H ANDBOOK OF D IFFERENTIAL E QUATIONS S TATIONARY PARTIAL D IFFERENTIAL E QUATIONS VOLUME I This Page Intentionally Left Blank H ANDBOOK OF D IFFERENTIAL E QUATIONS S TATIONARY PARTIAL D IFFERENTIAL E QUATIONS Volume I Edited by M CHIPOT Institute of Mathematics, University of Zurich, Zurich, Switzerland P QUITTNER Institute of Applied Mathematics, Comenius University, Bratislava, Slovak Republic 2004 ELSEVIER Amsterdam • Boston • Heidelberg • London • New York • Oxford • Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo ELSEVIER B.V Sara Burgerhartstraat 25 P.O Box 211, 1000 AE Amsterdam, The Netherlands ELSEVIER Inc 525 B Street, Suite 1900 San Diego, CA 92101-4495 USA ELSEVIER Ltd The Boulevard, Langford Lane Kidlington, Oxford OX5 1GB UK ELSEVIER Ltd 84 Theobalds Road London WC1X 8RR UK © 2004 Elsevier B.V All rights reserved This work is protected under copyright by Elsevier B.V., and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use Permissions may be sought directly from Elsevier’s Rights Department in Oxford, UK: phone (+44) 1865 843830, fax (+44) 1865 853333, e-mail: permissions@elsevier.com Requests may also be completed on-line via the Elsevier homepage (http://www.elsevier.com/locate/permissions) In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 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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 First edition 2004 Library of Congress Cataloging in Publication Data A catalog record is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record is available from the British Library ISBN: 444 51126 Set ISBN: 444 51743 x The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper) Printed in The Netherlands Preface This handbook is Volume I in a multi-volume series devoted to stationary partial differential equations It is a collection of self contained, state-of-the-art surveys written by well-known experts in the field The authors have made an effort to achieve readability for mathematicians and scientists from other fields, and we hope that this series of handbooks will become a new reference for research, learning and teaching Partial differential equations represent one of the most rapidly developing topics in mathematics This is due to their numerous applications in science and engineering on one hand and to the challenge and beauty of associated mathematical problems on the other This volume consists of eight chapters covering a variety of elliptic problems and explaining many useful ideas, techniques and results Although the central theme is the mathematically rigorous analysis, many of the contributions are enriched by a plenty of figures originating in numerical simulations We thank all the contributors for their clearly written and elegant articles, and Arjen Sevenster at Elsevier for efficient collaboration M Chipot and P Quittner v This Page Intentionally Left Blank List of Contributors Bandle, C., Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland (Ch 1) Galdi, G.P., University of Pittsburgh, 15261 Pittsburgh, USA (Ch 2) Ni, W.-M., University of Minnesota, Minneapolis, MN 55455, USA (Ch 3) Pedregal, P., Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain (Ch 4) Reichel, W., Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland (Ch 1) Shafrir, I., Technion, Israel Institute of Technology, 32000 Haifa, Israel (Ch 5) Takáˇc, P., Universität Rostock, D-18055 Rostock, Germany (Ch 6) Tarantello, G., Università di Roma ‘Tor Vergata’, Dipartimento di Matematica, Via della Ricerca Scientifica, 1, 00133 Rome, Italy (Ch 7) Véron, L., Université de Tours, Parc de Grandmont, 37200 Tours, France (Ch 8) vii This Page Intentionally Left Blank Contents Preface List of Contributors v vii Solutions of Quasilinear Second-Order Elliptic Boundary Value Problems via Degree Theory C Bandle and W Reichel Stationary Navier–Stokes Problem in a Two-Dimensional Exterior Domain G.P Galdi Qualitative Properties of Solutions to Elliptic Problems W.-M Ni On Some Basic Aspects of the Relationship between the Calculus of Variations and Differential Equations P Pedregal On a Class of Singular Perturbation Problems I Shafrir Nonlinear Spectral Problems for Degenerate Elliptic Operators P Takáˇc Analytical Aspects of Liouville-Type Equations with Singular Sources G Tarantello Elliptic Equations Involving Measures L Véron Author Index Subject Index 71 157 235 297 385 491 593 713 721 ix Elliptic equations involving measures 711 [51] I Iscoe, On the support of measure-valued critical branching Brownian motion, Ann Probab 16 (1988), 200–221 [52] N.J Kalton and I.E Verbitsky, Nonlinear equations and weighted nor inequalities, Trans Amer Math Soc., 351 (1999), 3341–3397 [53] J.B Keller, On solutions of u = f (u), Comm Pure Appl Math 10 (1957), 503–510 [54] V.P Khavin and V.G Maz’ya, Nonlinear Potential Theory, Russian Math Surveys 27 (1972), 71–148 [55] V Kondratiev and A Nikishkin, On positive solutions of singular value problems for the equation u = uk , Russ J Math Phys (1993), 123–138 [56] N.V Krylov, Nonlinear Elliptic and Parabolic Equations of the Second Order, Reidel, Dordrecht–Boston– Lancaster–Tokyo (1987) [57] S.E Kuznetsov, σ -moderate solutions of Lu = uα and fine trace on the boundary, C R Math Acad Sci Paris Sér I 326 (1998), 1189–1194 [58] S.E Kuznetsov, On uniqueness of a solution of Lu = uα with given trace, Electron Comm Probab., to appear [59] D Labutin, Wiener regularity for large solutions of nonlinear equations, Ark Math 41 (2003), 307–339 [60] N.S Landkof, Foundation of Modern Potential Theory, Springer-Verlag, London–Berlin–Heidelberg–New York (1972) [61] J.F Le Gall, Les solutions positives de u = u2 dans le disque unité, C R Math Acad Sci Paris Sér I 317 (1993), 873–878 [62] J.F Le Gall, The Brownian snake and solutions