H ANDBOOK OF D IFFERENTIAL E QUATIONS S TATIONARY PARTIAL D IFFERENTIAL E QUATIONS VOLUME V This page intentionally left blank H ANDBOOK OF D IFFERENTIAL E QUATIONS S TATIONARY PARTIAL D IFFERENTIAL E QUATIONS Volume V Edited by M CHIPOT Institute of Mathematics, University of Zürich, Zürich, Switzerland Amsterdam • Boston • Heidelberg • London • New York • Oxford Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo North-Holland is an imprint of Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands Linacre House, Jordan Hill, Oxford OX2 8DP, UK First edition 2008 Copyright © 2008 Elsevier B.V All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0)1865 853333; email: permissions@elsevier.com Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-444-53217-6 Set ISBN: 0-444-51743-x For information on all North-Holland publications visit our website at books.elsevier.com Printed and bound in Hungary 08 09 10 11 12 10 Preface This handbook is the fifth volume in the series devoted to stationary partial differential equations As the preceding volumes, it is a collection of self-contained, state-of-the-art surveys written by well-known experts in the field The topics covered by this volume include in particular semilinear and superlinear elliptic systems, the fibering method for nonlinear variational problems, some nonlinear eigenvalue problems, the studies of the stationary Boltzmann equation and the Gierer–Meinhardt system I hope that these surveys will be useful for both beginners and experts and help to the diffusion of these recent deep results in mathematical science I would like to thank all the contributors for their elegant articles I also thank Lauren Schultz Yuhasz and Mara Vos-Sarmiento at Elsevier for the excellent editing work of this volume M Chipot v This page intentionally left blank List of Contributors de Figueiredo, D.G., IMECC-Unicamp, C.P 6065 Campinas, S Paulo 13081-970, Brazil djairo@ime.unicamp.br (Ch 1) Pohozaev, S.I., Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina str 8, 119991 Moscow, Russia (Ch 2) Ruf, B., Dipartimento de Matematica, Università degli Studi di Milano, Via Saldini 50, I-20133 Milano, Italy (Ch 3) Suzuki, T., Division of Mathematical Science, Department of Systems Innovation, Graduate School of Science, Osaka University, Japan (Ch 4) Takahashi, F., Department of Mathematics, Faculty of Science, Osaka City University, Japan (Ch 4) Ukai, S., Department of Mathematics and Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong (Ch 5) Wei, J., Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong (Ch 6) Yang, T., Department of Mathematics and Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong (Ch 5) vii This page intentionally left blank Contents Preface List of Contributors Contents of Volume I Contents of Volume II Contents of Volume III Contents of Volume IV v vii xi xiii xv xvii Semilinear Elliptic Systems: Existence, Multiplicity, Symmetry of Solutions D.G de Figueiredo Nonlinear Variational Problems via the Fibering Method S.I Pohozaev Superlinear Elliptic Equations and Systems B Ruf Nonlinear Eigenvalue Problem with Quantization T Suzuki and F Takahashi Stationary Problem of Boltzmann Equation S Ukai and T Yang Existence and Stability of Spikes for the Gierer–Meinhardt System J Wei 49 211 277 371 487 Author Index 587 Subject Index 593 ix 584 J Wei [57] W.-M Ni and I Takagi, On the shape of least energy solution to a semilinear Neumann problem, Comm Pure Appl Math 41 (1991), 819–851 [58] W.-M Ni and I Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math J 70 (1993), 247–281 [59] W.-M Ni, I Takagi and E Yanagida, Stability analysis of point-condensation solutions to a reaction– diffusion system proposed by Gierer and Meinhardt, Tohoku Math J., in press [60] W.-M Ni, I Takagi and E Yanagida, Stability of least energy patterns of the shadow system for an activatorinhibitor mo del, Japan J Industr Appl Math 18 (2) (2001), 259–272 [61] W.-M Ni and J Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm Pure Appl Math 48 (1995), 731–768 [62] W.-M Ni and J Wei, On positive solutions concentrating on spheres for the Gierer–Meinhardt system, J Diff Eqns 221 (2006), 158–189 [63] J.E Pearson, Complex patterns in a simple system, Science 261 (1993), 189–192 [64] O Rey, The question of interior blow-up points for an elliptic Neumann problem: the critical case, J Math Pures Appl 81 (2002) 655–696 [65] O Rey and J Wei, Blow-up solutions for an elliptic Neumann problem with sub-or-supcritical nonlinearity, II: N 4, Ann Inst H Poincaré Anal Non Linéaire 22 (4) (2005), 459–484 [66] O Rey and J Wei, Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity, J Eur Math Soc (4) (2005) 449–476 [67] I Takagi, Point-condensation for a reaction–diffusion system, J Diff Eqns 61 (1986), 208–249 [68] A M Turing, The chemical basis of morphogenesis, Phil Trans Roy Soc Lond B 237 (1952), 37–72 [69] X.J Wang, Neumann problem of semilinear elliptic equations involving critical Sobolev exponents, J Diff Equ 93 (1991) 283–310 [70] M.J Ward and J Wei, Asymmetric spike