Ebook Functional analysis, sobolev spaces and partial differential equations Part 1

(BQ) Part 1 book Functional analysis, sobolev spaces and partial differential equations has contents: The hahn–banach theorems introduction to the theory of conjugate convex functions; the uniform boundedness principle and the closed graph theorem; compact operators spectral decomposition of self adjoint compact operators,...and other contents.

Universitext For other titles in this series, go to www.springer.com/series/223 Haim Brezis Functional Analysis, Sobolev Spaces and Partial Differential Equations 1C Haim Brezis Distinguished Professor Department of Mathematics Rutgers University Piscataway, NJ 08854 USA brezis@math.rutgers.edu and Professeur émérite, Université Pierre et Marie Curie (Paris 6) and Visiting Distinguished Professor at the Technion Editorial board: Sheldon Axler, San Francisco State University Vincenzo Capasso, Università degli Studi di Milano Carles Casacuberta, Universitat de Barcelona Angus MacIntyre, Queen Mary, University of London Kenneth Ribet, University of California, Berkeley Claude Sabbah, CNRS, École Polytechnique Endre Süli, University of Oxford Wojbor Woyczyński, Case Western Reserve University ISBN 978-0-387-70913-0 e-ISBN 978-0-387-70914-7 DOI 10.1007/978-0-387-70914-7 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010938382 Mathematics Subject Classification (2010): 35Rxx, 46Sxx, 47Sxx © Springer Science+Business Media, LLC 2011 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 is part of Springer Science+Business Media (www.springer.com) To Felix Browder, a mentor and close friend, who taught me to enjoy PDEs through the eyes of a functional analyst Preface This book has its roots in a course I taught for many years at the University of Paris It is intended for students who have a good background in real analysis (as expounded, for instance, in the textbooks of G B Folland [2], A W Knapp [1], and H L Royden [1]) I conceived a program mixing elements from two distinct “worlds”: functional analysis (FA) and partial differential equations (PDEs) The first part deals with abstract results in FA and operator theory The second part concerns the study of spaces of functions (of one or more real variables) having specific differentiability properties: the celebrated Sobolev spaces, which lie at the heart of the modern theory of PDEs I show how the abstract results from FA can be applied to solve PDEs The Sobolev spaces occur in a wide range of questions, in both pure and applied mathematics They appear in linear and nonlinear PDEs that arise, for example, in differential geometry, harmonic analysis, engineering, mechanics, and physics They belong to the toolbox of any graduate student in analysis Unfortunately, FA and PDEs are often taught in separate courses, even though they are intimately connected Many questions tackled in FA originated in PDEs (for a historical perspective, see, e.g., J Dieudonné [1] and H Brezis–F Browder [1]) There is an abundance of books (even voluminous treatises) devoted to FA There are also numerous textbooks dealing with PDEs However, a synthetic presentation intended for graduate students is rare and I have tried to fill this gap Students who are often fascinated by the most abstract constructions in mathematics are usually attracted by the elegance of FA On the other