The interplay between analysis and topology in some nonlinear PDE problems
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-
1 G. Alberti, S. Baldo and G. Orlandi, Functions with prescribed singularities (to appear).
- Frederick J. Almgren Jr. and Elliott H. Lieb, Singularities of energy minimizing maps from the ball to the sphere: examples, counterexamples, and bounds, Ann. of Math. (2) 128 (1988), no. 3, 483–530. MR 970609, DOI 10.2307/1971434
- Thierry Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9) 55 (1976), no. 3, 269–296. MR 431287
- Fabrice Bethuel, The approximation problem for Sobolev maps between two manifolds, Acta Math. 167 (1991), no. 3-4, 153–206. MR 1120602, DOI 10.1007/BF02392449
- F. Bethuel, A characterization of maps in $H^1(B^3,S^2)$ which can be approximated by smooth maps, Ann. Inst. H. Poincaré C Anal. Non Linéaire 7 (1990), no. 4, 269–286 (English, with French summary). MR 1067776, DOI 10.1016/S0294-1449(16)30292-X
- Fabrice Bethuel, On the singular set of stationary harmonic maps, Manuscripta Math. 78 (1993), no. 4, 417–443. MR 1208652, DOI 10.1007/BF02599324
- F. Bethuel, H. Brezis, and J.-M. Coron, Relaxed energies for harmonic maps, Variational methods (Paris, 1988) Progr. Nonlinear Differential Equations Appl., vol. 4, Birkhäuser Boston, Boston, MA, 1990, pp. 37–52. MR 1205144, DOI 10.1007/978-1-4757-1080-9_{3}
- Fabrice Bethuel, Haïm Brezis, and Frédéric Hélein, Ginzburg-Landau vortices, Progress in Nonlinear Differential Equations and their Applications, vol. 13, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1269538, DOI 10.1007/978-1-4612-0287-5
- Fabrice Bethuel and Xiao Min Zheng, Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal. 80 (1988), no. 1, 60–75. MR 960223, DOI 10.1016/0022-1236(88)90065-1
- Jean Bourgain, Haim Brezis, and Petru Mironescu, Lifting in Sobolev spaces, J. Anal. Math. 80 (2000), 37–86. MR 1771523, DOI 10.1007/BF02791533 11 —, Another look at Sobolev spaces, Optimal Control and Partial Differential Equations (J. L. Menaldi, E. Rofman and A. Sulem, eds.), a volume in honor of A. Bensoussan’s 60th birthday, IOS Press, 2001, pp. 439–455.
- Jean Bourgain, Haïm Brezis, and Petru Mironescu, On the structure of the Sobolev space $H^{1/2}$ with values into the circle, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), no. 2, 119–124 (English, with English and French summaries). MR 1781527, DOI 10.1016/S0764-4442(00)00513-9
- Haïm Brezis, $S^k$-valued maps with singularities, Topics in calculus of variations (Montecatini Terme, 1987) Lecture Notes in Math., vol. 1365, Springer, Berlin, 1989, pp. 1–30. MR 994017, DOI 10.1007/BFb0089176 14 —, How to recognize constant functions. Connections with Sobolev spaces, Russian Math. Surveys, volume in honor of M. Vishik (to appear).