patterns for the one-dimensional Gierer–Meinhardt model: equilibria and stability, Europ J Appl Math 13 (2002), 283–320 [71] L Wang and J Wei, Interior bubbles for a singularly perturbed problem in lower dimensions, preprint [72] J Wei, On the boundary spike layer solutions of singularly perturbed semilinear Neumann problem, J Diff Eqns 134 (1997), 104–133 [73] J Wei, On the interior spike layer solutions of singularly perturbed semilinear Neumann problem, Tohoku Math J 50 (2) (1998), 159–178 [74] J Wei, On the interior spike layer solutions for some singular perturbation problems, Proc Royal Soc Edinburgh Sect A (Mathematics) 128 (1998), 849–874 [75] J Wei, Uniqueness and eigenvalue estimates of boundary spike solutions, Proc Royal Soc Edin A 131 (2001), 1457–1480 [76] J Wei, On single interior spike solutions of Gierer–Meinhardt system: uniqueness and spectrum estimates, Eur J Appl Math 10 (1999), 353–378 [77] J Wei, Existence, stability and metastability of point condensation patterns generated by Gray–Scott system, Nonlinearity 12 (1999), 593–616 [78] J Wei, On the construction of interior spike layer solutions to a singularly perturbed semilinear Neumann problem, Partial Differential Equations: Theory and Numerical Solution, CRC Press LLC (1998), 336–349 [79] J Wei, On a nonlocal eigenvalue problem and its applications to point-condensations in reaction–diffusion systems, Int J Bifur Chaos 10 (6) (2000), 1485–1496 [80] J Wei and L Zhang, On a nonlocal eigenvalue problem, Ann Sc Norm Sup Pisa Cl Sci XXX (2001), 41–62 [81] J Wei and M Winter, Stationary solutions for the Cahn–Hilliard equation, Ann Inst H Poincaré Anal Non Linéaire 15 (1998), 459–492 [82] J Wei and M Winter, Multiple boundary spike solutions for a wide class of singular perturbation problems, J London Math Soc 59 (1999), 585–606 [83] J Wei and M Winter, Existence, classification and stability analysis of multiple-peaked solutions for the Gierer–Meinhardt system in R , preprint (2001) [84] J Wei and M Winter, On the two-dimensional Gierer–Meinhardt system with strong coupling, SIAM J Math Anal 30 (1999), 1241–1263 [85] J Wei and M Winter, On multiple spike solutions for the two-dimensional Gierer–Meinhardt system; the strong coupling case, J Diff Eqns 178 (2002), 478–518 Existence and stability of spikes 585 [86] J Wei and M Winter, On multiple spike solutions for the two-dimensional Gierer–Meinhardt system; the weak coupling case, J Nonlinear Sci (2001), 415–458 [87] J Wei and M Winter, Asymmetric Patterns for the Gierer–Meinhardt system, J Math Pures Appl (9) 83 (4) (2004), 433–476 [88] J Wei and M Winter, Stability of monotone solutions for the shadow Gierer–Meinhardt system with finite diffusivity, Diff Int Eqns 16 (2003), 1153–1180 [89] J Wei and M Winter, Higher-order energy expansions and spike locations, Cal Var PDE 20 (2004), 403– 430 [90] J Wei, M Winter and W Yeung, A higher-order energy expansion to two-dimensional singularly perturbed Neumann problems, Asymptotic Analysis 43 (1–2) (2005), 75–110 [91] J Wei and S Yan, Solutions with interior bubble and boundary layer for an elliptic Neumann problem with critical nonlinearity, C R Math Acad Sci Paris 343 (5) (2006), 311–316 [92] J Wei and J Yang, Concentration on curves for a singularly perturbed Neumann problem in twodimensional domain, Indiana Univ Math J., in press [93] J Wei and J Yang, Clustered line condensations for a singularly perturbed Neumann problem in twodimensional domain, preprint (2006) This page intentionally left blank Author Index Roman numbers refer to pages on which the author (or his/her work) is mentioned Italic numbers refer to reference pages Numbers between brackets are the reference numbers No distinction is made between the first author and co-author(s) Adams, R.A 84, 138, 207 [1]; 238, 241, 245, 250, 252, 255, 275 [1]; 466, 482 [1] Adimurthi 223, 225, 275 [2]; 275 [3]; 513, 581 [1]; 581 [2] Alama, S 81, 207 [2] Alessandrini, G 338, 365 [1] Alexandre, R 380, 482 [2] Allegretto, W 99, 207 [3] Almeida, L 300, 365 [2]; 365 [3] Alves, C.O 6, 45 [1] Amann, H 21, 25, 45 [2]; 111, 207 [4] Ambrosetti, A 7, 45 [3]; 51, 207 [5]; 223, 275 [4] Anane, A 83, 166, 180, 207 [6] Aoki, K 391, 392, 482 [3]; 484 [55] Arcoya, D 45, 45 [4] Arkeryd, L 373, 375, 390, 482 [4]; 482 [5]; 482 [6]; 482 [7]; 483 [8] Asano, K 373, 374, 439–441, 470, 473, 483 [9]; 484 [53]; 485 [64]; 485 [65] Atkinson, F.V 227, 275 [5] Aubin, Th 219, 275 [6] Avila, A 45, 45 [5] Azizieh, C 45, 45 [6] Bebernes, J 302, 365 [11] Benci, V 12, 45 [11]; 74, 88, 208 [11] Benjamin, T.B 21, 45 [12] Berestycki, H 36, 41, 44, 45 [13]; 45 [14]; 51, 111, 112, 208 [7]; 208 [12]; 516, 582 [3] Bergh, J 384, 465, 483 [13] Bers, L 338, 365 [12] Bethuel, F 285, 289, 300–302, 365 [2]; 365 [3]; 365 [13]; 365 [14]; 366 [15]; 366 [16] Bidault-Veron, M.F 45, 45 [15] Birindelli, I 39, 45 [16] Boccardo, L 6, 7, 45 [17] Boltzmann, L 372, 379, 483 [14] Bozhkov, Yu 159, 179, 208 [13]; 208 [14] Brezis, H 22, 23, 46 [21]; 220, 255, 264, 275 [8]; 275 [9]; 275 [10]; 280, 282, 301, 302, 317, 319, 322, 336, 365 [14]; 366 [15]; 366 [17]; 366 [18]; 366 [19]; 366 [20]; 366 [21]; 366 [22]; 366 [23] Busca, J 8, 30, 37, 39–41, 44, 45, 46 [18]; 46 [19]; 46 [20] Caffarelli, L 178, 179, 208 [15] Caflisch, R.E 374, 390, 392, 397, 483 [10]; 483 [15] Caglioti, E 310, 364, 366 [24]; 366 [25] Capuzzo-Dolcetta, I 41, 45 [13]; 111, 112, 208 [12] Carleman, T 372, 377, 483 [16]; 483 [17] Carleson, L 224, 275 [11]; 352, 366 [26] Casten, R.G 489, 582 [5] Cerami, G 19, 46 [23] Cercignani, C 372, 373, 375–378, 383, 386, 389–392, 482 [4]; 483 [18]; 483 [19] Chabrowski, J 45, 46 [22] Chandrasekhar, S 302, 366 [27] Chang, A 224, 275 [11] Bahri, A 51, 208 [7]; 280, 365 [4] Bandle, C 32, 45 [7]; 111, 112, 117, 118, 208 [8]; 208 [9]; 306, 310–312, 321, 336, 337, 349, 355, 365, 365 [5]; 365 [6]; 365 [7]; 365 [8] Baraket, S 318, 365 [9] Bardos, C 374, 390, 392, 397, 448, 483 [10]; 483 [11]; 483 [12] Barles, G 83, 208 [10] Bartolo, P 74, 88, 208 [11] Bartolucci, P 326, 365 [10] Bartsch, T 4, 16, 19, 20, 25, 45 [8]; 45 [9]; 45 [10]; 229, 233, 275 [7] Bates, P 494, 582 [4] 587 588 Author Index Chang, K.C 45, 46 [24] Chang, S.-Y.A 336, 363, 364, 366 [28] Chanillo, S 289, 364, 366 [29]; 366 [30] Chau, H 490, 583 [38] Chen, C.C 312, 318, 326, 336, 362–364, 365 [10]; 366 [28]; 366 [31]; 366 [32]; 366 [33]; 366 [34]; 390, 483 [20] Chen, J.T 338, 366 [35] Chen, W 31, 46 [25]; 321, 366 [36] Chen, X 320, 366 [37]; 543, 580, 582 [6]; 582 [7] Cheng, K.-S 312, 364, 366 [39] Cheng, S.Y 344, 366 [38] Clapp, M 19, 20, 25, 45 [8] Clément, Ph 4, 8, 9, 22, 24, 32, 33, 45, 45 [6]; 46 [26]; 46 [27]; 46 [28]; 159, 160, 164, 177, 178, 180, 208 [16]; 208 [17] Coifman, R 286, 289, 366 [40] Coron, F 390, 392–394, 397, 483 [21] Coron, J.M 282, 366 [18]; 366 [19] Costa, D.G 17, 46 [29] Courant, R 344, 347, 366 [41] Crandall, M.G 310, 365, 366 [42]; 367 [43]; 367 [44] Dancer, E.N 31, 39, 46 [30]; 494, 519, 535, 545, 572, 573, 582 [4]; 582 [8]; 582 [9] de Figueiredo, D.G 4, 6–9, 11, 12, 14, 16, 19–22, 24, 25, 28, 30, 32, 33, 36, 39–45, 45 [9]; 45 [17]; 46 [26]; 46 [27]; 46 [31]; 46 [32]; 46 [33]; 46 [34]; 46 [35]; 46 [36]; 46 [37]; 46 [38]; 46 [39]; 46 [40]; 46 [41]; 46 [42]; 46 [43]; 46 [44]; 46 [45]; 47 [46]; 47 [47]; 159, 160, 164, 177, 178, 208 [16]; 223–226, 229, 230, 232, 233, 243, 246, 250, 251, 259, 267, 268, 272–275, 275 [13]; 275 [14]; 275 [15]; 275 [16]; 275 [17]; 275 [18]; 276 [19]; 317, 367 [45] de Morais Filho, D.C 6, 45, 45 [1]; 48 [76]; 48 [81] de Thélin, F 45, 47 [49]; 180, 181, 208 [17]; 209 [54] Deimling, K 6, 47 [48] del Pino, M 318, 367 [46]; 494, 509, 513, 514, 516, 543, 558, 559, 582 [7]; 582 [13]; 582 [14]; 582 [15]; 582 [16]; 582 [17]; 582 [18]; 582 [19] Desvillettes, L 373, 380, 482 [2]; 483 [22]; 483 [23] Diaz, J.I 349, 367 [47] DiBenedetto, E 47 [50] Dickstein, F 16, 47 [51] Dillon, A 516, 582 [10] Ding, W 279, 280, 283, 319, 367 [48]; 367 [49] Ding, Y 229, 233, 275 [7] Ding, Y.H 4, 21, 44, 46 [34]; 47 [52] Diperna, R 373, 379, 483 [24] Ó, J.M 14, 16, 47 [46]; 47 [53]; 224, 243, 246, 259, 267, 268, 272, 274, 275, 275 [15]; 275 [16]; 275 [17] Doelman, A 543, 544, 582 [11]; 582 [12] Doi, T 394, 484 [56] Dolbeault, J 514, 582 [15] Drábek, P 80, 82, 88, 99, 101, 124, 168, 178, 180, 188, 191, 208 [18]; 208 [19]; 208 [20]; 208 [21]; 208 [22]; 208 [23] Duan, R 474, 483 [25] Dunford, N 467, 468, 483 [26] Eberly, D 302, 365 [11] Ei, S.-I 581, 582 [20] Ekeland, I 264, 275 [12] Ellis, R.S 449, 450, 453, 483 [27] Esposito, P 318, 367 [51] Essen, M 32, 45 [7] Evans, L.C 285, 367 [50] Faber, G 346, 367 [52] Feireisl, E 473, 483 [28] Felmer, P.L 8, 11, 12, 14, 16, 36, 37, 46 [35]; 46 [40]; 47 [54]; 47 [55]; 47 [56]; 47 [57]; 229, 230, 232, 233, 243, 275 [13]; 494, 509, 513, 582 [13]; 582 [16] Fernandez, B 45, 47 [59] Figueiredo, G.M 16, 47 [60] Fleckinger, J 45, 47 [58]; 180, 208 [17] Fleming, W 351, 367 [53] Flucher, M 224, 276 [20] Fortunato, D 74, 88, 208 [11] Fournier, J.J 238, 241, 245, 250, 252, 255, 275 [1] Friedman, A 327, 367 [54] Fuˇcik, S 84, 208 [27] Garcia Azorero, J.P 180, 208 [24] Garcia-Huidobro, M 45, 47 [61] Gardner, R.A 543, 582 [11] Ge, Y 514, 582 [25] Gel’fand, I.M 302, 367 [55] Ghidaglia, J.M 289, 366 [16] Ghoussoub, N 513, 514, 582 [23]; 582 [24] Gidas, B 24, 31, 32, 43, 44, 47 [62]; 47 [63]; 47 [64]; 47 [65]; 227, 276 [21]; 307, 317, 367 [56]; 493, 497, 582 [22] Gierer, A 489–491, 582 [21] Gilbarg, D 43, 47 [66]; 88, 208 [25]; 287, 352, 367 [57] Gladiali, F 318, 367 [58] Author Index Golse, F 374, 382, 386, 390, 392–394, 397, 440, 448, 452, 483 [11]; 483 [12]; 483 [21]; 483 [29]; 483 [30]; 483 [31]; 483 [32]; 484 [56] Gossez, J.P 238, 240, 241, 276 [22] Grad, H 372, 376, 380, 382, 383, 385, 386, 483 [33]; 483 [34] Grillot, P 45, 45 [15] Grossi, M 318, 367 [51]; 367 [58]; 495, 582 [30] Guedda, M 165, 178, 180, 206, 208 [26] Gui, C 494, 495, 504, 511, 513, 514, 582 [23]; 582 [24]; 582 [26]; 582 [27]; 582 [28]; 582 [29] Guisti, E 351, 367 [59] Gulliver, R.D., II 338, 367 [60] Guo, Y 373, 382, 483 [35]; 484 [57] Hai, D.D 45, 47 [67] Halperin, B.I 315, 367 [61]; 367 [68] Hamdache, K 373, 484 [36] Han, P 45, 47 [68] Hardt, R 343, 367 [62] Hartman, P 338, 367 [63]; 367 [64] Heinonen, J 352, 354, 367 [65] Hélein, F 285, 301, 302, 365 [14]; 366 [15]; 367 [66]; 367 [67] Hilbert, D 344, 347, 366 [41] Hille, E 475, 484 [37] Hohenberg, P.C 315, 367 [61]; 367 [68] Holland, C.J 489, 582 [5] Holloway, D.M 491, 583 [31] Huang, F.M 374, 484 [38] Huang, W.A 338, 366 [35] Huang, Y 80, 99, 207 [3]; 208 [21]; 208 [22] Hulshof, J 8, 12, 45, 47 [69]; 47 [70]; 229, 230, 243, 264, 276 [23]; 276 [24]; 276 [25] Illner, R 372, 375–378, 383, 386, 389, 391, 483 [19] Il’yasov, Ya.Sh 51, 208 [28]; 208 [29]; 208 [30]; 208 [31] Imai, K 373, 484 [49] Iron, D 544, 550, 556, 557, 572, 580, 581, 583 [32]; 583 [33]; 583 [34]; 583 [35] Jäger, W 364, 367 [69] Jing, R 514, 582 [25] Jost, J 279, 280, 319, 367 [48]; 367 [70] Kalita, E.A 45, 47 [71] Kaper, T.J 543, 544, 582 [11]; 582 [12] Kato, T 385, 386, 449, 452, 458, 466, 467, 475, 484 [40] Kavian, O 213, 276 [26] Kawashima, S 390, 484 [48] 589 Kawohl, B 349, 368 [71] Kiessling, M 364, 366 [29] Kilpeläinen, T 352, 354, 367 [65] Koch, A.J 490, 583 [36] Kowalczyk, M 318, 367 [46]; 516, 543, 558, 559, 580, 582 [6]; 582 [7]; 582 [17]; 582 [18]; 582 [19]; 583 [37] Krahn, E 346, 368 [72] Krasnoselski˘ı, M.A 21, 47 [72]; 238, 276 [27] Kryszewski, W 229, 233, 276 [28] Kufner, A 84, 208 [27] Kuo, T 341, 368 [73] Kwong, M.K 493, 496, 583 [39]; 583 [40] Landau, L.D 374, 484 [41] Lanford, O., III 372, 484 [39] Laptev, G.G 32, 47 [73] Levermore, D 448, 483 [11] Li, C 31, 46 [25]; 321, 366 [36] Li, J 319, 367 [48] Li, S 13, 48 [74]; 48 [75]; 229, 233, 276 [30] Li, Y.-Y 289, 318, 319, 321, 322, 326, 366 [20]; 366 [30]; 368 [74]; 368 [75]; 494, 583 [41] Lifshitz, E.M 374, 484 [41] Lin, C.-S 312, 318, 326, 336, 362–364, 365 [10]; 366 [28]; 366 [31]; 366 [32]; 366 [33]; 366 [34]; 366 [39]; 368 [76]; 368 [77]; 520, 583 [42] Lin, C.S 513, 583 [43] Lin, F.H 285, 368 [78]; 368 [79]; 495, 498, 501, 583 [44] Lindqvist, P 83, 180, 208 [32] Lions, J.L 51, 52, 208 [33]; 230, 276 [32] Lions, P.L 21, 24, 46 [39]; 263, 276 [31]; 286, 289, 302, 310, 317, 364, 366 [24]; 366 [25]; 366 [40]; 367 [45]; 368 [80]; 373, 379, 440, 452, 483 [24]; 483 [30] Liu, J.Q 13, 48 [74] Liu, T.-P 373, 374, 382, 390, 394, 483 [20]; 484 [42]; 484 [43]; 484 [44]; 484 [45] Löfström, J 384, 465, 483 [13] Lucia, M 364, 368 [77] Luckhaus, S 364, 367 [69] Ma, L 318, 321, 368 [81] Ma, S.-K 315, 367 [61] Magalhães, C.A 12, 17, 46 [29]; 46 [36] Magenes, E 230, 276 [32] Mahmoudi, F 515, 583 [51] Maini, P 542, 583 [45] Maini, P.K 490, 516, 582 [10]; 583 [38] Malchiodi, A 490, 514, 515, 579, 583 [48]; 583 [49]; 583 [50]; 583 [51]; 583 [52] 590 Author Index Manásevich, R 8, 30, 37, 39–41, 45, 46 [18]; 46 [28]; 47 [61] Mancini, G 513, 581 [1] Marchioro, C 302, 310, 364, 366 [24]; 366 [25]; 368 [82] Maremonti, P 473, 484 [46] Martinez, S 14, 47 [56] Martio, O 352, 354, 367 [65] Matano, H 489, 583 [46] Matuš˚u-Neˇcasovà, Š 473, 483 [28] Maxwell, J.C 372, 484 [47] McOwen, R 51, 209 [35] Meinhardt, H 489–491, 582 [21]; 583 [36] Merle, F 319, 366 [21] Meyer, Y 286, 289, 366 [40] Mitidieri, E 4, 8, 9, 22, 24, 32, 33, 35, 36, 39, 45, 45 [16]; 46 [26]; 46 [27]; 46 [28]; 46 [37]; 47 [61]; 47 [70]; 48 [77]; 48 [78]; 159, 160, 164, 177–181, 208 [13]; 208 [14]; 208 [16]; 208 [17]; 208 [34]; 261, 264, 276 [25]; 276 [29] Miyagaki, O.H 223, 225, 273, 275 [14] Mizoguchi, N 336, 368 [83] Montenegro, M 514, 515, 583 [49]; 583 [50] Montenegro, M.S 25, 45, 48 [79]; 48 [80] Moseley, J.L 318, 368 [84] Moser, J 222, 276 [33]; 314, 368 [85] Mossino, J 349, 368 [86] Mou, L 279, 285, 368 [87] Murray, J 490, 583 [47] Musso, M 318, 367 [46]; 509, 513, 514, 582 [13]; 582 [14]; 582 [15] Nagasaki, K 316, 317, 363, 368 [88]; 369 [129] Naito, Y 321, 368 [89] Namba, T 111, 209 [36] Ni, W.-M 43, 44, 47 [63]; 227, 276 [21]; 280, 307, 317, 367 [56]; 368 [90]; 492, 493, 495–499, 501, 504, 513–515, 517, 520, 534, 538, 539, 572, 576, 578–580, 582 [22]; 583 [42]; 583 [44]; 583 [52]; 583 [53]; 583 [54]; 583 [55]; 583 [56]; 584 [57]; 584 [58]; 584 [59]; 584 [60]; 584 [61]; 584 [62] Nicolaenko, B 374, 390, 392, 397, 483 [10]; 483 [15] Nikuni, S 390, 484 [48] Nirenberg, L 36, 41, 43, 44, 45 [13]; 45 [14]; 47 [63]; 51, 111, 112, 178, 179, 208 [12]; 208 [15]; 209 [37]; 220, 227, 264, 275 [9]; 276 [21]; 280, 307, 317, 366 [22]; 367 [56]; 493, 497, 582 [22] Nishida, T 373, 484 [49] Nishino, K 391, 392, 482 [3] Nouri, A 375, 390, 482 [5]; 482 [6]; 482 [7]; 483 [8] Nussbaum, R.D 21, 24, 46 [39]; 317, 367 [45] Ohtsuka, H 321, 364, 368 [91]; 368 [92] Ohwada, T 394, 484 [56] Othmer, H.G 516, 582 [10] Pacard, F 318, 365 [9]; 514, 582 [25] Pacella, F 513, 581 [2] Painter, K.J 490, 583 [38] Palais, R 216, 276 [34] Palczewski, A 373, 374, 441, 484 [50] Pan, X.B 513, 583 [55] Parker, T.H 283, 285, 368 [93] Pearson, J.E 490, 584 [63] Peletier, L.A 4, 8, 32, 33, 48 [84]; 227, 275 [5] Peral Alonso, I 180, 208 [24] Perthame, B 382, 386, 390, 440, 452, 483 [30]; 483 [31]; 484 [51] Petzeltovà, H 473, 483 [28] Phillipes, R.S 475, 484 [37] Pierotti, D 111, 118, 209 [38] Pinasco, J.P 45, 47 [59] Pinsky, M.A 449, 450, 453, 483 [27] Pistoia, A 318, 367 [51]; 495, 514, 582 [14]; 582 [30] Plotnikov, P.I 51, 209 [39] Pohozaev, S.I 14, 48 [83]; 51, 52, 82, 103, 107, 111, 114, 168, 177–180, 187, 188, 191, 206, 208 [23]; 209 [40]; 209 [41]; 209 [42]; 209 [43]; 209 [44]; 209 [45]; 209 [46]; 209 [47]; 209 [48]; 209 [49]; 209 [50]; 209 [51]; 216, 218, 221, 253, 276 [35]; 276 [36]; 280, 368 [95] Polacik, P 538, 539, 583 [56] Pólya, G 349, 368 [94] Poupaud, F 390, 392, 483 [32] Pozio, M.A 111, 112, 117, 118, 208 [8]; 208 [9] Preijel, Å 346, 368 [96] Protter, M.H 45, 48 [85] Pucci, P 33, 37, 48 [86]; 206, 209 [52]; 261, 276 [37] Pulvirenti, M 302, 310, 364, 366 [24]; 366 [25]; 368 [82]; 372, 375–378, 383, 386, 389, 391, 483 [19] Qing, J 279, 280, 283, 285, 294, 368 [97]; 368 [98]; 368 [99] Quittner, P 24, 48 [82] Rabinowitz, P.H 7, 12, 45 [3]; 45 [11]; 48 [87]; 51, 207 [5]; 223, 247, 272, 275 [4]; 276 [38]; Author Index 280, 310, 365, 366 [42]; 367 [43]; 367 [44]; 368 [100] Ramos, M 12, 46 [38] Rao, M.M 238, 240, 242, 244, 276 [39] Ren, Z.D 238, 240, 242, 244, 276 [39] Renardy, M 111, 207 [4] Rey, O 509, 511, 513, 514, 584 [64]; 584 [65]; 584 [66] Rishel, R.I 351, 367 [53] Rivière, T 285, 368 [79]; 368 [101]; 368 [102] Rossi, J.D 45, 47 [59] Ruf, B 4, 14, 46 [44]; 47 [46]; 223–226, 243, 246, 250, 251, 258, 259, 267, 268, 272–275, 275 [14]; 275 [15]; 275 [16]; 275 [17]; 275 [18]; 276 [19]; 276 [40] Ruiz, D 45, 45 [4] Runst, T 51, 208 [30] Ruticki˘ı, J.B 238, 276 [27] Sacks, J 286, 368 [103] Sakaguchi, S 338, 352, 369 [104]; 369 [105] Sato, T 318, 321, 352, 369 [106]; 369 [107] Schoen, R 285, 369 [113] Schwartz, J 467, 468, 483 [26] Semmes, S 286, 289, 366 [40] Senba, T 336, 364, 369 [108]; 369 [109] Sentis, R 440, 452, 483 [30] Serrin, J 33, 35, 37, 41, 48 [86]; 48 [88]; 48 [89]; 101, 206, 209 [52]; 209 [53]; 261, 276 [37]; 352, 369 [110]; 369 [111] Shafrir, I 319, 322, 366 [20]; 368 [75]; 369 [112] Shi, J 494, 582 [4] Shivaji, R 45, 47 [67] Shizuta, Y 373, 374, 383, 484 [52]; 484 [53] Silva, E.A.B 13, 48 [90] Simon, L 343, 367 [62] Sirakov, B 4, 25, 28, 30, 39–41, 43–45, 46 [19]; 46 [20]; 46 [42]; 47 [47]; 48 [91] Smale, S 216, 276 [34] Sone, Y 374, 390–392, 394, 482 [3]; 483 [12]; 484 [54]; 484 [55]; 484 [56] Souganidis, P.E 440, 452, 484 [51] Souplet, Ph 24, 48 [82] Souto, M.A.S 6, 8, 31, 32, 34, 41, 45, 45 [1]; 48 [81]; 48 [92] Spruck, J 24, 31, 47 [64]; 47 [65]; 178, 179, 208 [15] Stampacchia, G 317, 369 [114] Stein, E.M 287, 369 [115] Strain, G 373, 484 [57] Straškraba, I 473, 483 [28] Strauss, W.A 317, 366 [23] Strichartz, R.S 254, 276 [41] 591 Struwe, M 213, 276 [42]; 279, 281, 282, 302, 319, 369 [116]; 369 [117]; 369 [118]; 369 [119]; 369 [120]; 369 [121]; 369 [122] Sugimoto, H 391, 392, 482 [3] Sulem, C 382, 386, 390, 392–394, 397, 483 [21]; 483 [31] Suzuki, T 302, 310, 312, 314–319, 321, 327, 329, 331, 336, 352, 354, 359, 362–364, 368 [83]; 368 [88]; 368 [89]; 368 [91]; 368 [92]; 369 [105]; 369 [106]; 369 [107]; 369 [108]; 369 [109]; 369 [123]; 369 [124]; 369 [125]; 369 [126]; 369 [127]; 369 [128]; 369 [129]; 369 [130] Sweers, G 180, 181, 208 [34] Szegö, G 347, 349, 365, 368 [94]; 369 [131] Szulkin, A 229, 233, 276 [28] Takac, P 45, 47 [58] Takagi, I 492, 493, 496, 513, 517, 520, 534, 538, 543, 544, 549, 556, 572, 580, 583 [55]; 584 [57]; 584 [58]; 584 [59]; 584 [60]; 584 [67] Takahashi, F 352, 354, 369 [107]; 369 [130] Talenti, G 219, 276 [43]; 349, 369 [132] Tanaka, K 45, 48 [93] Tarantello, G 81, 207 [2]; 302, 310, 319, 326, 365 [10]; 369 [122]; 369 [133]; 370 [134] Temam, R 264, 275 [12] Terracini, S 111, 118, 209 [38] Tesei, A 107, 111, 112, 114, 117, 118, 208 [9]; 209 [49]; 209 [50] Tian, G 279, 280, 283, 285, 367 [49]; 368 [99] Toland, J.F 315, 370 [135]; 370 [136] Tolksdorf, P 48 [94]; 88, 124, 209 [55] Troy, W.C 44, 48 [95] Trudinger, N.S 14, 43, 47 [66]; 48 [96]; 88, 208 [25]; 209 [56]; 221, 253, 276 [44]; 287, 352, 367 [57] Turing, A.M 489, 584 [68] Turner, R.E.L 22, 23, 46 [21] Ubilla, P 16, 47 [53] Uhlenbeck, K 285, 286, 368 [103]; 369 [113] Ukai, S 372–374, 380, 383, 390, 395, 439–441, 446–450, 470, 473, 474, 476–480, 482, 483 [25]; 484 [58]; 484 [59]; 484 [60]; 484 [61]; 484 [62]; 484 [63]; 485 [64]; 485 [65]; 485 [66]; 485 [67]; 485 [68]; 485 [69] Valli, A 473, 485 [70] van der Ploeg, H 543, 544, 582 [12] van der Vorst, R.C.A.M 4, 8, 9, 12, 32, 33, 45, 47 [69]; 47 [70]; 48 [84]; 48 [97]; 180, 181, 208 [34]; 229, 230, 243, 261, 264, 276 [23]; 276 [24]; 276 [25]; 276 [45] 592 Author Index Vásquez, J.L 48 [98] Velin, J 180, 181, 209 [54] Véron, L 106, 107, 165, 178, 180, 206, 208 [26]; 209 [51]; 209 [57]; 209 [58]; 354, 370 [137] Villani, C 373, 380, 482 [2]; 483 [23] Wainger, S 255, 275 [10] Wang, C 279, 285, 368 [87] Wang, C.Y 280, 283, 370 [138] Wang, G 319, 329, 367 [48]; 370 [139] Wang, L 513, 514, 583 [43]; 584 [71] Wang, S 328, 370 [140] Wang, W.K 390, 485 [71] Wang, X.J 513, 584 [69] Wang, Z.Q 16, 47 [57] Ward, M.J 544, 549–552, 556, 572, 580, 581, 583 [32]; 583 [33]; 583 [34]; 584 [70] Wei, J 328, 370 [141]; 493–495, 499, 504, 509, 511, 513–517, 519–523, 533–535, 537, 538, 541, 542, 544, 548–552, 556–559, 561, 563, 564, 