Multiscale weak compactness in metric spaces. (English) Zbl 1392.46033
This survey paper deals with some problems concerning the setting of metric spaces profile decomposition properties. The content is based on some previous papers by the same author. The reviewer points out the final section of the paper under review, where the author provides a brief sketch of profile decomposition theorems by means of a polar profile reconstruction defined by means of a characterizing formula which does not require any linear structure.
Reviewer: Vicenţiu D. Rădulescu (Craiova)
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 35B33 | Critical exponents in context of PDEs |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |
References:
| [1] | T. Aubin, Nonlinear analysis on manifolds, Monge-Ampère equations. Grundlehren 252. Berlin Heidelberg New York: Springer 1982. · Zbl 0512.53044 · doi:10.1007/978-1-4612-5734-9 |
| [2] | H. Bahouri, A. Cohen and G. Koch, A general wavelet-based profile decomposition in the critical embedding of function spaces, Confluentes Math. 3 (2011) 387-411. · Zbl 1231.42034 · doi:10.1142/S1793744211000370 |
| [3] | H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc.88 (1983), 486-490. · Zbl 0526.46037 · doi:10.2307/2044999 |
| [4] | J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc., vol. 40 (1936), 396-414. · JFM 62.0460.04 · doi:10.1090/S0002-9947-1936-1501880-4 |
| [5] | G. Cerami, D. Passaseo, S. Solimini, Infinitely many positive solutions to some scalar field equations with non-symmetric coefficients, Comm. Pure Appl. Math. 66 (2013), no. 3, 372-413. · Zbl 1292.35128 · doi:10.1002/cpa.21410 |
| [6] | G. Cerami, D. Passaseo, S. Solimini, Nonlinear scalar field equations: existence of a positive solution with infinitely many bumps, Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 1, 23-40. · Zbl 1311.35081 · doi:10.1016/j.anihpc.2013.08.008 |
| [7] | G. Devillanova, S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth, Adv. Differential Equations 7 (2002), no. 10, 1257-1280. · Zbl 1208.35048 |
| [8] | G. Devillanova, S. Solimini, Infinitely many positive solutions to some nonsymmetric scalar field equations: the planar case. Calc. Var. Partial Differential Equations 52 (2015), no. 3-4, 857-898, DOI: 10.1007/s00526-014-0736-7. · Zbl 1318.35009 · doi:10.1007/s00526-014-0736-7 |
| [9] | Devillanova, G.; Solimini, S., Some remarks on profile decomposition theorems (2016) · Zbl 1361.46025 |
| [10] | G. Devillanova, S. Solimini, C. Tintarev, On weak convergence in metric spaces, in Nonlinear Analysis and Optimization, Contemporary Mathematics, vol. 659, Amer. Math. Soc., Providence, RI, 2016, 43-63. · Zbl 1356.46018 |
| [11] | G. Devillanova, S. Solimini, K. Tintarev, Profile decomposition in metric spaces, preprint. · Zbl 1356.46018 |
| [12] | P. Gérard, Description du défaut de compacité de l’injection de Sobolev. ESAIM: Control, Optimisation and Calculus of Variations 3 (1998), 213-233. · Zbl 0907.46027 · doi:10.1051/cocv:1998107 |
| [13] | S. Jaffard Analysis of the lack of compactness in the critical Sobolev embeddings J. Funct. Anal., 161 (1999), 384-396. · Zbl 0922.46030 · doi:10.1006/jfan.1998.3364 |
| [14] | G. S. Koch, Profile decompositions for critical Lebesgue and Besov space embeddings, Indiana Univ. Math. J. 59 (2010), no. 5, 1801-1830. · Zbl 1230.46030 · doi:10.1512/iumj.2010.59.4426 |
| [15] | G. Kyriasis, Nonlinear approximation and interpolation spaces, J. Approx. Theory 113 (2001) 110-126. · Zbl 0992.41006 · doi:10.1006/jath.2001.3625 |
| [16] | T.C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179-182. · Zbl 0346.47046 · doi:10.1090/S0002-9939-1976-0423139-X |
| [17] | P-L. Lions, The concentration-compactness principle in the calculus of variations. The Limit Case, Part 1, Revista Matematica Iberoamericana1.1 (1985), 145-201. · Zbl 0704.49005 |
| [18] | P-L. Lions, The concentration-compactness principle in the calculus of variations. The Limit Case, Part 2, Revista Matematica Iberoamericana 1.2 (1985), 45-121. · Zbl 0704.49006 · doi:10.4171/RMI/12 |
| [19] | R. Molle, D. Passaseo, Multiplicity of solutions of nonlinear scalar field equations, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 26 (2015), no. 1, 75-82. · Zbl 1312.35060 · doi:10.4171/RLM/693 |
| [20] | F. Maddalena, S. Solimini, Synchronic and asynchronic descriptions of irrigation problems. Adv. Nonlinear Stud. 13 (2013) no. 3, 583-623. · Zbl 1290.49094 · doi:10.1515/ans-2013-0303 |
| [21] | J. Lindenstrauss, L. Tzafriri, Classical Banach spaces II, Springer-Verlag, New York/Berlin, (1996) reprint. · Zbl 0852.46015 |
| [22] | Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597. · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0 |
| [23] | P. Sacks, K. Uhlenbeck On the existence of minimal immersions of 2-spheres. Ann. Math. 113 (1981), 1-24. · Zbl 0462.58014 · doi:10.2307/1971131 |
| [24] | S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space. Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995), 319-337. · Zbl 0837.46025 · doi:10.1016/S0294-1449(16)30159-7 |
| [25] | Solimini, S.; Tintarev, K., Concentration analysis in Banach spaces (2015) |
| [26] | S. Solimini, K. Tintarev, On defect of compactness in Banach Spaces. Comptes Rendus Mathematique, 353 (2015), no 10, 899-903. · Zbl 1338.46028 · doi:10.1016/j.crma.2015.07.011 |
| [27] | J. Staples, Fixed point theorems in uniformly rotund metric spaces, Bull. Austral. Math. Soc. 14 (1976), no. 2, 181-192. · Zbl 0353.54029 · doi:10.1017/S0004972700025028 |
| [28] | M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. 187 (1984), 511-517. · Zbl 0535.35025 · doi:10.1007/BF01174186 |
| [29] | T. Tao, Concentration compactness via nonstandard analysis. Therence Tao Blog: What’s new, http://terrytao.wordpress.com/2010/11/29/concentration-compactness-via-nonstandard-analysis/, November 10, 2010. · Zbl 1311.35077 |
| [30] | K. Tintarev, K-H. Fieseler, Concentration Compactness: Functional-Analitic Grounds and Applications Imperial College Press, London (2007). · Zbl 1118.49001 · doi:10.1142/p456 |
| [31] | D. van Dulst, Equivalent norms and the fixed point property for nonexpansive mappings, J. London Math. Soc. 25 (1982), no 2, 139-144. · Zbl 0453.46017 · doi:10.1112/jlms/s2-25.1.139 |
| [32] | J. Wei - S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in ℝN, Calc. Var. Partial Differential Equations, 37 (2010), 423-439. · Zbl 1189.35106 · doi:10.1007/s00526-009-0270-1 |
| [33] | A. Weiwei, J. Wei Infinitely many positive solutions for nonlinear equations with non-symmetric potential, Calc. Var. Partial Differential Equations 51 (2014), no 3, 761-798. · Zbl 1311.35077 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