- Haïm Brezis and Felix Browder, Partial differential equations in the 20th century, Adv. Math. 135 (1998), no. 1, 76–144. MR 1617413, DOI 10.1006/aima.1997.1713
- Haïm Brezis and Jean-Michel Coron, Large solutions for harmonic maps in two dimensions, Comm. Math. Phys. 92 (1983), no. 2, 203–215. MR 728866, DOI 10.1007/BF01210846
- Haïm Brezis and Jean-Michel Coron, Multiple solutions of $H$-systems and Rellich’s conjecture, Comm. Pure Appl. Math. 37 (1984), no. 2, 149–187. MR 733715, DOI 10.1002/cpa.3160370202
- Haïm Brezis, Jean-Michel Coron, and Elliott H. Lieb, Harmonic maps with defects, Comm. Math. Phys. 107 (1986), no. 4, 649–705. MR 868739, DOI 10.1007/BF01205490
- Haim Brezis and Yanyan Li, Topology and Sobolev spaces, J. Funct. Anal. 183 (2001), no. 2, 321–369. MR 1844211, DOI 10.1006/jfan.2000.3736
- Haïm Brezis, Yanyan Li, Petru Mironescu, and Louis Nirenberg, Degree and Sobolev spaces, Topol. Methods Nonlinear Anal. 13 (1999), no. 2, 181–190. MR 1742219, DOI 10.12775/TMNA.1999.009
- H. Brezis and L. Nirenberg, Degree theory and BMO. I. Compact manifolds without boundaries, Selecta Math. (N.S.) 1 (1995), no. 2, 197–263. MR 1354598, DOI 10.1007/BF01671566
- J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), no. 1, 1–68. MR 495450, DOI 10.1112/blms/10.1.1
- J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), no. 5, 385–524. MR 956352, DOI 10.1112/blms/20.5.385 24 J. Ericksen, Equilibrium of liquid crystals, in “Advances in Liquid Crystals 2” (G. Brown, ed.), Acad. Press, 1976, 233–299.
- J. L. Ericksen and D. Kinderlehrer (eds.), Theory and applications of liquid crystals, The IMA Volumes in Mathematics and its Applications, vol. 5, Springer-Verlag, New York, 1987. Papers from the IMA workshop held in Minneapolis, Minn., January 21–25, 1985. MR 900827, DOI 10.1007/978-1-4613-8743-5
- Lawrence C. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Rational Mech. Anal. 116 (1991), no. 2, 101–113. MR 1143435, DOI 10.1007/BF00375587
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. MR 717034
- Mariano Giaquinta and Stefan Hildebrandt, A priori estimates for harmonic mappings, J. Reine Angew. Math. 336 (1982), 124–164. MR 671325, DOI 10.1515/crll.1982.336.124
- P. Erdös, On the distribution of normal point groups, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 294–297. MR 2000, DOI 10.1073/pnas.26.4.294
- E. Giusti (ed.), Harmonic mappings and minimal immersions, Lecture Notes in Mathematics, vol. 1161, Springer-Verlag, Berlin, 1985. Lectures given at the first 1984 session of the Centro Internationale Matematico Estivo (CIME) held at Montecatini, June 24–July 3, 1984. MR 821967, DOI 10.1007/BFb0075135
- Richard S. Hamilton, Harmonic maps of manifolds with boundary, Lecture Notes in Mathematics, Vol. 471, Springer-Verlag, Berlin-New York, 1975. MR 0482822, DOI 10.1007/BFb0087227 33 F. B. Hang and F. H. Lin, Topology of Sobolev mappings (to appear).
- Robert Hardt, David Kinderlehrer, and Fang-Hua Lin, Existence and partial regularity of static liquid crystal configurations, Comm. Math. Phys. 105 (1986), no. 4, 547–570. MR 852090, DOI 10.1007/BF01238933
- Robert Hardt and Fang-Hua Lin, A remark on $H^1$ mappings, Manuscripta Math. 56 (1986), no. 1, 1–10. MR 846982, DOI 10.1007/BF01171029
- Robert Hardt and Fanghua Lin, Singularities for $p$-energy minimizing unit vectorfields on planar domains, Calc. Var. Partial Differential Equations 3 (1995), no. 3, 311–341. MR 1385291, DOI 10.1007/BF01189395
- Frédéric Hélein, Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 8, 591–596 (French, with English summary). MR 1101039 38 —, Harmonic maps, conservation laws and moving frames, 2nd ed., Cambridge Univ. Press, 2002. 39 D. Hilbert, Uber das Dirichletsche Prinzip, Jahresbericht Deut. Math.-Ver. VIII, 1900, 184–188 (also in J. Reine Angew. Math. 129 (1905), 63–67) and Math. Ann 59 (1904), 161–184.
- Stefan Hildebrandt, Nonlinear elliptic systems and harmonic mappings, Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3 (Beijing, 1980) Sci. Press Beijing, Beijing, 1982, pp. 481–615. MR 714341 41 R. L. Jerrard and H. M. Soner, Functions of bounded higher variation (to appear). 42 J. Jost, Harmonic mappings between surfaces, Lecture Notes in Math., vol. 1062, Springer, 1984.