hand, they are repelled by the neverending PDE formulas with their countless subscripts I have attempted to present a “smooth” transition from FA to PDEs by analyzing first the simple case of onedimensional PDEs (i.e., ODEs—ordinary differential equations), which looks much more manageable to the beginner In this approach, I expound techniques that are possibly too sophisticated for ODEs, but which later become the cornerstones of the PDE theory This layout makes it much easier for students to tackle elaborate higher-dimensional PDEs afterward A previous version of this book, originally published in 1983 in French and followed by numerous translations, became very popular worldwide, and was adopted as a textbook in many European universities A deficiency of the French text was the vii viii Preface lack of exercises The present book contains a wealth of problems I plan to add even more in future editions I have also outlined some recent developments, especially in the direction of nonlinear PDEs Brief user’s guide Statements or paragraphs preceded by the bullet symbol • are extremely important, and it is essential to grasp them well in order to understand what comes afterward Results marked by the star symbol can be skipped by the beginner; they are of interest only to advanced readers In each chapter I have labeled propositions, theorems, and corollaries in a continuous manner (e.g., Proposition 3.6 is followed by Theorem 3.7, Corollary 3.8, etc.) Only the remarks and the lemmas are numbered separately In order to simplify the presentation I assume that all vector spaces are over R Most of the results remain valid for vector spaces over C I have added in Chapter 11 a short section describing similarities and differences Many chapters are followed by numerous exercises Partial solutions are presented at the end of the book More elaborate problems are proposed in a separate section called “Problems” followed by “Partial Solutions of the Problems.” The problems usually require knowledge of material coming from various chapters I have indicated at the beginning of each problem which chapters are involved Some exercises and problems expound results stated without details or without proofs in the body of the chapter Acknowledgments During the preparation of this book I received much encouragement from two dear friends and former colleagues: Ph Ciarlet and H Berestycki I am very grateful to G Tronel, M Comte, Th Gallouet, S Guerre-Delabrière, O Kavian, S Kichenassamy, and the late Th Lachand-Robert, who shared their “field experience” in dealing with students S Antman, D Kinderlehrer, and Y Li explained to me the background and “taste” of American students C Jones kindly communicated to me an English translation that he had prepared for his personal use of some chapters of the original French book I owe thanks to A Ponce, H.-M Nguyen, H Castro, and H Wang, who checked carefully parts of the book I was blessed with two extraordinary assistants who typed most of this book at Rutgers: Barbara Miller, who is retired, and now Barbara Mastrian I not have enough words of praise and gratitude for their constant dedication and their professional help They always found attractive solutions to the challenging intricacies of PDE formulas Without their enthusiasm and patience this book would never have been finished It has been a great