∈ epi f ∞ } |z|=1 |z|=1 ∞ = inf inf {t ∈ R : (z, t) ∈ (epi f ) } |z|=1 SinhVienZone.com https://fb.com/sinhvienzonevn 4.7 Approximation of Convex Functions Since f is convex and lower semicontinuous, by Propositions 3.10 and 4.22, epi f is closed and convex, and by Theorem 4.67 (iii) we have ∞ inf f ∞ (z) = inf inf {t ∈ R : (z, t) ∈ (epi f ) } |z|=1 = inf inf t ∈ R : there are (wn , tn ) ∈ epi f , θn → 0+ , |z|=1 |z|=1 such that θn (wn , tn ) → (z, t)} = inf t ∈ R : there are (wn , tn ) ∈ epi f , θn → 0+ , such that |θn wn | → 1, θn tn → t} om = inf t ∈ R : there are wn ∈ Rm , θn → 0+ , such that |θn wn | → 1, θn f (wn ) → t} f (wn ) m →t = inf t ∈ R : there is wn ∈ R , such that |wn | → ∞, |wn | f (z) = lim inf |z|→∞ |z| This concludes the proof .C Remark 4.74 The previous theorem continues to hold for nonconvex functions provided f ∞ is defined as in Remark 4.71 ne 4.7 Approximation of Convex Functions en Zo In this section we show that every convex lower semicontinuous function f may be written as the supremum of a sequence of affine functions If V is a topological vector space, an affine continuous function g : V → R is a function of the form g (v) = α + v , v V ,V , where v ∈ V and α ∈ R Proposition 4.75 Let V be a locally convex topological vector space and let f : V → (−∞, ∞] be a convex and lower semicontinuous function Then nh Vi Si 293 (i) there exists an affine continuous function g such that g ≤ f ; (ii) f (v) = sup {g (v) : g affine continuous, g ≤ f } Proof (i) If f ≡ ∞, then the result is immediate So assume that there exists v0 ∈ dome f , and fix t0 < f (v0 ) By the second geometric form of the Hahn– Banach theorem, with C := epi f , K := {(v0 , t0 )}, there exist a continuous linear functional L : V × R → R and two numbers α ∈ R and ε > such that L (v, f (v)) ≥ α + ε SinhVienZone.com for all v ∈ dome f and L (v0 , t0 ) ≤ α − ε https://fb.com/sinhvienzonevn ...Si nh Vi en Zo ne C om Modern Methods in the Calculus p of Variations: L Spaces SinhVienZone. com https://fb .com/ sinhvienzonevn Irene Fonseca Giovanni Leoni Si nh Vi en Zo ne C om Modern Methods. .. Methods in the Calculus p of Variations: L Spaces SinhVienZone. com https://fb .com/ sinhvienzonevn Giovanni Leoni Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213... in the one-dimensional setting for integrands f = f (x, u, u ), we work in N dimensions and no derivatives are present In addition, it is self-contained in SinhVienZone. com https://fb .com/ sinhvienzonevn