567, 569–572, 576, 578–581, 582 [3]; 582 [16]; 582 [17]; 582 [18]; 582 [19]; 582 [20]; 582 [27]; 582 [28]; 582 [29]; 582 [30]; 583 [34]; 583 [35]; 583 [43]; 583 [45]; 583 [52]; 584 [61]; 584 [62]; 584 [65]; 584 [66]; 584 [70]; 584 [71]; 584 [72]; 584 [73]; 584 [74]; 584 [75]; 584 [76]; 584 [77]; 584 [78]; 584 [79]; 584 [80]; 584 [81]; 584 [82]; 584 [83]; 584 [84]; 584 [85]; 585 [86]; 585 [87]; 585 [88]; 585 [89]; 585 [90]; 585 [91]; 585 [92]; 585 [93] Wei, J.C 318, 321, 368 [81]; 495, 498, 501, 583 [44] Weinberger, H.F 45, 48 [85]; 347, 365, 370 [142] Wennberg, B 380, 482 [2] Wente, H.C 286, 338, 370 [143]; 370 [144] Weston, V.H 318, 370 [145] Willem, M 13, 19, 45 [10]; 48 [75]; 213, 229, 233, 276 [30]; 276 [46] Winter, M 493, 494, 511, 537, 538, 542, 548, 551, 556, 557, 561, 563, 564, 567, 569–571, 582 [29]; 583 [35]; 583 [45]; 584 [81]; 584 [82]; 584 [83]; 584 [84]; 584 [85]; 585 [86]; 585 [87]; 585 [88]; 585 [89]; 585 [90] Wintner, A 338, 367 [63]; 367 [64] Wolansky, G 328, 359, 370 [146]; 370 [147] Yadava, S.L 223, 275 [3]; 513, 581 [2] Yamashita, I 391, 392, 484 [55] Yan, S 494, 514, 582 [9]; 585 [91] Yanagida, E 534, 538, 539, 572, 580, 583 [56]; 584 [59]; 584 [60] Yang, J 25, 42, 43, 45, 45 [5]; 46 [22]; 46 [43]; 46 [45]; 515, 516, 585 [92]; 585 [93] Yang, T 373, 374, 382, 390, 395, 448, 474, 476–480, 482, 483 [20]; 483 [25]; 484 [38]; 484 [42]; 484 [43]; 485 [66]; 485 [67]; 485 [68]; 485 [69]; 485 [71] Yang, X 390, 485 [71] Yang, Y 302, 310, 370 [148] Yarur, C.S 45, 47 [61]; 48 [99] Ye, D 329, 370 [139] Yeung, W 493, 585 [90] Yoshida, K 321, 368 [89]; 475, 485 [72] Yu, S.-H 373, 374, 382, 390, 394, 395, 484 [42]; 484 [43]; 484 [44]; 484 [45]; 485 [68]; 485 [69] Zhang, L 496, 534, 583 [40]; 584 [80] Zhao, H 374, 474, 483 [25]; 484 [43] Zhu, M 514, 582 [24] Ziemer, W.P 354, 370 [149] Zou, H 25, 33, 35, 37, 41, 48 [88]; 48 [89]; 48 [100] Subject Index A a priori estimate 391, 424 absence of solution 77–80 admissible – boundary data 391 – condition 390, 394 – couples 28 albedo problem 373 algebraic – decay 431, 434, 436, 437 – factor 55, 72 – rate 432 Ambrosetti–Rabinowitz method 51 associated Legendre equation 311, 358, 361 asymptotic – expansion 447, 448, 450, 455, 470 – relation 376 – stability 372, 373, 392, 473, 480 – – global 372, 373 – – local 373 – with damping 399, 403–405, 415, 418, 420, 423 bootstrap argument 392, 423, 424, 427, 436, 446, 460, 470, 477, 478 boundary – condition 372, 373, 378, 387, 388 – – absorption 388 – – accommodation 389 – – bounce-back reflection 373 – – conservative 460 – – diffuse reflection 388 – – Dirichlet 373, 388, 391 – – dissipative 460 – – inhomogeneous 373 – – Maxwell 389 – – nonisothermal 373, 375 – – periodic 474 – – reverse reflection 373, 388 – – space-periodic 372 – – specular reflection 373, 388, 390 – – time-periodic 374 – data 410, 423, 431, 438 – layer 373, 390, 392, 394, 405, 409, 412, 423– 425, 432–434, 437 – operator 439 – value 387 – – problem 373, 374 – – – time-periodic 474 bounded perturbation 386 Bromwich integration path 477 bubble 291 bulk velocity 373, 377, 378, 438, 439 B balance law 375 Bandle mean value theorem 312 Bandle spherically decreasing rearrangement 306, 307, 311, 355 bifurcation equation 57–60, 62, 66, 72, 75, 84, 113, 115, 116, 119, 123, 125, 128, 148, 152 bifurcation manifold 51 bifurcation-fibering equation 72 blow-up analysis 318 blow-up method 24 Bol’s inequality 306, 313, 355 Boltzmann constant 377 Boltzmann equation 372, 375, 377, 390 – conservation law 377 – inhomogeneous 374 – L1 theory 379 – linearized 415 – nonlinear 390, 404, 420, 423 – spatially homogeneous 372 – spatially inhomogeneous 372 C Cauchy problem 372, 399, 481 Chapman–Enskog 390 chemotaxis 315, 364 co-area formula 350 co-dimension 391, 392, 395, 404, 406, 410, 423 collapse formation 315 collision – binary 375 – cross section 376 593 594 Subject Index – elastic 375, 376 – frequency 382, 409, 431 – grazing 380 – invariant 376, 377, 393 – invariants 385 – kernel 380, 401 – operator 375, 460 – – linearized 380, 382, 411, 432, 440, 441 – – modified 380 – – nonlinear 376, 424 compact perturbation 386, 440, 449, 460 compactness 440, 441, 460, 465–467 complex Ginzburg–Landau equation 299 concentration function 290 condensation–evaporation 390 conservation law 376, 381 – kinetic energy 376 – momentum 376 – total energy 378 – total mass 378 – total momenta 378 constrained extremum point 113 contraction mapping principle 404, 423, 441, 474, 480 convection 395 convergence 372 – almost exponential 373 – exponential 373 – rate 410, 423, 435 critical – growth 216, 217, 223, 225 – hyperbola 8, 213, 229, 243 – Orlicz pair 242, 245 – point 55, 56, 58, 60, 62, 64, 65, 67–72, 80, 84, 85, 88, 95, 106, 112, 113, 129, 144, 147, 149, 150, 152, 153, 166, 168, 170, 184, 187, 188, 191, 214 – – conditional 188 – – conditionally 55, 56, 58–60, 65, 67–74, 86, 87, 95 – – regular 70 – – unconditionally 71 critical set 65 critical value 148 cutoff function 397, 401, 412, 445, 467, 469 D -regular 239 damping 395, 404, 413, 416 decreasing rearrangement 347 DeGiorgi isoperimetric