pleasure, as Preface ix ever, to work with Ann Kostant at Springer on this project I have had many opportunities in the past to appreciate her long-standing commitment to the mathematical community The author is partially supported by NSF Grant DMS-0802958 Haim Brezis Rutgers University March 2010 186 But3 The Hille–Yosida Theorem d |u(t) − u(t)|2 = dt d (u(t) − u(t)), u(t) − u(t) dt Thus, the function t → |u(t) − u(t)| is nonincreasing on [0, +∞) Since |u(0) − u(0)| = 0, it follows that |u(t) − u(t)| = ∀t ≥ The main idea in order to prove existence is to replace A by Aλ in (6), to apply Theorem 7.3 on the approximate problem, and then to pass to the limit as λ → using various estimates that are independent of λ So, let uλ be the solution of the problem ⎧ ⎨ duλ + Aλ uλ = on [0, +∞), (7) dt ⎩u (0) = u ∈ D(A) λ Step 2: We have the estimates |uλ (t)| ≤ |u0 | ∀t ≥ 0, (8) ∀λ > 0, duλ (t) = |Aλ uλ (t)| ≤ |Au0 | ∀t ≥ 0, dt (9) ∀λ > They follow directly from the next lemma and the fact that |Aλ u0 | ≤ |Au0 | Lemma 7.1 Let w ∈ C ([0, +∞); H ) be a function satisfying (10) dw + Aλ w = on [0, +∞) dt Then the functions t → |w(t)| and t → [0, +∞) = |Aλ w(t)| are nonincreasing on dw dt (t) Proof We have dw , w + (Aλ w, w) = dt d By Proposition 7.2(e) we know that (Aλ w, w) ≥ and thus 21 dt |w|2 ≤ 0, so that |w(t)| is nonincreasing On the other hand, since Aλ is a linear bounded operator, we deduce (by induction) from (10) that w ∈ C ∞ ([0, +∞); H ) and also that d dt dw dt Applying the preceding fact to any order k, the function dk w dt k dw dt , + Aλ dw dt we see that = dw dt (t) is nonincreasing In fact, at (t) is nonincreasing Keep in mind that if ϕ ∈ C ([0, +∞); H ), then |ϕ|2 ∈ C ([0, +∞); R) and d dt |ϕ| = 2( dϕ dt , ϕ) 7.2 Solution of the Evolution Problem 187 Step 3: We will prove here that for every t ≥ 0, uλ (t) converges, as λ → 0, to some limit, denoted by u(t) Moreover, the convergence is uniform on every bounded interval [0, T ] For every λ, μ > we have duμ duλ − + Aλ uλ − Aμ uμ = dt dt and thus (11) d |uλ (t) − uμ (t)|2 + (Aλ uλ (t) − Aμ uμ (t), uλ (t) − uμ (t)) = dt Dropping t for simplicity, we write (Aλ uλ − Aμ uμ ,uλ − uμ ) (12) = (Aλ uλ − Aμ uμ , uλ − Jλ uλ + Jλ uλ − Jμ uμ + Jμ uμ − uμ ) = (Aλ uλ − Aμ uμ , λAλ uλ − μAμ uμ ) + (A(Jλ uλ − Jμ uμ ), Jλ uλ − Jμ uμ ) ≥ (Aλ uλ − Aμ uμ , λAλ uλ − μAμ uμ ) It follows from (9), (11), and (12) that d |uλ − uμ |2 ≤ 2(λ + μ)|Au0 |2 dt Integrating this inequality, we obtain |uλ (t) − uμ (t)|2 ≤ 4(λ + μ)t|Au0 |2 , i.e., (13) |uλ (t) − uμ (t)| ≤ (λ + μ)t|Au0 | It follows that for every fixed t ≥ 0, uλ (t) is a Cauchy sequence as λ → and thus it converges to a limit, denoted by u(t) Passing to the limit in (13) as μ → 0, we have √ |uλ (t) − u(t)| ≤ λt|Au0 | Therefore, the convergence is uniform in t on every bounded interval [0, T ] and so u ∈ C([0, +∞); H ) Step 4: Assuming, in addition, that u0 ∈ D(A2 ), i.e., u0 ∈ D(A) and Au0 ∈ D(A), λ we prove here that du dt (t) converges, as λ → 0, to some limit and that the convergence is uniform on every bounded interval [0, T ] dvλ λ Set vλ = du dt , so that dt + Aλ vλ = Following the same argument as in Step 3, we see that 188 (14) The Hille–Yosida Theorem d |vλ − vμ |2 ≤ (|Aλ vλ | + |Aμ vμ |)(λ|Aλ vλ | + μ|Aμ vμ |) dt By Lemma 7.1 we have |Aλ vλ (t)| ≤ |Aλ vλ (0)| = |Aλ Aλ u0 | (15) and similarly (16) |Aμ vμ (t)| ≤ |Aμ vμ (0)| = |Aμ Aμ u0 | Finally, since Au0 ∈ D(A), we obtain Aλ Aλ u0 = Jλ AJλ Au0 = Jλ Jλ AAu0 = Jλ2 A2 u0 and thus (17) |Aλ Aλ u0 | ≤ |A2 u0 |, |Aμ Aμ u0 | ≤ |A2 u0 | Combining (14), (15), (16), and (17), we are led to d |vλ − vμ |2 ≤ 2(λ + μ)|A2 u0 |2 dt λ We conclude, as in Step 3, that vλ (t) = du dt (t) converges, as λ → 0, to some limit and that the convergence is uniform on every bounded interval [0, T ] Step 5: Assuming that u0 ∈ D(A2 ) we prove here that u is a solution of (6) By Steps and we know that for all T < ∞, ⎧ ⎨uλ (t) → u(t), as λ → 0, uniformly on [0, T ], du ⎩ λ (t) converges, as λ → 0, uniformly on [0, T ] dt It follows easily that u ∈ C ([0, +∞); H ) and that uniformly on [0, T ] Rewrite (7) as (18) duλ dt (t) → du dt (t), as λ → 0, duλ (t) + A(Jλ uλ (t)) = dt Note that Jλ uλ (t) → u(t) as λ → 0, since |Jλ uλ (t) − u(t)| ≤ |Jλ uλ (t) − Jλ u(t)| + |Jλ u(t) − u(t)| ≤ |uλ (t) − u(t)| + |Jλ u(t) − u(t)| → Applying the fact that A has a closed graph, we deduce from (18) that u(t) ∈ D(A) ∀t ≥ 0, and that du (t) + Au(t) = dt 7.2 Solution of the Evolution Problem 189 Finally, since u ∈ C ([0, +∞); H ), the function t → Au(t) is continuous from [0, +∞) into H and thus u ∈ C([0, +∞); D(A)) Hence we have obtained a solution of (6) satisfying, in addition, |u(t)| ≤ |u0 | ∀t ≥ and du (t) = |Au(t)| ≤ |Au0 | ∀t ≥ dt Step 6: We conclude here the proof of the theorem We shall use the following lemma Lemma 7.2 Let u0 ∈ D(A) Then ∀ε > ∃ u0 ∈ D(A2 ) such that |u0 − u0 | < ε and |Au0 − Au0 | < ε In other words, D(A2 ) is dense in D(A) (for the graph norm) Proof of Lemma 7.2 Set u0 = Jλ u0 for some appropriate λ > to be fixed later We have u0 ∈ D(A) and u0 + λAu0 = u0 Thus Au0 ∈ D(A), i.e., u0 ∈ D(A2 ) On the other hand, by Proposition 7.2, we know that lim |Jλ u0 − u0 | = 0, lim |Jλ Au0 − Au0 | = 0, λ→0 λ→0 and Jλ Au0 = AJλ u0 The desired conclusion follows by choosing λ > small enough We now turn to the proof of Theorem 7.4 Given u0 ∈ D(A) we construct (using Lemma 7.2) a sequence (u0n ) in D(A2 ) such that u0n → u0 and Au0n → Au0 By Step we know that there is a solution un of the problem ⎧ ⎨ dun + Aun = on [0, +∞), (19) dt ⎩u (0) = u n 0n We have, for all t ≥ 0, |un (t) − um (t)| ≤ |u0n − u0m | −→ 0, m,n→∞ dun dum (t) − (t) ≤ |Au0n − Au0m | −→ m,n→∞ dt dt Therefore un (t) → u(t) uniformly on [0, +∞), dun du (t) → (t) uniformly on [0, +∞), dt dt with u ∈ C ([0, +∞); H ) Passing to the limit in (19)—using the fact that A is a closed operator—we see that u(t) ∈ D(A) and u satisfies (6) From (6) we deduce that u ∈ C([0, +∞); D(A)) 190 The Hille–Yosida Theorem Remark Let uλ be the solution of (7): (a) Assume u0 ∈ D(A) We know (by Step 3) that as λ → 0, uλ (t) converges, for every t ≥ 0, to some limit u(t) One can prove directly that u ∈ C ([0, +∞); H ) ∩ C([0, +∞); D(A)) and that it satisfies (6) (b) Assume only that u0 ∈ H One can still prove that as λ → 0, uλ (t) converges for every t ≥ 0, to some limit, denoted by u(t) But it may happen that this limit u(t) does not belong to D(A) ∀t > and that u(t) is nowhere differentiable on [0, +∞) Hence u(t) is not a “classical” solution of (6) In fact, for such a u0 , problem (6) has no classical solution Nevertheless, we may view u(t) as a “generalized” solution of (6) We shall see in Section 7.4 that this does not happen when A is self-adjoint: in this case u(t) is a “classical” solution of (6) for