of u = u2 in a domain, Probab Theory Related Fields 102 (1995), 393–432 [63] J.F Le Gall, A probabilistic Poisson representation for positive solutions of u = u2 in a domain, Comm Pure Appl Math 50 (1997), 69–103 [64] J.F Le Gall, A probabilistic approach to the trace on the boundary for solutions of semilinear parabolic partial differential equations, J Appl Math Stochastics Anal (1996), 399–414 [65] J.F Le Gall, Spatial Branching Processes, Random Snakes and Partial Differential Equations, Birkhäuser, Boston, MA (1999) [66] P.L Lions, Isolated singularities in semilinear problems, J Differential Equations 38 (1980), 441–450 [67] C Loewner and L Nirenberg, Partial differential equations invariant under conformal or projective transformations, Contributions to Analysis, L Ahlfors et al., eds, Academic Press, London–New York (1972), 245–272 [68] M Marcus and L Véron, Traces au bord des solutions positives d’équations elliptiques non-linéaires, C R Math Acad Sci Paris Sér I 321 (1995), 179–184 [69] M Marcus and L Véron, Traces au bord des solutions positives d’équations elliptiques et paraboliques non-linéaires: Résultats d’existence et d’unicité, C R Math Acad Sci Paris Sér I 323 (1996), 603–608 [70] M Marcus and L Véron, The boundary trace of positive solutions of semilinear elliptic equations: The subcritical case, Arch Ration Mech Anal 144 (1998), 201–231 [71] M Marcus and L Véron, The boundary trace of positive solutions of semilinear elliptic equations: The supercritical case, J Math Pures Appl 77 (1998), 481–524 [72] M Marcus and L Véron, Removable singularities and boundary traces, J Math Pures Appl 80 (2001), 879–900 [73] M Marcus and L Véron, The boundary trace and generalized B.V.P for semilinear elliptic equations with coercive absorption, Comm Pure Appl Math 56 (2003), 689–731 [74] M Marcus and L Véron, Boundary trace of positive solutions of nonlinear elliptic inequalities, Preprint (2003) [75] M Marcus and L Véron, On a new characterization of Besov spaces with negative exponents, Preprint (2001) [76] R.S Martin, Minimal positive harmonic function, Trans Amer Math Soc 49 (1941), 137–172 [77] V.G Maz’ya, Beurling’s theorem on a minimum principle for positive harmonic functions, Zap Nauchn Sem LOMI 30 (1972), 76–90 (in Russian); English transl.: J Soviet Math (1976), 367–379 [78] C Miranda, Partial Differential Equations of Elliptic Type, Springer-Verlag, Berlin–New York (1970) [79] J Moser, On Harnack’s theorem for elliptic differential equations, Comm Pure Appl Math 14 (1961), 577–591 712 L Véron [80] B Mselati, Classification et représentation probabiliste des solutions positives de u = u2 dans un domaine, Ph.D Thesis, Université Paris (2002) [81] B Mselati, Classification and Probabilistic Interpretation of the Positive Solutions of a Semilinear Elliptic Equation, Mem Amer Math Soc., Vol 168, Providence, RI (2004) [82] W.M Ni, On the elliptic equation u + K(x)u(n+2)/(n−2) , Indiana Univ Math J 31 (1982), 493–539 [83] R Osserman, On the inequality u f (u), Pacific J Math (1957), 1641–1647 [84] M Pierre, Problèmes semi-linéaires avec données mesures, Séminaire Goulaouic–Meyer–Schwartz XIII (1982–1983) [85] Y Pinchover, On positive solutions of second-order elliptic equations, stability results, and classification, Duke Math J 57 (1988), 955–980 [86] H Rademacher, Einige besondere Probleme partieller Differentialgleichungen, [43], 838–845 [87] A Ratto, M Rigoli and L Véron, Scalar curvature and conformal deformation of hyperbolic space, J Funct Anal 121 (1994), 543–572 [88] Y Richard and L Véron, Isotropic singularities of nonlinear elliptic inequalities, Ann Inst H Poincaré Anal Non Linéaire (1989), 37–72 [89] G Stampacchia, Le problème de Dirichlet pour des équations elliptiques du second ordre coefficients discontinus, Ann Inst Fourier (Grenoble) 15 (1965), 189–258 [90] G Stampacchia, Equations elliptiques du second ordre coefficients discontinus, Séminaires de Math Sup 16, Les Presses de l’Université de Montréal (1966) [91] G Stampacchia, Some limit cases of Lp -estimates for solutions of second order elliptic equations, Comm Pure Appl Math 16 (1963), 505–510 [92] E Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math Ser., Vol 30, Princeton Univ Press (1970) [93] H Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam (1978) [94] J.L Vazquez, On a semilinear equation in R2 involving bounded measures, Proc Roy Soc Edinburgh 95A (1983), 181–202 [95] J.L Vazquez, An a priori interior estimate for the solution of a nonlinear problem representing weak diffusion, Nonlinear Anal (1981), 119–135 [96] L Véron, Singularities of Solutions of Second Order Quasilinear Equations, Pitman Research Notes in Math., Vol 353, Longman, Harlow (1996) [97] L Véron, Semilinear elliptic equations with uniform blow-up on the boundary, J Anal Math 59 (1992), 231–250 [98] L Véron, Weak and strong singularities of nonlinear elliptic equations, Proc Symp Pure Math 45(2) (1986), 477–495 [99] L Véron, Generalized boundary value problems for nonlinear elliptic equations, Electron J Differ Equ Conf (2000), 313–342 [100] L Véron, Comportement asymptotique des solutions d’équations elliptiques semi-linéaires dans RN , Ann Mat Pura Appl 127 (1981), 25–50 [101] L Véron, Singular solutions of some nonlinear elliptic equations, Nonlinear Anal (1981), 225–242 [102] L Véron, Singularités éliminables d’équations elliptiques non linéaires, J Differential Equations 41 (1981), 87–95 [103] K Yosida, Functional Analysis, 2nd Edition, Springer-Verlag, London–Berlin–Heidelberg–New York (1968) 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 first and co-author(s) Acerbi, E 272, 293 [1] Adams, D.R 510, 587 [Ad]; 597, 627, 628, 671, 672, 677, 700, 709 [1]; 709 [2] Aftalion, A 47, 68 [1] Akhiezer, N.I 242, 293 [2] Allaire, G 81, 154 [1] Allegretto, W 402, 406, 487 [1] Amann, H 5, 12, 15, 23, 28, 32, 49, 50, 68 [2]; 68 [3]; 68 [4]; 182, 219, 228 [A1]; 228 [A2]; 228 [A3]; 228 [AC]; 709 [3] Ambrosetti, A 32, 50, 65, 68 [4]; 68 [5]; 68 [6]; 174–177, 219, 228 [AMN1]; 228 [AMN2]; 228 [AMN3]; 228 [AR]; 500, 586, 587 [AMN] Ambrosio, L 212, 229 [AmC] Amick, C.J 76, 78, 79, 114, 115, 117, 123, 124, 133, 138, 142, 146, 154 [2]; 154 [3] Anane, A 391, 392, 395, 435, 441, 443, 470, 487 [2]; 487 [3]; 487 [4] Ancona, A 609, 709 [4] André, N 301, 302, 339, 340, 342–348, 352, 355, 356, 359, 363, 373, 380, 382 [2]; 382 [3]; 382 [4]; 382 [5] Angenent, S.B 348, 382 [1] Arcoya, D 409, 459, 487 [5] Armitage, D.H 709 [5] Artino, R.A 52, 54, 58, 68 [7] Atkinson, K 289, 293 [3] Aubin, T 495, 510, 578, 579, 582, 587 [Au] Aviles, P 679, 709 [6] Avudainayagam, A 81, 154 [4]; 154 [5] Ball, J.M 242, 245, 247, 257, 287, 293 [6]; 293 [7]; 293 [8]; 293 [9]; 293 [10]; 293 [11]; 293 [12]; 293 [13]; 293 [14]; 294 [15] Bandle, C 45, 68 [8]; 543, 586, 587 [B]; 709 [7]; 709 [8] Baraket, S 500, 586, 587 [BP] Baras, P 596, 597, 632, 635, 639, 662, 666, 677, 709 [9]; 709 [10] Bartolucci, D 493, 497, 499, 523, 527, 543, 559, 570, 586, 587 [B1]; 587 [B2]; 587 [BCLT]; 588 [BT1]; 588 [BT2] Bebernes, J 493, 588 [BE] Beckner, W 510, 588 [Be] Benilan, Ph 596, 616, 617, 645, 709 [11]; 709 [12] Benjamin, T.B 15, 68 [9] Berestycki, H 211, 212, 229 [BCN1]; 229 [BCN2]; 229 [BN] Berger, M.S 340, 382 [7]; 697, 709 [13] Berker, R 81, 92, 154 [7] Bers, L 218, 229 [Be] Bethuel, F 163, 229 [BBH]; 300–302, 321, 323–325, 327, 336–338, 343, 354, 355, 382 [8]; 382 [9]; 382 [11]; 382 [12]; 382 [13]; 499, 588 [BBH] Bhattacharya, T 213, 218, 229 [B] Bidaut-Véron, M.F 609, 616, 675, 709 [14]; 709 [15] Bieberbach, L 219, 229 [Bi]; 709 [16] Birindelli, I 409, 487 [6] Blake, J.R 77, 89, 92, 154 [8]; 154 [9] Boccardo, L 709 [17] Bogomolnyi, E.B 493, 588 [Bo] Böhme, R 50, 57, 68 [10] Boissonade, J 187, 229 [CDBD] Bourgain, J 338, 382 [10] Boutet de Monvel-Berthier, A 318, 382 [14] Babenko, K.I 113, 154 [6] Bahri, A 493, 586, 587 [BC1]; 587 [BC2]; 587 [Ba] Bakelman, I.J 276, 278, 293 [4] Balder, E.J 256, 293 [5] Baldo, S 315, 382 [6] 713 714 Author Index Brascamp, H.J 213, 229 [BL] Brennen, C 77, 154 [10] Brezis, H 16, 33, 50, 68 [5]; 68 [11]; 68 [12]; 68 [13]; 68 [14]; 68 [15]; 163, 208, 229 [BBH]; 229 [BrN]; 300–302, 318, 321, 323–325, 327, 330, 336–339, 343, 346, 354, 355, 382 [8]; 382 [9]; 382 [10]; 382 [11]; 382 [12]; 382 [15]; 382 [16]; 382 [17]; 382 [18]; 493, 497–499, 505, 512, 513, 515, 524, 530, 533, 538, 539, 543, 588 [BBH]; 588 [BLS]; 588 [BM]; 588 [BV]; 596, 605, 616, 617, 638, 645, 673, 679, 709, 709 [11]; 709 [12]; 709 [18]; 709 [19]; 709 [20]; 709 [21]; 710 [22]; 710 [23] Brouwer, L.E.J 3, 4, 68 [16] Browder, F.E 483, 487 [7] Brown, B.M 409, 487 [8] Buffoni, B 58, 68 [17] Busca, J 216, 217, 229 [BS] Cabré, X 212, 229 [AmC]; 586, 588 [CLS]; 673, 709 [20] Caffarelli, L.A 211–214, 229 [BCN1]; 229 [BCN2]; 229 [CF]; 229 [CGS]; 493, 503, 588 [CGS]; 588 [CY]; 679, 710 [24] Caglioti, E 493, 586, 588 [CLMP1]; 588 [CLMP2] Calderon, A.P 710 [25] Callahan, T.K 188, 229 [CK] Cantrell, R.S 182, 229 [CC] Casten, R.G 162, 192, 229 [CH] Castets, V 187, 229 [CDBD] Castro, A 162, 229 [CCN] Cerami, G 50, 68 [5] Cesari, L 242, 248, 294 [16] Chae, D 493, 500, 588 [CI1]; 588 [CI2]; 588 [CI3]; 588 [CT]; 588 [ChK1]; 588 [ChK2] Chan, H 493, 500, 588 [CFL] Chandrasekhar, S 159, 229 [Ch]; 588 [Cha] Chang, A 493, 494, 510, 511, 588 [ChY1]; 588 [ChY2]; 588 [ChY3] Chang, I.-D 81, 154 [11] Chang, K.C 219, 229 [C]; 494, 588 [CL] Chanillo, S 493, 494, 500, 588 [CK1]; 588 [CK2]; 589 [CK3] Chen, C.C 493, 494, 498–500, 515, 530, 559, 570, 586, 587 [BCLT]; 589 [ChL1]; 589 [ChL2]; 589 [ChL3]; 589 [ChL4] Chen, G 219, 221, 222, 226, 228, 229 [CDNZ]; 229 [CEZ]; 229 [CNPZ]; 229 [CNZ]; 230 [DCC]; 230 [DCNZ] Chen, W.X 34, 68 [18]; 211, 229 [CL]; 493, 494, 496, 503, 510, 582, 589 [CD]; 589 [Ch]; 589 [CL1]; 589 [CL2]; 589 [CL3] Chen, X 493, 589 [CHMY] Chen, X.X 553, 589 [Che] Chen, X.Y 642, 643, 645, 694, 710 [27] Cheng, K.-S 217, 229 [CN] Chipot, M 261, 294 [17]; 500, 589 [CSW] Choi, Y.S 219, 230 [CM] Choquet, G 627, 710 [28] Chou, K.S 496, 503, 589 [CW] Chow, S.N 51, 68 [19] Christov, C 81, 154 [12] Ciarlet, P.G 268, 269, 272, 289, 294 [18]; 294 [19] Cignoli, R 616, 710 [26] Cildress, S 493, 589 [CP] Clarke, F.H 294 [20] Clément, Ph 407, 409, 487 [9]; 487 [10] Coddington, E.A 199, 230 [CoL] Coffman, C.V 208, 230 [Co] Cohen, D.S 32, 69 [42] Comte, M 338, 382 [19] Coron, M 586, 587 [BC1]; 587 [BC2] Cosner, C 182, 229 [CC] Cossio, J 162, 229 [CCN] Costa, D 219, 230 [DCC] Cottlar, M 616, 710 [26] Courant, R 242, 275, 294 [21] Crandall, M 616, 709 [12] Crandall, M.G 48, 50, 58, 68 [20]; 68 [21]; 68 [22]; 68 [23]; 219, 228 [AC]; 493, 589 [CR] Cuesta, M 397, 409, 487 [11]; 487 [12]; 487 [13] Dacorogna, B 242, 250, 253, 271, 278, 284, 294 [22]; 294 [23]; 294 [24]; 294 [25] Dal Maso, G 632, 710 [29] Damascelli, L 215, 230 [DPR] Dancer, E.N 477, 487 [14] Dautray, R 613, 676, 710 [30] De Coster, C 32, 69 [24]; 475, 478, 487 [15] de Figueiredo, D.G 33, 69 [25]; 409, 487 [11] De Kepper, P 187, 229 [CDBD] de la Pradelle, A 629, 710 [42] de Thélin, F 390, 391, 403, 406, 407, 409, 414, 487 [10]; 488 [27]; 488 [28]; 488 [29]; 488 [30]; 488 [31] Deimling, K 4, 5, 16, 36, 69 [26]; 483, 487 [16] del Pino, M 334, 356, 382 [20]; 390, 391, 403, 422, 451, 489 [48]; 489 [49] Demoulini, S 289, 294 [26] Deng, Y 219, 229 [CDNZ]; 230 [DCNZ] Dhersin, J.L 710 [31] Díaz, J.I 390, 399, 401, 403, 487 [17] DiBenedetto, E 395, 443, 470, 488 [18] Ding, W.