inequality 351 Dirichlet – boundary condition 74, 82, 178, 179 – condition 90 – data 390–392, 394 – norm 215 – principle 214, 215 – problem 80 discrete velocity model 390 dissipation 395, 411, 414, 432 dissipative equation 379 distribution function 347 domain of definition 441, 442, 446, 447 Duhamel formula 474, 480 E ε-regularity 287 eigenfunction 335, 394, 442, 449, 468 eigenprojection 395, 442, 447 eigenvalue 335, 336, 385, 393–395, 449, 452– 454 – accumulation point 385 – discrete 442, 447, 448, 468 – semi-simple 385, 447, 450 elliptic systems 228 Emden–Fowler equation 302 Emden–Fowler type 51 energy – concentration 286 – decay estimate 286 – density 377 – estimate 388, 395, 398, 401, 405, 412, 413, 418, 434, 436 – identity 280, 281 – method 382, 387, 392, 394, 423, 424 – renormalized 300, 301 – rescaled 284 entropy 379 – dissipation integral 379 equation of state 377 equilibrium state 372, 378, 391, 438, 448 Euler equations 374, 376, 390, 473 Euler–Lagrange principle 214 existence theorem 393, 395, 398, 404, 423 exponential critical exponent 274 exponential decay 413, 414, 416, 424, 431, 434, 436, 477, 482 exterior domain 438, 445 exterior problem 373 external force 372, 373, 375, 378, 388, 438 external source 372, 373, 375, 388, 395, 438, 473 – time-periodic 374 F far field 374, 390–392, 394, 406, 423, 424, 439 fibering – constraint 54, 85, 95, 119, 120, 123 Subject Index – functional 54, 56, 66, 68, 69, 75, 84–86, 91, 95, 125, 138 – – of norm type 69, 119 – method 51, 54, 57, 58, 60–62, 64, 66, 71, 77, 80, 82, 84, 87, 98, 99, 106, 118, 119, 121, 123, 124, 128, 129, 159, 162, 166, 168, 180, 183, 184 – – k-parametric 54 – – minimax realization 60, 62, 64 – – one-parameter 55 – – one-parametric 54, 55 – – parameter-free realization 68 – technique 156 fibration 54 finite-dimensional approximation 247, 259, 272 fixed point 431, 438, 475, 478, 480 – theorem 415, 424, 426, 435 Fleming–Rishel’s co-area formula 351 flow – Benard 373 – Couette 373, 375 – subsonic 391 – supersonic 374, 391 – time-periodic 373 – transonic 374, 394, 439 flow past an obstacle 373, 374, 438 fluid dynamical equation 374, 376, 473 force-free space 373 Fourier multiplier 454 Fourier transform 445, 447, 454, 465, 466 – inverse 454 fractional Sobolev space 10, 229, 230 Fréchet derivative 408 free boundary problem 326 – higher-dimensional case 327 Fubini’s theorem 444 function space 377, 383, 440, 454, 475, 482 functions – spherically sub-harmonic 312 – spherically super-harmonic 312 G gas dynamics 373, 438 ghost effect 374 global existence 395, 423, 426, 435 Grad angular cutoff 395 – assumption 380 Grad cutoff assumption 375 Green formula 387, 446, 468, 469 Green function 394 H H-function 378 H -systems 281 595 H-theorem 372, 373, 378, 379, 396 half-space problem 373, 390, 394 hard sphere 376, 380, 383 – model 380, 395, 396, 404, 410, 413, 415, 420, 423, 424, 432, 434, 436, 437 Hardy–Littlewood inequality 23, 348, 350 Hardy-BMO structure 285 harmonic map – equation 283 – minimizing 284 – stationary 284 – weakly 283 Hartman–Wintner’s theorem 338 heat diffusivity 448 Helmholtz free energy 314 Hilbert expansion 390 I ideal gas 377 ill-posed 388, 391, 440 implicit function theorem 409, 440 incoming particle 387, 388, 390, 391 initial boundary value problem 373, 423, 426, 431 integral kernel 385, 441 interior problem 373, 374 interpolation 384 inverse 440–442, 446, 453, 454, 460, 462, 468, 477 – bounded 440 – unbounded 441 isoperimetric inequality 346, 347, 350 K Kelvin transformation 328 kinetic equation 372 kinetic theory 372, 377 Knudsen number 376, 390 Kramer problem 373, 390 Krasnoselski˘ı theorem 21 Kuo’s theorem 341 L Laplace transform 473, 475, 476 Laplace–Beltrami operator 310 layer 390 – boundary 390 – initial 390 – shock 390 limiting absorption 440 – principle 441, 446, 454 linearized – Boltzmann operator 441, 474 596 Subject Index – problem 393 – stability 310 linking structure 259, 268 linking theorem 13 Liouville integral 309, 311 Liouville theorems 31–41 local linking 247 Localized Energy Method 491 Lorentz space 251, 254 Luxemburg norm 239 Lyapunov–Schmidt method 51, 57, 58 Lyusternik–Shnirel’man theory 59, 64, 65, 74, 88 M Mach number 374, 390, 391, 393, 406, 424, 431, 439 macroscopic component 405, 413, 414, 416, 432 manifold 391, 394, 404, 410, 466 map 389, 406 – C 404, 410 – contraction 431, 438, 472, 475, 479, 481 – nonlinear 431, 474 mass density 377, 378 – macroscopic 377 – microscopic 377 maximal growth 221 Maxwellian 372, 373, 390, 439 – background 411 – definition 378 – local 378 – standard 382, 392, 460 – uniform 372 – uniform (global, absolute) 378 mean field equation 358, 364 mean free path 376 method of fibering 51 method of symmetrization – Green’s function 331 – rearrangement 365 microscopic component 411, 414 microscopic description 377 Milne problem 373, 390 min–max principle 336, 337 minimax realization 64, 66 monotonicity formula 284 Mountain-Pass Theorem N N -function 238 Navier–Stokes equations – compressible 374 Nehari functional 72 Nehari manifold 51 374, 376, 390, 473 Neumann – boundary