every u0 ∈ H , even when u0 ∈ / D(A) Remark (Contraction semigroups) For each t ≥ consider the linear map u0 ∈ D(A) → u(t) ∈ D(A), where u(t) is the solution of (6) given by Theorem 7.4 Since |u(t)| ≤ |u0 | and since D(A) is dense in H , we may extend this map by continuity as a bounded operator from H into itself, denoted by SA (t).4 It is easy to check that SA (t) satisfies the following properties: (a) for each t ≥ 0, SA (t) ∈ L(H ) and SA (t) L(H ) ≤ 1, SA (t1 + t2 ) = SA (t1 ) ◦ SA (t2 ) SA (0) = I, (b) (c) lim |SA (t)u0 − u0 | = t→0 t>0 ∀t1 , t2 ≥ 0, ∀u0 ∈ H Such a family {S(t)}t≥0 of operators (from H into itself) depending on a parameter t ≥ and satisfying (a), (b), (c) is called a continuous semigroup of contractions A remarkable result due to Hille andYosida asserts that conversely, given a continuous semigroup of contractions S(t) on H there exists a unique maximal monotone operator A such that S(t) = SA (t) ∀t ≥ This establishes a bijective correspondence between maximal monotone operators and continuous semigroups of contractions (For a proof see the references quoted in the comments on Chapter 7.) • Remark Let A be a maximal monotone operator and let λ ∈ R The problem ⎧ ⎨ du + Au + λu = on [0, +∞), dt ⎩u(0) = u , reduces to problem (6) using the following simple device Set v(t) = eλt u(t) Then v satisfies Alternatively one may use Remark 4(b) to define S u(t) ∈ H A (t) on H directly as being the map u0 ∈ H → 7.3 Regularity 191 ⎧ ⎨ dv + Av = dt ⎩v(0) = u on [0, +∞), 7.3 Regularity We shall prove here that the solution u of (6) obtained in Theorem 7.4 is more regular than just C ([0, +∞); H ) ∩ C([0, +∞); D(A)) provided one makes additional assumptions on the initial data u0 For this purpose we define by induction the space D(Ak ) = {v ∈ D(Ak−1 ); Av ∈ D(Ak−1 )}, where k is any integer, k ≥ It is easily seen that D(Ak ) is a Hilbert space for the scalar product k (u, v)D(Ak ) = (Aj u, Aj v); j =0 the corresponding norm is ⎛ k |u|D(Ak ) = ⎝ ⎞1/2 |Aj u|2 ⎠ j =0 Theorem 7.5 Assume u0 ∈ D(Ak ) for some integer k ≥ Then the solution u of problem (6) obtained in Theorem 7.4 satisfies u ∈ C k−j ([0, +∞); D(Aj )) ∀j = 0, 1, , k Proof Assume first that k = Consider the Hilbert space H1 = D(A) equipped with the scalar product (u, v)D(A) It is easy to check that the operator A1 : D(A1 ) ⊂ H1 → H1 defined by D(A1 ) = D(A2 ), A1 u = Au for u ∈ D(A1 ), is maximal monotone in H1 Applying Theorem 7.4 to the operator A1 in the space H1 , we see that there exists a function u ∈ C ([0, +∞); H1 ) ∩ C([0, +∞); D(A1 )) such that ⎧ ⎨ du + A1 u = dt ⎩u(0) = u on [0, +∞), In particular, u satisfies (6); by uniqueness, this u is the solution of (6) It remains only to check that u ∈ C ([0, +∞); H ) Since 192 The Hille–Yosida Theorem A ∈ L(H1 , H ) and u ∈ C([0, +∞); H1 ), it follows that Au ∈ C ([0, +∞); H ) and du d (Au) = A dt dt (20) Applying (6), we see that and that (21) d dt du dt du dt ∈ C ([0, +∞); H ), i.e., u ∈ C ([0, +∞); H ) +A du dt =0 on [0, +∞) We now turn to the general case k ≥ We argue by induction on k: assume that the result holds up to order (k − 1) and let u0 ∈ D(Ak ) By the preceding analysis we know that the solution u of (6) belongs to C ([0, +∞); H ) ∩ C ([0, +∞); D(A)) and that u satisfies (21) Letting du v= , dt we have v ∈ C ([0, +∞); H ) ∩ C([0, +∞); D(A)), ⎧ ⎨ dv + Av = on [0, +∞), dt ⎩v(0) = −Au In other