-Y 177, 208, 214, 230 [DN]; 493, 494, 499, 510, 586, 589 [CD]; 589 [DJLPW]; 589 [DJLW1]; 589 [DJLW2]; 589 [DJLW3] Author Index Ding, Z 219, 230 [DCC] Doelman, A 189, 230 [DGK] Doob, J 613, 710 [32] Drábek, P 391, 392, 409, 410, 422, 433, 437, 438, 440–442, 446, 451, 456, 458, 469, 470, 478, 480, 483–487, 488 [19]; 488 [20]; 488 [21]; 488 [22]; 488 [23]; 488 [24]; 488 [25]; 488 [26]; 489 [48] Du, Y 708, 710 [33] Dugundji, J 8, 14, 69 [27] Dulos, E 187, 229 [CDBD] Dunne, G 493, 589 [D1]; 589 [D2] Dynkin, E.B 597, 609, 683, 696, 705, 706, 710 [34]; 710 [35]; 710 [36]; 710 [37]; 710 [38]; 710 [39]; 710 [40] Eberly, D 493, 588 [BE] Ekeland, I 242, 293, 294 [27]; 294 [28]; 294 [29] Elgueta, M 390, 403, 489 [49] Englert, B.-G 221, 229 [CEZ] Epstein, I.R 187, 231 [LE1]; 231 [LE2] Esposito, P 500, 589 [E] Evans, L.C 272, 294 [30]; 303, 360, 382 [21]; 382 [22] Fabbri, J 710 [41] Faxén, H 74, 154 [13] Federer, H 309, 382 [23] Felmer, P.L 334, 356, 382 [20] Feyel, D 629, 710 [42] Fife, P.C 177, 230 [F] Finn, R 75, 77, 78, 81, 106, 117, 135, 154 [11]; 154 [14]; 154 [15]; 154 [16] Fleckinger, J 390, 391, 403, 406, 407, 409, 414, 427–430, 433, 469, 487 [10]; 488 [27]; 488 [28]; 488 [29]; 488 [30]; 488 [31]; 488 [32]; 488 [33]; 488 [34] Flucher, M 499, 589 [FGS]; 589 [FS]; 589 [Fl] Fomin, S.V 242, 294 [32] Fonseca, I 303, 311, 359, 382 [24] Fontana, L 494, 510, 589 [F] Fraenkel, G 697, 709 [13] Fraenkel, L.E 340, 382 [7] Franchi, B 215, 230 [FL] Frank, P 710 [43] Friedman, A 213, 214, 229 [CF]; 261, 294 [31] Fu, C.C 493, 500, 588 [CFL] Fuˇcík, S 390–392, 417, 422, 453, 488 [35]; 488 [36]; 488 [37]; 489 [44] Fujita, H 75, 78, 117, 154 [17] Fusco, N 272, 293 [1] 715 Galdi, G.P 75–80, 82, 83, 85, 86, 89, 94–97, 100–102, 106, 111, 114–120, 123, 132–135, 137, 147, 148, 150, 151, 154 [18]; 154 [19]; 154 [20]; 154 [21]; 154 [22]; 154 [23]; 154 [24]; 154 [25]; 154 [26]; 155 [27]; 155 [28] Gallouët, T 709 [17]; 710 [44]; 710 [45] Gámez, J.L 409, 459, 487 [5]; 488 [38] Garcia-Prada, O 494, 589 [G] Gardiner, S.J 709 [5] Gardner, R.A 189, 230 [DGK] Gariepy, R.F 272, 294 [30]; 303, 382 [21] Garroni, A 499, 589 [FGS]; 589 [GS] Gazzola, F 48, 69 [28] Geetha, J 81, 154 [5] Gelfand, I.M 242, 294 [32]; 493, 590 [Ge] Georgescu, V 318, 382 [14] Ghergu, M 50, 69 [29] Ghoussoub, N 212, 213, 230 [GG1]; 230 [GG2]; 293, 294 [29] Giaquinta, M 247, 272, 292, 293, 294 [33]; 294 [34] Gidas, B 26, 45, 69 [30]; 69 [31]; 159, 173, 174, 176, 204–206, 208, 211, 229 [CGS]; 230 [GNN1]; 230 [GNN2]; 499, 503, 588 [CGS]; 590 [GNN]; 679, 710 [24]; 710 [46] Gierer, A 164, 230 [GM] Gilbarg, D 76, 78, 114, 123, 125, 126, 133, 139, 150, 155 [29]; 155 [30]; 247, 274, 294 [35]; 308, 346, 359, 369, 382 [26]; 393, 418, 419, 428, 488 [39]; 507–510, 575, 590 [GT]; 595, 609, 646, 710 [47] Gilbarg, G 17, 19, 20, 22, 31, 69 [32] Girg, P 391, 409, 410, 433, 442, 446, 451, 456, 458, 480, 483, 485–487, 488 [22]; 488 [23]; 488 [38] Giusti, E 303, 304, 316, 382 [25] Gmira, A 597, 617, 681, 692, 694, 695, 710 [48] Gossez, J.-P 32, 69 [33]; 391, 406, 407, 409, 414, 487 [11]; 488 [27]; 488 [28] Gray, P 188, 230 [GS] Grillot, M 706, 710 [49] Grossi, M 170, 172, 207, 230 [GPW]; 230 [Gr] Grun-Rehomme, M 628, 710 [50] Gui, C 161, 170, 172, 177, 189, 211–213, 230 [G]; 230 [GG1]; 230 [GG2]; 230 [GNW1]; 230 [GNW2]; 230 [GW1]; 230 [GW2]; 230 [GWW] Guo, Z 708, 710 [33] Gutiérrez, C.E 278, 294 [36] Hale, J.K 51, 68 [19]; 193, 197, 230 [HS]; 230 [HV] Hamel, G 106, 139, 155 [31] 716 Author Index Han, J 493, 590 [Ha] Han, W 289, 293 [3] Han, Z.C 494, 590 [H] Hastings, S 493, 589 [CHMY] Hedberg, L.I 627, 628, 672, 700, 709 [1] Heinz, E 16, 69 [34] Hélein, F 163, 229 [BBH]; 300–302, 321, 323–325, 327, 336–338, 343, 354, 355, 382 [8]; 382 [9]; 499, 588 [BBH] Henrard, M 32, 69 [24]; 475, 478, 487 [15] Hernández, J 390, 403, 409, 488 [29]; 488 [30]; 488 [31] Hess, P 29, 31, 33, 67, 69 [35]; 69 [36]; 69 [37]; 69 [38]; 477, 487 [14]; 488 [40] Hestenes, H.R 242, 292, 294 [37] Hilbert, D 242, 243, 294 [38] Hildebrandt, S 292, 293, 294 [34] Hofer, H 11, 69 [39] Holland, C.J 162, 192, 229 [CH] Holmes, P.J 293 [10] Holubová, G 409, 410, 433, 437, 438, 440, 441, 469, 478, 480, 488 [24] Hong, C.W 510, 590 [Ho] Horstmann, D 186, 230 [Ho] Huang, Y.-X 402, 406, 487 [1] Hutchinson, G.E 182, 230 [Hu] Huy, C.U 219, 231 [HMW] Idogawa, T 402, 488 [41] Imanuvilov, O 493, 500, 588 [CI1]; 588 [CI2]; 588 [CI3] Ioffe, A.D 293, 294 [39] Iscoe, I 711 [51] Ishige, K 315, 382 [27] Ishihara, K 219, 231 [I] Jaffe, A 493, 590 [JT] Jäger, W 58, 69 [40] James, R.D 293 [10]; 293 [11]; 293 [12] Jang, J 188, 212, 231 [J]; 231 [JNT] Jeanjean, L 583, 590 [J] Jerrard, R 338, 383 [28]; 383 [29] Jimbo, S 193, 231 [JM] John, A.O 392, 422, 453, 489 [44] Joseph, D.D 38, 69 [41] Jost, J 493, 494, 499, 500, 510, 586, 589 [DJLPW]; 589 [DJLW1]; 589 [DJLW2]; 589 [DJLW3]; 590 [JoW1]; 590 [JoW2] Jothiram, B 81, 154 [4] Judd, S.L 188, 231 [JS] Kabeya, Y 231 [KN] Kalton, N.J 597, 673, 675, 677, 711 [52] Kaper, T.J 189, 230 [DGK] Kato, T 31, 69 [38]; 104, 155 [32]; 413, 417, 488 [42] Kawasaki, K 182, 233 [SKT] Kazdan, J 493, 494, 586, 590 [K]; 590 [KW1]; 590 [KW2] Keller, E.F 186, 231 [KS]; 493, 590 [KS] Keller, H.B 32, 69 [42] Keller, J.B 47, 69 [43]; 638, 711 [53] Kennington, A 213, 231 [Ke] Khavin, V.P 672, 711 [54] Kiessling, M 493, 494, 500, 586, 588 [CK1]; 588 [CK2]; 589 [CK3]; 590 [Ki1]; 590 [Ki2] Kim, N 493, 588 [ChK1]; 588 [ChK2] Kinderlehrer, D 261, 283, 289, 294 [40]; 294 [41]; 294 [42]; 294 [43] Kirchheim, B 263, 287, 293 [13]; 294 [44] Knobloch, E 188, 229 [CK] Kondratiev, V 711 [55] Korevaar, N 213, 214, 231 [K]; 231 [KL]; 500, 586, 590 [KMPS] Krasnosel’skii, M.A 14–16, 51, 57, 58, 69 [44] Krejˇcí, P 488 [25] Kristensen, J 287, 293 [13] Krylov, N.V 609, 711 [56] Kuˇcera, M 391, 488 [36] Kufner, A 392, 422, 423, 453, 488 [43]; 489 [44] Kuznetsov, S.E 683, 696, 705, 706, 710 [36]; 710 [37]; 710 [38]; 710 [39]; 710 [40]; 711 [57]; 711 [58] Labutin, D 711 [59] Laetsch, Th 49, 69 [45] Lanconelli, E 215, 230 [FL] Landesman, E 65, 66, 69 [46] Landkof, N.S 613, 711 [60] Lassoued, L 339, 383 [30] Lazer, A.C 45, 46, 65–67, 69 [46]; 69 [47]; 69 [48] Le Gall, J.F 597, 598, 683, 696, 700, 706, 710 [31]; 711 [61]; 711 [62]; 711 [63]; 711 [64]; 711 [65] Ledyaev, Y.S 294 [20] Lee, K.J 188, 231 [LMPS] Lengyel, I 187, 231 [LE1]; 231 [LE2] Leray, J 3, 5, 22, 69 [49]; 74, 75, 78, 114–116, 155 [33]; 155 [34] Levinson, N 199, 230 [CoL] Lewis, J 213, 214, 231 [KL] Li, C 34, 68 [18]; 211, 229 [CL]; 493, 496, 503, 582, 589 [CL1]; 589 [CL2]; 589 [CL3] Li, J 493, 494, 499, 510, 586, 589 [DJLPW]; 589 [DJLW1]; 589 [DJLW2]; 589 [DJLW3] Li, Y 219, 221, 231 [LZ1]; 231 [LZ2]; 231 [LZ3] Author Index Li, Y.Y 208, 231 [L]; 493, 494, 498, 499, 515, 527, 530, 533, 538, 539, 543, 577, 588 [BLS]; 590 [L1]; 590 [L2]; 590 [LS] Li, Yi 160, 209, 211, 215, 217, 231 [Li]; 231 [LiN] Licois, J.R 710 [41] Lieb, E 213, 229 [BL] Lieberman, G 395, 443, 470, 489 [45] Lin, C.-S 165, 177, 194, 207, 208, 231 [LNT]; 231 [LT]; 231 [LW]; 231 [LinN]; 493, 494, 498–500, 515, 530, 559, 570, 586, 587 [BCLT]; 588 [CFL]; 589 [ChL1]; 589 [ChL2]; 589 [ChL3]; 589 [ChL4]; 590 [Li] Lin, F.H 163, 231 [LR]; 231 [Lin]; 338, 383 [31]; 500, 590 [Lin] Lindqvist, P 391, 401, 489 [46] Lions, J.L 613, 676, 710 [30] Lions, P.-L 33, 69 [25]; 493, 586, 588 [CLMP1]; 588 [CLMP2]; 597, 679, 709 [21]; 711 [66] Liouville, J 493, 501, 590 [Lio] Liu, J.Q 494, 588 [CL] Lizorkin, P.I 97, 155 [35] Loewner, C 597, 711 [67] Lou, Y 182–185, 231 [LN1]; 232 [LN2]; 232 [LNY] Lucia, M 493, 500, 586, 588 [CLS]; 590 [LN] Lundgren, T.S 38, 69 [41] Luskin, M 294 [45] Ma, L 499, 590 [ML] Maginu, K 194, 232 [Mg] Malchiodi, A 48, 69 [28]; 173–176, 228 [AMN1]; 228 [AMN2]; 228 [AMN3]; 232 [M]; 232 [MM1]; 232 [MM2]; 232 [MNW]; 500, 586, 587 [AMN] Malek, J 287, 294 [46] Manásevich, R.F 390–392, 403, 409, 422, 433, 451, 480, 487 [10]; 488 [22]; 489 [47]; 489 [48]; 489 [49] Mancini, G 32, 68 [4] Marcellini, P 271, 294 [25] Marchioro, C 493, 586, 588 [CLMP1]; 588 [CLMP2] Marcus, M 598, 610, 683, 690, 696, 697, 700, 704–707, 709 [7]; 709 [8]; 711 [68]; 711 [69]; 711 [70]; 711 [71]; 711 [72]; 711 [73]; 711 [74]; 711 [75] Martin, R.S 613, 711 [76] Matano, H 162, 189, 190, 192–194, 232 [Ma1]; 232 [Ma2]; 642, 643, 645, 694, 710 [27] Maz’ya, V.G 672, 711 [54]; 711 [77] Mazzeo, R 500, 586, 590 [KMPS]; 590 [MP1]; 590 [MP2] McCormick, W.D 188, 231 [LMPS] 717 McKenna, P.J 45, 46, 67, 69 [47]; 69 [48]; 219, 230 [CM]; 231 [HMW] McLeod, J 493, 589 [CHMY] McShane, E.J 242, 245, 295 [47] Meinhardt, H 164, 230 [GM] Merle, F 330, 382 [16]; 493, 497, 505, 512, 513, 515, 524, 588 [BM] Miranda, C 17, 69 [50]; 609, 711 [78] Mironescu, P 338, 339, 382 [10]; 382 [19]; 383 [30]; 383 [32]; 383 [33] Mizel, V.J 247, 293 [14]; 294 [15] Modica, L 299, 300, 303, 308, 310, 383 [34] Montenegro, M 173, 174, 232 [MM1]; 232 [MM2] Morel, J.M 710 [44]; 710 [45] Morita, Y 193, 231 [JM] Morrey, Ch.B 242, 244, 269, 295 [48]; 295 [49] Moseley, J.L 500, 590 [M] Moser, J 493, 510, 590 [Mo]; 618, 711 [79] Mselati, B 706, 712 [80]; 712 [81] Müller, S 256, 263, 279, 294 [44]; 295 [50]; 295 [51]; 499, 589 [FGS]; 589 [FS]; 589 [GS] Murrey, J.D 493, 590 [Mu] Nagai, T 186, 232 [NS] Nagasaki, K 208, 232 [NSu] Nagashi, K 499, 500, 586, 591 [NS] Nagumo, M 16, 32, 69 [51]; 69 [52]; 69 [53] Neˇcas, J 287, 294 [46]; 390, 391, 417, 422, 488 [36]; 488 [37] Nehari, Z 177, 232 [Ne] Neuberger, J.M 162, 229 [CCN] Ni, W.-M 45, 69 [30]; 159–162, 165, 167, 169, 170, 173–179, 182–185, 188, 189, 194, 197, 199–209, 211, 214, 215, 217, 219, 222, 226, 228, 228 [AMN1]; 228 [AMN2]; 228 [AMN3]; 229 [CDNZ]; 229 [CN]; 229 [CNPZ]; 229 [CNZ]; 230 [DCNZ]; 230 [DN]; 230 [GNN1]; 230 [GNN2]; 230 [GNW1]; 230 [GNW2]; 231 [JNT]; 231 [KN]; 231 [LN1]; 231 [LNT]; 231 [LiN]; 231 [LinN]; 232 [LN2]; 232 [LNY]; 232 [MNW]; 232 [N1]; 232 [N2]; 232 [NPY]; 232 [NT1]; 232 [NT2]; 232 [NT3]; 232 [NT4]; 232 [NTY1]; 232 [NTY2]; 232 [NTa]; 232 [NW]; 232 [NY]; 494, 499, 500, 503, 579, 586, 587 [AMN]; 590 [GNN]; 591 [N]; 591 [NW]; 660, 712 [82] Nikishkin, A 711 [55] Nirenberg, L 16, 45, 68 [13]; 68 [14]; 69 [30]; 159, 173, 174, 176, 204–206, 208, 211, 212, 229 [BCN1]; 229 [BCN2]; 229 [BN]; 229 [BrN]; 230 [GNN1]; 230 [GNN2]; 318, 382 [17]; 499, 503, 590 [GNN]; 597, 711 [67] 718 Author Index Nolasco, M 493, 494, 500, 510, 581, 586, 587, 590 [LN]; 591 [NT1]; 591 [NT2]; 591 [NT3]; 591 [No] Novotný, A 75, 154 [25] Nussbaum, R.D 33, 69 [25] Obata, M 494, 591 [Ob] Odqvist, F.K.G 116, 155 [36] Ohtsuka, H 576, 591 [OS] Olesen, P 494, 591 [Ol] Omari, P 32, 69 [33] Onofri, E 493, 510, 591 [On] Oprea, J 252, 278, 295 [52] Orlandi, G 338, 382 [11]; 382 [12] Oseen, C.W 74, 92, 155 [37]; 155 [38] Osserman, R 47, 70 [54]; 638, 712 [83] Oswald, L 339, 346, 382 [18] Ôtani, M 402, 488 [41] Otto, S.R 77, 154 [9] Owen, N.C 315, 316, 383 [35] Pacard, F 338, 383 [36]; 500, 586, 587 [BP]; 590 [KMPS]; 590 [MP1]; 590 [MP2]; 591 [PR]; 591 [Pa] Pacella, F 215, 230 [DPR] Padula, M 75, 154 [25] Pan, X 163, 232 [Pan] Pao, C.V 219, 232 [P1]; 232 [P2]; 232 [P3] Parter, S.V 218, 233 [P] Pearson, J.E 188, 231 [LMPS]; 233 [Pe] Pedregal, P 256, 258, 279, 282–284, 289, 294 [40]; 294 [41]; 294 [42]; 295 [53]; 295 [54]; 295 [55] Pego, R.L 293 [10] Peitgen, H 36, 70 [55] Peletier, L.A 407, 487 [9] Peng, X 493, 499, 589 [DJLPW] Percus, J.K 493, 589 [CP] Perronnet, A 219, 228, 229 [CNPZ] Petryshyn, W.V 483, 487 [7] Pierre, M 596, 597, 632, 635, 639, 662, 666, 671, 672, 677, 709 [2]; 709 [9]; 709 [10]; 712 [84] Pinchover, Y 609, 712 [85] Pistoia, A 170, 172, 230 [GPW] Pohozaev, S.I 489 [50] Poincaré, H 493, 591 [P] Polacik, P 162, 189, 197, 199, 211, 232 [NPY]; 233 [PY] Polya, G 204, 233 [PS] Prajapat, J 496, 503, 504, 591 [PT] Prodi, G 65, 68 [6] Protter, M.H 31, 70 [56]; 216, 233 [PW] Pulvirenti, M 493, 586, 588 [CLMP1]; 588 [CLMP2] Purice, R 318, 382 [14] Quittner, P 34, 70 [57]; 709 [3] Rabier, P.J 106, 114, 154 [26] Rabinowitz, P.H 11, 48, 50, 54, 55, 58, 60, 68 [20]; 68 [21]; 68 [22]; 68 [23]; 70 [58]; 70 [59]; 70 [60]; 70 [61]; 177, 219, 228 [AR]; 437, 438, 440, 462, 464, 465, 489 [51]; 493, 584, 589 [CR]; 591 [R] Rademacher, H 712 [86] Rado, T 16, 70 [62] R˘adulescu, V 50, 69 [29] Ramakrishna, J 81, 154 [4] Ramaswamy, M 215, 230 [DPR] Ratto, A 597, 660, 706, 712 [87] Reichel, W 47, 68 [1]; 409, 487 [8] Reichelderfer, P.V 16, 70 [62] Rey, O 499, 591 [Re] Ricciardi, T 493, 586, 591 [RT1]; 591 [RT2]; 591 [Ri] Richard, Y 712 [88] Rigoli, M 597, 660, 706, 712 [87] Rivière, T 163, 231 [LR]; 330, 338, 355, 382 [13]; 382 [16]; 383 [31]; 383 [36]; 383 [37]; 500, 591 [PR] Robinson, S.B 392, 488 [26] Rockafellar, R.T 242, 293, 294 [39]; 295 [56] Rokyta, M 287, 294 [46] Rubinstein, J 301, 315, 316, 383 [35]; 383 [38] Rudin, W 256, 295 [57] Ruzicka, M 287, 294 [46] Saa, J.E 390, 399, 401, 403, 487 [17] Sakamoto, K 197, 230 [HS] Saloff-Coste, L 510, 591 [Sa] Sanchon, M 586, 588 [CLS] Sandier, E 338, 383 [39]; 383 [40]; 383 [41]; 383 [42] Sattinger, D.H 12, 32, 51, 55, 70 [63]; 70 [64]; 196, 219, 233 [Sa] Savin, O 212, 233 [S] Sazonov, L.I 78, 133, 137, 138, 155 [39] Scarpellini, B 45, 68 [8] Schaefer, H.H 14, 70 [65]; 254, 295 [58] Schaeffer, D.G 208, 233 [Sc] Schauder, J 3, 5, 22, 69 [49]; 116, 155 [34] Schmitt, K 36, 58, 62, 64, 69 [40]; 70 [55]; 70 [66] Schoen, R 16, 70 [67]; 500, 586, 590 [KMPS]; 591 [Sc1]; 591 [Sc2] Author Index Scott, S.K 188, 230 [GS] Segel, L.A 186, 231 [KS]; 493, 590 [KS] Senba, T 186, 232 [NS] Serfaty, S 338, 383 [40]; 383 [41]; 383 [42]; 383 [43]; 383 [44] Serrin, J 215, 217, 233 [SZ]; 233 [Se] Shafrir, I 301, 302, 339, 340, 342–348, 352, 355, 356, 359, 363, 373, 380, 382 [2]; 382 [3]; 382 [4]; 382 [5]; 493, 498, 500, 515, 530, 533, 538, 539, 543, 588 [BLS]; 589 [CSW]; 590 [LS]; 591 [SW1]; 591 [SW2]; 591 [Sh] Shigesada, N 182, 233 [SKT] Silber, M 188, 231 [JS] Simader, C.G 80, 83, 85, 155 [27]; 155 [40] Sirakov, B 216, 217, 229 [BS] Skrypnik, I.V 483, 484, 489 [52] Smith, D.R 75, 77, 78, 106, 134, 135, 154 [15]; 154 [16]; 155 [41] Sohr, H 78, 80, 133, 135, 137, 155 [28]; 155 [40] Soner, M 338, 383 [29] Souˇcek, J 390, 417, 422, 488 [37] Souˇcek, V 390, 417, 422, 488 [37] Souplet, P 34, 70 [57] Spruck, J 26, 69 [31]; 211, 229 [CGS]; 493, 500, 503, 588 [CGS]; 592 [SY1]; 592 [SY2]; 592 [SY3]; 592 [Sp]; 679, 710 [24]; 710 [46] Stampacchia, G 261, 294 [43]; 605, 712 [89]; 712 [90]; 712 [91] Stein, E 712 [92] Stern, R.J 294 [20] Sternberg, P 300, 303, 308, 310, 311, 315, 316, 359, 383 [35]; 383 [45]; 383 [46] Stokes, G.G 73, 74, 155 [42] Strauss, W 605, 617, 710 [22] Struwe, M 16, 70 [68]; 293, 295 [59]; 301, 327, 332, 337, 343, 355, 383 [47]; 383 [48]; 390, 403, 462, 489 [53]; 499, 510, 583, 584, 586, 592 [ST]; 592 [St1]; 592 [St2]; 592 [St3]; 592 [St4] Suzuki, T 208, 232 [NSu]; 493, 499, 500, 576, 586, 591 [NS]; 591 [OS]; 592 [Su1]; 592 [Su2] Sverak, V 245, 263, 271, 294 [44]; 295 [60] Swart, P.J 293 [10] Sweers, G 194, 233 [Sw]; 409, 489 [54] Swinney, H.L 188, 231 [LMPS] Szegö, G 204, 233 [PS] Takáˇc, P 15, 70 [69]; 390–392, 397, 399, 402, 403, 406, 407, 409, 410, 414, 415, 418, 422–424, 427–430, 433, 441–443, 446, 451, 454, 456–461, 464, 465, 469, 470, 472, 473, 477, 480, 483, 485–487, 487 [12]; 487 [13]; 488 [23]; 488 [25]; 488 [27]; 488 [28]; 488 [29]; 719 488 [32]; 488 [33]; 488 [34]; 489 [47]; 489 [55]; 489 [56]; 489 [57]; 489 [58]; 489 [59]; 489 [60] Takagi, I 159, 161, 165, 167, 169, 170, 177–179, 200–204, 207, 222, 226, 231 [LNT]; 231 [LT]; 232 [NT1]; 232 [NT2]; 232 [NT3]; 232 [NT4]; 232 [NTY1]; 232 [NTY2]; 233 [T] Taliaferro, S.D 217, 233 [Ta] Tang, M 188, 231 [JNT]; 232 [NTa] Tarantello, G 493, 494, 496–500, 503, 504, 510, 523, 527, 530, 543, 559, 570, 581, 586, 587, 587 [BCLT]; 588 [BT1]; 588 [BT2]; 588 [CT]; 591 [NT1]; 591 [NT2]; 591 [NT3]; 591 [PT]; 591 [RT1]; 591 [RT2]; 592 [ST]; 592 [T1]; 592 [T2]; 592 [T3]; 592 [T4] Tartar, L 50, 68 [23]; 303, 311, 359, 382 [24] Taubes, C 493, 590 [JT]; 592 [Ta] Tello, L 390, 391, 399, 403, 489 [60] Temam, R 242, 294 [28] Teramoto, E 182, 233 [SKT] Toland, J 58, 68 [17] Tolksdorf, P 395–397, 443, 470, 489 [61]; 489 [62] Tonelli, L 242, 244, 295 [61] Trembley, A 163, 233 [Tr] Triebel, H 423, 489 [63]; 685, 690, 712 [93] Troianiello, G.M 420, 489 [64] Troutman, J.L 242, 295 [62] Troy, W.C 215, 233 [Ty] Trudinger, N.S 17, 19, 20, 22, 31, 69 [32]; 247, 274, 294 [35]; 308, 346, 359, 369, 382 [26]; 393, 418, 419, 428, 488 [39]; 507–510, 575, 590 [GT]; 592 [Tr]; 595, 609, 646, 710 [47] Tsouli, N 392, 435, 487 [4] Turing, A.M 163, 187, 233 [Tu] Turner, R.E.L 33, 68 [15] Uhlenbeck, K 16, 70 [67] Ulm, M 390, 391, 399, 403, 410, 442, 446, 451, 456, 458, 483, 485–487, 488 [23]; 489 [60] Valadier, M 256, 295 [63] van Baalen, G 78, 111, 155 [43] Vázquez, J.L 397, 489 [65]; 493, 588 [BV]; 645, 649, 659, 712 [94]; 712 [95] Vegas, J.M 193, 230 [HV] Verbitsky, I.E 597, 673, 675, 677, 711 [52] Véron, L 597, 598, 610, 617, 638, 641–643, 645, 660, 681, 683, 690, 692, 694–697, 700, 704–707, 709, 710 [23]; 710 [27]; 710 [48]; 710 [49]; 711 [68]; 711 [69]; 711 [70]; 711 [71]; 711 [72]; 711 [73]; 711 [74]; 711 [75]; 712 [87]; 712 [88]; 712 [96]; 712 [97]; 712 [98]; 712 [99]; 712 [100]; 712 [101]; 712 [102] Vinter, R 293, 295 [64] 720 Author Index Vivier, L 609, 616, 709 [14] von Mises, R 710 [43] Walter, W 219, 231 [HMW] Waltman, P.E 182, 233 [Wm] Wan, T.Y.H 496, 503, 589 [CW] Wang, G 493, 494, 499, 500, 510, 579, 586, 589 [DJLPW]; 589 [DJLW1]; 589 [DJLW2]; 589 [DJLW3]; 590 [JoW1]; 590 [JoW2]; 592 [W1]; 592 [WW1]; 592 [WW2] Wang, R 493, 592 [Wa] Wang, S 493, 494, 592 [WY] Wang, X 189, 211, 230 [GNW1]; 230 [GNW2]; 233 [Wa] Warner, F 494, 590 [KW1]; 590 [KW2] Wei, J 161, 169, 170, 172–174, 177, 181, 189, 207, 208, 218, 222, 230 [GPW]; 230 [GW1]; 230 [GW2]; 230 [GWW]; 231 [LW]; 232 [MNW]; 232 [NW]; 233 [W1]; 233 [W2]; 233 [W3]; 233 [W4]; 233 [WW1]; 233 [WW2]; 233 [WW3]; 499, 500, 579, 586, 590 [ML]; 591 [NW]; 592 [WW1]; 592 [WW2] Weinberger, H.F 31, 70 [56]; 76, 78, 114, 123, 125, 126, 133, 139, 150, 155 [29]; 155 [30]; 216, 233 [PW] Wente, H.C 500, 592 [We] Weston, V.H 592 [W2] Whyburn, G.T 36, 42, 70 [70]; 70 [71] Winet, H 77, 154 [10] Winter, M 172, 181, 189, 230 [GWW]; 233 [WW1]; 233 [WW2]; 233 [WW3] Wittwer, P 78, 111, 155 [44]; 155 [45] Wolansky, G 493, 500, 589 [CSW]; 591 [SW1]; 591 [SW2]; 592 [Wo] Wolenski, P.R 294 [20] Yanagida, E 162, 165, 189, 197, 199–204, 211, 214, 232 [NPY]; 232 [NTY1]; 232 [NTY2]; 233 [PY]; 233 [Y] Yang, P 493, 494, 510, 511, 588 [ChY1]; 588 [ChY2]; 588 [ChY3] Yang, Y 493, 494, 501, 588 [CY]; 589 [CHMY]; 592 [SY1]; 592 [SY2]; 592 [SY3]; 592 [WY]; 592 [Y] Yarur, C 675, 709 [15] Yosida, K 429, 442, 489 [66]; 685, 712 [103] Yotsutani, S 184, 214, 232 [LNY]; 232 [NY] Young, L.C 242, 245, 295 [65]; 295 [66] Zabreiko, P.P 14–16, 51, 57, 58, 69 [44] Zheng, H 293, 295 [64] Zhou, J 219, 221, 222, 226, 228, 229 [CDNZ]; 229 [CEZ]; 229 [CNPZ]; 229 [CNZ]; 230 [DCNZ]; 231 [LZ1]; 231 [LZ2]; 231 [LZ3] Ziemer, W 303, 304, 383 [49] Zou, H 215, 233 [SZ] Subject Index 3-G inequality, 676 − S n−1 , 641 S n−1 , 641, 692, 694, 695 S n−1 the Laplace–Beltrami, 641 Σ , 697 Δ2 -condition, 683,683 ∇ n−1 , 693, 694 S (g, ˜ 0) – admissibility, 625 – admissible, 623 – boundary-admissible, 683 (g, ˜ k) – admissible, 622 – boundary-admissible, 683 (g, r0 )-admissibility, 623 (g, ˜ r0 ) – admissible, 622 – boundary-admissible Radon, 683 -convergence, 304, 306, 316 (n, 0)-weak-singularity assumption, 626, 645 (n, α)-weak-singularity, 621 – assumption, 617, 621 blow-up – analysis, 498, 499, 557, 577 – behavior, 500, 557 – point, 497, 513–516, 521–525, 532, 533, 538, 544, 547, 555, 558, 571, 573 – profile, 496, 559 – sequence, 498, 499 – technique, 516 boundary – layer, 301, 316, 339, 340, 357 – trace, 612, 700–704, 707, 708 boundary-q-admissible, 683, 684, 692 – measure, 688 boundary-weak-singularity assumption, 681, 682 bounded variation, 303 brachistochrone, 239 Brouwer degree, 318, 367 Brouwer’s fixed point theorem, 6, 7, 16 BV space, 303, 304 capacitary measure, 672 capacity, 627–629, 631–635, 637, 640, 683, 691, 699, 700 Carathéodory – function, 237, 284 – integrand, 258 center of mass, 580, 582 Chern–Simons – theory, 493 – vortex, 494 circular-well potential, 302, 358 class of variations, 246 coercive, 708 – nonlinearity, 707, 708 coercivity, 239, 242 compact – map, 7, 51, 52 – perturbation, 7, 8, 53 complex plane, 500 concentration, 497 – phenomena, 495, 496, 498, 499, 500, 502, 529 a priori bounds, 26, 33, 34, 44, 67 Abrikosov’s mixed states, 493, 494 absorbing nonlinearity, 614 absorption principle, 595 admissible “variation”, 238 algebraic multiplicity, 9, 10, 52, 54, 59, 60 anti-maximum principle, 407, 409 arbitrary Reynolds number, 114 augmented functional, 262 auxiliary Stokes fields, 83 bad discs, 356, 380 Besov space, 685, 690, 711 Bessel – capacities, 626, 669, 671 – kernel, 626, 672 – potential, 627 bi-harmonic equation, 241, 291 bifurcation from infinity, 391 721 722 Subject Index concentration–compactness, 496, 497, 512, 523, 577, 579 concentrations of solutions, 163, 177 condition (H), 601, 602, 605, 606, 609, 611, 613, 614, 616, 617, 622, 632, 644, 667, 680, 681, 683 conditionally q-removable, 696, 699, 700 conformal geometry, 493, 574 conservation laws, 293 continuous compact map, continuum, 35, 54 convexity, 239, 242, 249 critical point, 239, 293, 576 D-solution, 75, 78, 122, 128, 130–132, D-solutions, asymptotic structure, 133 degenerate metric, 358, 359 derivative at infinity, 12 diffusion system, 203, 240, 262 Dirac measure, 493, 500, 501 direct method, 242 Dirichlet – boundary condition, 579 – problem, 506, 508, 513 discretization (or semidiscretization) in time, 288 double-well potential, 299, 300, 310, 315 drag, 111 Dugundji’s theorem, 8, 14, 37 eikonal-type equation, 359 elastic energy, 240 electroweak theory, 493 elliptic – estimates, 507 – regularity, 508, 517, 578, 579 energy equality, 110, 140, 141 equivalence, 609 Euler characteristic, 499, 586 Euler–Lagrange equation, 237 example of nonuniqueness, 107 exponential – nonlinearity, 493, 510 – order of growth, 645, 646, 648, 659 field theories, 292 finite – perimeter, 304 – Reynolds number, 111 first eigenvalue, 396–398, 417, 427, 430, 435, 441 flat two-torus, 494, 586, 587 Fredholm – alternative, 10, 57, 58, 386, 390, 391, 409, 427, 433, 440, 447, 460, 482 – map, 499, 586 Fredholm–Riesz–Schauder theory, Fujita method, 117 Galerkin – approximation, 117 – method, 75 gas combustion, 493 gauge field theories, 493 Gauss curvature, 494 generalized – nullspace, 9, 58 – range, 9, 58 – variational principle, 285 geometric multiplicity, Ginzburg–Landau – energy, 301, 315, 338, 358 – equations, 493 – model, 493, 494 – potential, 301–303 global minimizer, 239, 254, 259, 390, 391, 403–406, 437, 464–466, 468, 469 Green – function, 579, 608 – kernel, 675 – potential, 622 – potential of λ, 609 – representation formula, 499, 527, 560, 572, 581, 608 Green–Gauss theorem, 511 Hamilton–Jacobi theory, 292 Hamiltonian system, 292 harmonic map, 337, 338 Harnack-type inequality, 498, 548 Harnack’s inequality, 498, 506, 507, 526, 530, 544 heat equation, 288 homogeneous Sobolev space, 80 Hopf “cut-off” function, 151 improved Poincaré inequality, 427, 428, 432, 433, 436 inequalities of John–Nirenberg, 505 inf + sup estimate, 498, 537, 559, 570 inf + sup inequality, 498, 530, 543 invading domains, 116 isoperimetrical inequality, 543 Jensen’s inequality, 256, 258, 505, 524, 580 joint convexity, 255 Korn’s inequality, 269 Krein–Rutman theorem, 15, 23, 37, 55, 56, 64 Subject Index L-harmonic, 611 lack of – existence, 242 – minimizer, 244 