- Jürgen Jost, The Dirichlet problem for harmonic maps from a surface with boundary onto a $2$-sphere with nonconstant boundary values, J. Differential Geom. 19 (1984), no. 2, 393–401. MR 755231
- S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), no. 9, 1221–1268. MR 1231427, DOI 10.1002/cpa.3160460902
- S. Klainerman and M. Machedon, Smoothing estimates for null forms and applications, Duke Math. J. 81 (1995), no. 1, 99–133 (1996). A celebration of John F. Nash, Jr. MR 1381973, DOI 10.1215/S0012-7094-95-08109-5
- M. Kléman, Points, lines and walls. In liquid crystals, magnetic systems and various ordered media, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.
- P. Erdös, On the distribution of normal point groups, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 294–297. MR 2000, DOI 10.1073/pnas.26.4.294
- Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511, DOI 10.1007/978-3-540-69952-1 49 H. Poincaré, Sur les équations aux dérivées partielles de la physique mathématique, Amer. J. Math. 12 (1980), 211–294. 50 —, Théorie du potential newtonien, Carré et Naud, 1899; reprinted J. Gabay, 1990.
- Tristan Rivière, Everywhere discontinuous harmonic maps into spheres, Acta Math. 175 (1995), no. 2, 197–226. MR 1368247, DOI 10.1007/BF02393305
- Tristan Rivière, Line vortices in the $\textrm {U}(1)$-Higgs model, ESAIM Contrôle Optim. Calc. Var. 1 (1995/96), 77–167. MR 1394302, DOI 10.1051/cocv:1996103
- Jacob Rubinstein and Peter Sternberg, Homotopy classification of minimizers of the Ginzburg-Landau energy and the existence of permanent currents, Comm. Math. Phys. 179 (1996), no. 1, 257–263. MR 1395224, DOI 10.1007/BF02103722
- J. Sacks and K. Uhlenbeck, The existence of minimal immersions of $2$-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1–24. MR 604040, DOI 10.2307/1971131
- Rudolph E. Langer, The boundary problem of an ordinary linear differential system in the complex domain, Trans. Amer. Math. Soc. 46 (1939), 151–190 and Correction, 467 (1939). MR 84, DOI 10.1090/S0002-9947-1939-0000084-7
- Richard Schoen and Karen Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps, J. Differential Geom. 18 (1983), no. 2, 253–268. MR 710054
- Richard Schoen and Karen Uhlenbeck, Regularity of minimizing harmonic maps into the sphere, Invent. Math. 78 (1984), no. 1, 89–100. MR 762354, DOI 10.1007/BF01388715
- R. Schoen and S. T. Yau, Lectures on harmonic maps, Conference Proceedings and Lecture Notes in Geometry and Topology, II, International Press, Cambridge, MA, 1997. MR 1474501
- Henry C. Wente, An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl. 26 (1969), 318–344. MR 243467, DOI 10.1016/0022-247X(69)90156-5 60 H. Weyl, The method of orthogonal projection in potential theory, Duke Math. J. 7 (1940), 411–440.
- Brian White, Homotopy classes in Sobolev spaces and the existence of energy minimizing maps, Acta Math. 160 (1988), no. 1-2, 1–17. MR 926523, DOI 10.1007/BF02392271
Additional Information
- Haim Brezis
- Affiliation: Department of Mathematics, Rutgers University, Piscataway, NJ 08854; Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Boîte courrier 187, 4 place Jussieu, 75252 Paris cedex 05, France
- MR Author ID: 41485
- Email: brezis@math.rutgers.edu, brezis@ann.jussieu.fr, brezis@ccr.jussieu.fr
- Received by editor(s): October 23, 2002
- Published electronically: February 12, 2003
- Additional Notes: This text is an expanded version of the invited address at the AMS Meeting “Mathematical Challenges of the 21st Century”, UCLA (2000)
- © Copyright 2003 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 40 (2003), 179-201
- MSC (2000): Primary 35A15, 35A20, 35J50, 35J65, 35Qxx, 46Txx, 47Hxx, 47Jxx, 55Pxx, 58E15
- DOI: https://doi.org/10.1090/S0273-0979-03-00976-5
- MathSciNet review: 1962295