words, v is the solution of (6) corresponding to the initial data v0 = −Au0 Since v0 ∈ D(Ak−1 ), we know, by the induction assumption, that (22) v ∈ C k−1−j ([0, +∞); D(Aj )) ∀j = 0, 1, , k − 1, i.e., u ∈ C k−j ([0, +∞); D(Aj )) ∀j = 0, 1, , k − It remains only to check that (23) u ∈ C([0, +∞); D(Ak )) Applying (22) with j = k − 1, we see that (24) du ∈ C([0, +∞); D(Ak−1 )) dt It follows from (24) and equation (6) that Au ∈ C([0, +∞); D(Ak−1 )), i.e., (23) 7.4 The Self-Adjoint Case 193 7.4 The Self-Adjoint Case Let A : D(A) ⊂ H → H be an unbounded linear operator with D(A) = H Identifying H with H , we may view A as an unbounded linear operator in H Definition One says that • • A is symmetric if (Au, v) = (u, Av) ∀u, v ∈ D(A), A is self-adjoint if D(A ) = D(A) and A = A Remark For bounded operators the notions of symmetric and self-adjoint operators coincide However, if A is unbounded there is a subtle difference between symmetric and self-adjoint operators Clearly, any self-adjoint operator is symmetric The converse is not true: an operator A is symmetric if and only if A ⊂ A , i.e., D(A) ⊂ D(A ) and A = A on D(A) It may happen that A is symmetric and that D(A) = D(A ) Our next result shows that if A is maximal monotone, then (A is symmetric) ⇔ (A is self-adjoint) Proposition 7.6 Let A be a maximal monotone symmetric operator Then A is selfadjoint Proof Let J1 = (I + A)−1 We will first prove that J1 is self-adjoint Since J1 ∈ L(H ) it suffices to check that (25) (J1 u, v) = (u, J1 v) ∀u, v ∈ H Set u1 = J1 u and v1 = J1 v, so that u1 + Au1 = u, v1 + Av1 = v Since by assumption, (u1 , Av1 ) = (Au1 , v1 ), it follows that (u1 , v) = (u, v1 ), i.e., (25) Let u ∈ D(A ) and set f = u + A u We have (f, v) = (u, v + Av) ∀v ∈ D(A), i.e., (f, J1 w) = (u, w) ∀w ∈ H Therefore u = J1 f and thus u ∈ D(A) This proves that D(A ) = D(A) and hence A is self-adjoint Remark One has to be careful that if A is a monotone operator (even a symmetric monotone operator) then A need not be monotone However, one can prove that the following properties are equivalent: 194 The Hille–Yosida Theorem A is maximal monotone ⇐⇒ A is maximal monotone ⇐⇒ A is closed, D(A) is dense, A and A are monotone A more general version of this result appears in Problem 16 • Theorem 7.7 Let A be a self-adjoint maximal monotone operator Then for every u0 ∈ H there exists a unique function5 u ∈ C([0, +∞); H ) ∩ C ((0, +∞); H ) ∩ C((0, +∞); D(A)) ⎧ ⎨ du + Au = dt ⎩u(0) = u such that on (0, +∞), Moreover, we have |u(t)| ≤ |u0 | and (26) du (t) = |Au(t)| ≤ |u0 | ∀t > 0, dt t u ∈ C k ((0, +∞); D(A )) ∀k, integers Proof Uniqueness Let u and u be two solutions By the monotonicity of A we see that ϕ(t) = |u(t) − u(t)|2 is nonincreasing on (0, +∞) On the other hand, ϕ is continuous on [0, +∞) and ϕ(0) = Thus ϕ ≡ Existence The proof is divided into two steps: Step Assume first that u0 ∈ D(A2 ) and let u be the solution of (6) given by Theorem 7.4 We claim that (27) du (t) ≤ |u0 | ∀t > dt t As in the proof of Proposition 7.6 we have Jλ = Jλ and Aλ = Aλ ∀λ > We go back to the approximate problem introduced in the proof of Theorem 7.4: (28) duλ + Aλ uλ = on [0, +∞), uλ (0) = u0 dt Taking the scalar product of (28) with uλ and integrating on [0, T ], we obtain Let us emphasize the difference between Theorems 7.4 and 7.7 Here u0 ∈ H (instead of u0 ∈ D(A)); the conclusion is that there