Laplace – equation, 289 – operator, 500, 501 Laplace–Beltrami, 693, 697 – operator, 494, 575 Laplacian, 239 Lavrentiev phenomenon, 247 Legendre transformation, 292 Legendre–Hadamard condition, 271 Legendre–Jacobi theory, 292 Leray method, 115 Leray–Schauder – alternative, 13 – degree, 499, 586 lift, 111 line singularities, 358 linear – elasticity, 240, 268 – second-order differential operators in divergence form, 598 Liouville – equation, 496, 501, 543 – formula, 503, 543, 552 Liouville-type equation, 493, 500, 502, 505, 512 local minima, 239 lower semicontinuity, 252 Lyapunov–Schmidt reduction, 56, 57 M(∂Ω), 605, 616 M(Ω), 605 α ), 605, 616, 621 M(Ω; ρ∂Ω making variations, 292 Marcinkiewicz spaces, 615, 616 maximum principle, 18, 19, 23, 24, 29–31, 50, 56, 57, 61, 123, 125, 320, 325, 374, 505, 534 mean field equations, 574 mean value theorem, 508 minimal – solution, 28, 48, 49 – solution u, 24 – surface, 240, 277 minimizer, 238 Monge–Ampère equation, 241, 275, 278 Moser–Trudinger inequality, 494, 500, 510, 576, 582 mountain pass type, 11 moving plane technique, 499, 503, 533, 543 multi-peak profile, 558 multiple ‘peak’ concentration, 496 723 multiplier, 262 multipliers theory, 97 Navier–Stokes – equations, 73 – problem, 105 Neumann boundary conditions, 579 Noether equations, 293 nonconvex problems, 242 nonconvexity, 279 nondivergence form, 600 nonlinear eigenvalue problem, 386, 408 nonlinear elasticity, 240, 268, 279 nonsmooth analysis, 293 norm, 500 null-Lagrangians, 265 obstacle problem, 240, 260, 290 odd multiplicity, 53 Ogden materials, 267 one-dimensional, 238 oscillatory minimizing sequences, 286 Oseen – approximation, 74, 77, 80, 92, 101 – approximation, a variant to, 103 – fundamental solution, 93 – operator, 99, 100 p-Laplacian, 240, 290, 386, 390, 391, 402, 409, 441, 475 Palais–Smale sequence, 576, 583, 584 parabolic wake, 95 paradigmatic problem of the CV, 237 penalization, 301, 355 phase transition, 299 physically reasonable, 138 physically reasonable solutions, 75, 106, 135 plate equation, 241 Pohozaev-identity, 528 Pohozaev-type identity, 497, 499, 511, 527, 567 Poincaré inequality, 120, 148 point – of mountain pass type, 11, 12 – singularities, 301, 358 pointwise estimate, 499, 559, 569 Poisson – kernel, 609, 613 – potential, 609, 683, 684 polyconvex integrands, 266 polyconvexity, 267 potential – operator, 11, 57 – well, 281 724 Subject Index principle of a priori bounds, 13 product formula, 9, 53 profile of the blow-up sequence, 499 q-accumulation point, 705 q-admissible, 677–679 q-removable, 696 quantization, 497, 498, 529, 530, 544 quasicontinuous, 628 quasiconvexity, 244, 270 quasieverywhere, 628, 629, 635, 636, 640 radially symmetric, 503 rank-one convexity, 245 regular – part, 703 – point, 707 regularity of the minimizer, 245, 247 relaxation theorem, 284 renormalized energy, 338, 343 retract, 7, 8, 14, 32 retraction, 7, 8, 15, 37 Reynolds – number, 73, 139 – number, arbitrary large, 139 – number, small, 107 Riemannian surface, 494, 510, 574, 585 saddle point, 390, 403, 436–438, 440, 463–465, 468, 470–475 Sard’s lemma, scalar, 238 scalar, multidimensional variational problem, 238 scalar product, 500, 576 Schauder’s – estimates, 509 – fixed point theorem, 7, 8, 32 second variation, 292 second-order – problems, 242 – variational problem, 273 self-propelled, 92 – motion, 77, 80, 89 selfdual – gauge field theories, 493 – vortices, 493 singular – boundary of A relative, 705 – Liouville equations, 493 – part, 703 – perturbation, 299, 300, 366 – set, 707 – sources, 493, 501 smallest eigenvalue, 12, 26, 31, 55 Sobolev – critical exponent, 586 – space, 500 Sobolev’s estimates, 509 stability, 159, 161, 162, 165, 189, 190, 194–197, 199, 200, 201–204 standard two-sphere, 494 – S , 586 statistical mechanics, 493 steady states, 164, 165, 178, 182–189, 192, 200 Stokes – approximation, 73, 80, 81, 92 – fundamental solution, 81 – paradox, 74, 80, 81, 87, 100 stress tensor, 82 stretching tensor, 82 strict convexity, 259 strictly positive operator, 19, 20, 23, 31, 32, 48, 50 strong – barrier property, 708 – maximum principle, 600 strongly positive operator, 15, 23, 37, 64 sub- and supersolutions, 24–26, 28, 32, 33, 46, 67, 386, 441, 475, 476, 478, 479 subcritical with respect, 649 – to g, 658 super-L-harmonic, 613 symmetric – domains, 88 – Leray solution, 141, 142, 153 – solutions, 115, 122, 140 – solutions, existence for arbitrary large Reynolds number, 149 symmetries, 159, 160, 204–207 symmetry breaking bifurcation, 58 Toda system, 500 transversality condition, 58 two-dimensional weak-singularity assumption, 645, 646, 649 uniqueness, 259 vanishing Reynolds number, 101, 111 variational – formulation, 580 – problem, 238 vector, one-dimensional variational problem, 238 vector – problems, 242 – variational problems, 238, 244, 250, 268 very weak solution, 602, 606, 611, 613, 614 visualization, 218, 222 Subject Index vortex, 354, 493 – point, 493, 497, 500 vorticity, 141 wake structure, 96 wave equation, 240, 287 weak – continuity property, 266 – lower semicontinuity, 255 – stability, 607 – subsolution, 600 weakly L-harmonic, 610, 611 Weierstrass field theory, 292 Wirtinger inequality, 126 Young measures, 242, 279, 282 725 ... k ∂x1k1 ∂x2k2 · · · ∂xNN with |k| = kj j =1 Moreover, we set, for α ∈ [0, 1] and m ∈ N, Mm+α (f ) = sup M D k f , |k| =m m f m = Mj (f ), j =0 m f m+ α = Mj (f ) + Mm+α (f ) j =0 Let C m (D) denote... = Hofer [39] has extended Theorem 1. 11 to critical points of mountain pass type T HEOREM 1. 13 (Hofer [39 ]) Let g be as in Theorem 1. 11 Suppose in addition that it is in C (U, R) for some open... assumption that (i) does not hold T HEOREM 1. 18 (Principle of a priori bounds) For t ∈ [0, 1] let F (t, ) : X → X be a family of compact operators with F (0, ) ≡ Assume, moreover, that F (t, x)