condition 108, 118 – problem 124, 364 – – nonhomogeneous 124 – – positive solutions 106, 107 – series 453 nodal – domain 344, 346 – line 344 – set 343 nondegeneracy condition 56, 86, 95 nonexistence result 102, 159, 177–180, 206 nonlinear – boundary condition 107, 156 – Fredholm alternative 52, 53 – hyperbolic equation 51 – monotone coercive operators 52 Nonlocal Eigenvalue Problem 491 nonpositivity 379 null space 385, 403, 420, 442, 448 O operator – adjoint 413, 462 – bilinear symmetric 379 – boundary 439, 440 – bounded 385, 386, 403, 411, 415, 418, 419, 448, 454 – closed 385, 446 – compact 385, 399, 415, 426, 427, 436, 441, 442, 449, 452, 462, 466–468 – compact integral 441 – densely defined 446 – Hilbert–Schmidt type 385 – integral 383, 388 – linearized 379 – multiplication 386, 448 – negative definite 425 – nonlinear pseudo-differential 380 – nonpositive 385, 453 – norm 385, 448, 450, 454, 457, 462, 463, 466, 470 – positive definite 413 – quadratic 379 – reflection 388 – self-adjoint 385, 386, 393, 440, 442, 453 – solution 404, 423, 426, 436 – trace 439 – transport 446, 452 – unbounded 440 Orlicz – class 238 – norm 240 Subject Index – space 238 – – criticality 242 Orlicz–Sobolev space 240 outgoing macroscopic information outgoing particle 387 395 P p-Laplace operator 52, 80, 82, 106, 112, 122, 124, 133, 141, 159, 179, 180, 187, 188, 195 (p, q)-Laplacian operator 206 Palais–Smale property 216 Palais–Smale sequence 281 Parseval relation 445 periodic solution 51, 482 phase space 372 – one-particle 377 Pohozaev identity 216, 218, 300 Poincaré inequality 214 potential 376 – cutoff hard 380, 391, 395, 409, 410, 412, 423, 431, 434 – hard 376, 395 – interaction 376, 387 – inverse power law 376, 380 – singularity 380 – soft 376 Q quantization – energy 279, 288 – mass 299 quantized blowup mechanism 326, 327 279, 315, 316, R range space 441, 446 Rayleigh quotient 337 reduced functional 113–115, 129, 146, 149, 150, 153 reflection 388 – law 388 – – bounced back 388 – – diffuse 388 – – nondeterministic 388 – – random 388 – – reverse 388 – – specular 388 regular extremal point 70 regularizing property 426, 430, 436 Rellich theorem 216, 452 renormalized solution 373 residual vanishing 319, 323 resolvent 440, 441, 447, 449, 453, 454, 473, 475 597 – equation 453 – set 449, 477 Riesz representation – theorem 397, 407 – theory 396 rotation invariant 471 S scattering theory 440, 441, 446 Schwarz inequality 384, 411, 428, 469 Schwarz symmetrization 311, 347, 348 semi-group 424–426, 429, 430, 437, 473–475, 477 – C0 476 shock profile 374 shooting method 227 singularity 380, 441, 454, 457, 470–472 Smoluchowski–Poisson equation 315 smoothing effect 446, 468, 477, 480 Sobolev conjugate 241 Sobolev embedding 216, 219 Sobolev exponent 280 Sobolev–Lorentz space 213, 266 Sobolev–Rellich theorem 466 solution 372 – global 372, 373, 437, 481 – local 372 solvability condition 374, 391, 395, 404–406, 415, 423 sound speed 391, 448 spectral analysis 357, 440, 441, 446, 447, 482 spectral gap 386 spectrum 385, 442, 446–450, 468 – continuous 440 – essential 448, 449 spherical fibering 54, 57, 58, 60, 61, 64, 68, 69 – method 57, 58 stability 374, 394, 434, 439, 475, 482 – boundary layer 423, 424, 431, 432 – stationary solution 374 – steady solution 390 stationary – point 63 – – conditionally 63 – problem 373, 376, 439 – – linear 440 – solution 372, 378, 424, 439, 474, 475, 482 – – asymptotically stable 372 – – non-Maxwellian 372, 374, 439 – – stable 374 – state 373 strongly indefinite 229 – functionals 228 598 sub-linear growth 431 subcritical growth 216, 223, 245 sup + inf inequality 321, 322 systems – gradient 3, 5–7 – Hamiltonian 3, 7–13 – strongly coupled 25, 26 – weakly coupled 25–27 T θ -regular 242 Talenti theorem 349 tilde map 246, 267 time sequence 373 time-asymptotic wave profile 374 time-decay estimate 474 time-periodic solution 374, 473, 474, 479 Toland duality 315 torus 372, 373, 482 trace 387, 397, 443, 445, 465 – operator 387 – space 443, 473 – theorem 443 transport 375, 376 traveling wave 374 Subject Index Trudinger inequality 221 Trudinger–Moser functional 314, 362 U unfolding Legendre transformation uniqueness 373, 375, 397 314 V velocity averaging 374, 440, 441, 450, 452, 460, 465 velocity distribution function 372, 389 viscosity coefficient 448 W wave – contact discontinuity 374 – pattern 374 – rarefaction 374 – shock 374 – sound 374, 473 weak Lp space 352 weight function 392, 395, 401, 409, 410, 432, 434, 443 Weyl theorem 386, 449 ... List of Contributors Contents of Volume I Contents of Volume II Contents of Volume III Contents of Volume IV v vii xi xiii xv xvii Semilinear Elliptic Systems: Existence, Multiplicity, Symmetry of. .. Contents of Volume IV Preface List of Contributors Contents of Volume I Contents of Volume II Contents of Volume III v vii xi xiii xv Rearrangements and Applications to Symmetry Problems in PDE... (v), − v = g(u) in Ω (3. 3) t In this case the Hamiltonian is H (u, v) = F (v) + G(u), where F (t) = f (s) ds, and similarly G is a primitive of g However, the treatment given in [26] of system