is a solution of (6), which is smooth away from t = However, | du dt (t)| may possibly “blow up” as t → 7.4 The Self-Adjoint Case 195 T |uλ (T )|2 + (29) (Aλ uλ , uλ )dt = |u0 |2 λ Taking the scalar product of (28) with t du dt and integrating over [0, T ], we obtain T (30) duλ (t) t dt + dt T Aλ uλ (t), duλ (t) t dt = dt But d duλ duλ , uλ + Aλ uλ , (Aλ uλ , uλ ) = Aλ dt dt dt = Aλ uλ , duλ dt , since Aλ = Aλ Integrating the second integral in (30) by parts, we are led to T Aλ uλ (t), (31) duλ (t) t dt = dt T d [(Aλ uλ , uλ )]t dt dt 1 = (Aλ uλ (T ), uλ (T )) T − 2 T (Aλ uλ , uλ ) dt λ On the other hand, since the function t → | du dt (t)| is nonincreasing (by Lemma 7.1), we have duλ duλ (t) t dt ≥ (T ) dt dt T (32) T2 Combining (29), (30), (31), and (32), we obtain duλ |uλ (T )|2 + T (Aλ uλ (T ), uλ (T )) + T (T ) dt ≤ |u0 |2 ; it follows, in particular, that (33) duλ (T ) ≤ |u0 | ∀T > dt T Finally, we pass to the limit in (33) as λ → This completes the proof of (27), since duλ du dt → dt (see Step in the proof of Theorem 7.4) Step Assume now that u0 ∈ H Let (u0n ) be a sequence in D(A2 ) such that u0n → u0 (recall that D(A2 ) is dense in D(A) and that D(A) is dense in H ; thus D(A2 ) is dense in H ) Let un be the solution of ⎧ ⎨ dun + Aun = on [0, +∞), dt ⎩u (0) = u n We know (by Theorem 7.4) that 0n 196 The Hille–Yosida Theorem |un (t) − um (t)| ≤ |u0n − u0m | ∀m, n, ∀t ≥ 0, and (by Step 1) that dun dum (t) − (t) ≤ |u0n − u0m | dt dt t ∀m, n, ∀t > n It follows that un converges uniformly on [0, +∞) to some limit u(t) and that du dt (t) du converges to dt (t) uniformly on every interval [δ, +∞), δ > The limiting function u satisfies u ∈ C([0, +∞); H ) ∩ C ((0, +∞); H ), u(t) ∈ D(A) ∀t > and du (t) + Au(t) = dt ∀t > (this uses the fact that A is closed) We now turn to the proof of (26) We will show by induction on k ≥ that (34) u ∈ C k−j ((0, +∞); D(Aj )) ∀j = 0, 1, , k Assume that (34) holds up to order k − In particular, we have (35) u ∈ C((0, +∞); D(Ak−1 )) In order to prove (34) it suffices (in view of Theorem 7.5) to check that (36) u ∈ C((0, +∞), D(Ak )) ˜ ⊂ H˜ → H˜ Consider the Hilbert space H˜ = D(Ak−1 ) and the operator A˜ : D(A) defined by ˜ = D(Ak ), D(A) A˜ = A It is easily seen that A˜ is maximal monotone and symmetric in H˜ ; thus it is selfadjoint Applying the first assertion of Theorem 7.7 in the space H˜ to the operator ˜ we obtain a unique solution v of the problem A, ⎧ ⎨ dv + Av = on (0, +∞), (37) dt ⎩v(0) = v , given any v0 ∈ H˜ Moreover, ˜ v ∈ C([0, +∞); H˜ ) ∩ C ((0, +∞); H˜ ) ∩ C((0, +∞); D(A)) Choosing v0 = u(ε)(ε > 0)—we already know by (35) that v0 ∈ H˜ —we conclude that u ∈ C((ε, +∞); D(Ak )), and this completes the proof of (36) 7.4 Comments on Chapter 197 Comments on Chapter The Hille–Yosida theorem in Banach spaces The Hille–Yosida theorem extends to Banach spaces The precise statement is the following Let E be a Banach space and let A : D(A) ⊂ E → E be an unbounded linear operator One says that A is m-accretive if D(A) = E and for every λ > 0, I + λA is bijective from D(A) onto E with (I + λA)−1 L(E) ≤ Theorem 7.8 (Hille–Yosida) Let A be m-accretive Then given any u0 ∈ D(A) there exists a unique function u ∈ C ([0, +∞); E) ∩ C([0, +∞); D(A)) such that ⎧ ⎨ du + Au = dt ⎩u(0) = u (38) on [0, +∞), Moreover, u(t) ≤ u0 and du (t) = Au(t) ≤ Au0 dt ∀t ≥ The map u0 → u(t) extended by continuity to all of E is denoted by SA (t) It is a continuous semigroup of contractions on E Conversely, given any continuous semigroup of contractions S(t), there exists a unique m-accretive operator A such that S(t) = SA (t) ∀t ≥ For the proof, see, e.g., P Lax [1], A Pazy [1], J Goldstein [1], E Davies [1], [2], K Yosida [1], M Reed–B Simon [1], Volume 2, H Tanabe [1], N Dunford– J T Schwartz [1] Volume 1, M Schechter [1], A Friedman [2], R Dautray–J.L Lions [1], Chapter XVII, A Balakrishnan [1], T Kato [1], W Rudin [1] These references present extensive developments on the theory of semigroups The exponential formula There are numerous iteration techniques for solving (38) Let us mention a basic method Theorem 7.9 Assume that A is m-accretive Then for every u0 ∈ D(A) the solution u of (38) is given by the “exponential formula” (39) u(t) = lim n→+∞ I+ t A n −1 n u0 For a proof see, e.g., K.Yosida [1] andA Pazy [1] Formula (39) corresponds, in the language of numerical analysis, to the convergence of an implicit time discretization scheme for (38) (see, e.g., K W Morton–D F Mayers [1]) More precisely, one 198 The Hille–Yosida Theorem divides the interval [0, t] into n intervals of equal length inductively the equations uj +1 − uj + Auj +1 = 0, t t = t/n and one solves j = 0, 1, , n − 1, starting with u0 In other words, un is given by tA)−n u0 = I + un = (I + As n → ∞ (i.e., t A n −n u0 t → 0) it is “intuitive” that un converges to u(t) Theorem 7.7 is a first step toward the theory of analytic semigroups On this subject see, e.g., K Yosida [1], T Kato [1], M Reed–B Simon [1], Volume 2, A Friedman [2], A Pazy [1], and H Tanabe [1] Inhomogeneous equations Nonlinear equations Consider, in a Banach space E, the problem ⎧ ⎨ du (t) + Au(t) = f (t) on [0, T ], (40) dt ⎩u(0) = u The following holds Theorem 7.10 Assume that A is m-accretive Then for every u0 ∈ D(A) and every f ∈ C ([0, T ]; E) there exists a unique solution u of (40) with u ∈ C ([0, T ]; E) ∩ C([0, T ]; D(A)) Moreover, u is given by the formula (41) t u(t) = SA (t)u0 + SA (t − s)f (s)ds, where SA (t) is the semigroup introduced in Comment Note that if one assumes just f ∈ L1 ((0, T ); E), formula (41) still makes sense and provides a generalized solution of (40) On these questions see, e.g., T Kato [1], A Pazy [1], R H Martin [1], H Tanabe [1] In physical applications one encounters many “semilinear” equations of the form du + Au = F (u), dt where F is a nonlinear map from E into E On these questions see, e.g., R H Martin [1], Th Cazenave–A Haraux [1], and the comments on Chapter 10 7.4 Comments on Chapter 199 Let us also mention that some results of Chapter have nonlinear extensions It is useful to consider nonlinear m-accretive operators A : D(A) ⊂ E → E On this subject, see, e.g., H Brezis [1] and V Barbu [1] ... satisfying (1) and (2) is sometimes called a Minkowski functional H Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, DOI 10 .10 07/978-0-387-70 914 -7 _1, © Springer... Folland [2], A W Knapp [1] , and H L Royden [1] ) I conceived a program mixing elements from two distinct “worlds”: functional analysis (FA) and partial differential equations (PDEs) The first part. .. www.springer.com/series/223 Haim Brezis Functional Analysis, Sobolev Spaces and Partial Differential Equations 1C Haim Brezis Distinguished Professor Department of Mathematics Rutgers University