|
The interplay between analysis and topology in some nonlinear PDE problems
Author(s):
Haim
Brezis
Journal:
Bull. Amer. Math. Soc.
40
(2003),
179-201.
MSC (2000):
Primary 35A15, 35A20, 35J50, 35J65, 35Qxx, 46Txx, 47Hxx, 47Jxx, 55Pxx, 58E15
Posted:
February 12, 2003
Retrieve article in:
PDF
References |
Similar articles |
Additional information
References:
-
- 1.
- G. Alberti, S. Baldo and G. Orlandi, Functions with prescribed singularities (to appear).
- 2.
- F. Almgren and E. Lieb, Singularities of energy minimizing maps from the ball to the sphere: Examples, counterexamples, and bounds, Ann.of Math. 128 (1988), 483-530. MR 91a:58049
- 3.
- Th. Aubin, Equations différentielles nonlinéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures et Appl. 55 (1976), 269-296. MR 55:4288
- 4.
- F. Bethuel, The approximation problem for Sobolev maps between two manifolds, Acta Math. 167 (1991), 153-206. MR 92f:58023
- 5.
- -, A characterization of maps in
which can be approximated by smooth maps, Ann. Inst. H. Poincaré, Anal. Non Linéaire 7 (1990), 269-286. MR 91f:58013 - 6.
- -, On the singular set of stationary harmonic maps, Manuscripta Math. 78 (1993), 417-443. MR 94a:58047
- 7.
- F. Bethuel, H. Brezis and J.-M. Coron, Relaxed energies for harmonic maps, in ``Variational Methods'' (H. Berestycki, J.-M. Coron and I. Ekeland, eds.), Birkhäuser, 1990. MR 94a:58046
- 8.
- F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices, Birkhäuser, 1994. MR 95c:58044
- 9.
- F. Bethuel and X. Zheng, Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal. 80 (1988), 60-75. MR 89i:58015
- 10.
- J. Bourgain, H. Brezis and P. Mironescu, Lifting in Sobolev spaces, J. Analyse Math. 80 (2000), p. 37-86. MR 2001h:46044
- 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.
- 12.
- -,
maps with values into the circle: Minimal connections, lifting and the Ginzburg-Landau equation (to appear). Part of the results were announced in a note by the same authors: On the structure of the Sobolev space with values into the circle, C. R. Acad. Sci. Paris Ser. I Math. 331 (2000), 119-124. MR 2001m:46068 - 13.
- H. Brezis,
-valued maps with singularities, Topics in the Calculus of Variations (M. Giaquinta, ed.), Lecture Notes in Math., vol. 1365, Springer, 1989, 1-30. MR 90f:58029 - 14.
- -, How to recognize constant functions. Connections with Sobolev spaces, Russian Math. Surveys, volume in honor of M. Vishik (to appear).
- 15.
- H. Brezis and F. Browder, Partial differential equations in the 20th century, Advances in Math. 135 (1998), 76-144, and Enciclopedia Italiana (to appear). MR 99i:35001
- 16.
- H. Brezis and J.-M. Coron, Large solutions for harmonic maps in two dimensions, Comm. Math. Phys. 92 (1983), 203-215. MR 85a:58022
- 17.
- -, Multiple solutions of
-systems and Rellich's conjecture, Comm. Pure Appl. Math. 37 (1984), 149-187. MR 85i:53010 - 18.
- H. Brezis, J.-M. Coron and E. Lieb, Harmonic maps with defects, Comm. Math. Phys. 107 (1986), 649-705. MR 88e:58023
- 19.
- H. Brezis and Y. Li, Topology and Sobolev spaces, J. Funct. Anal. 183 (2001), 321-369. MR 2002h:58009
- 20.
- H. Brezis, Y. Li, P. Mironescu and L. Nirenberg, Degree and Sobolev spaces, Topological Methods in Nonlinear Analysis 13 (1999), 181-190. MR 2001a:47065
- 21.
- H. Brezis and L. Nirenberg, Degree theory and BMO, Part I : compact manifolds without boundaries, Selecta Math. 1 (1995), 197-263. MR 96g:58023
- 22.
- J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1-68. MR 82b:58033
- 23.
- -, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), 385-524. MR 89i:58027
- 24.
- J. Ericksen, Equilibrium of liquid crystals, in ``Advances in Liquid Crystals 2'' (G. Brown, ed.), Acad. Press, 1976, 233-299.
- 25.
- J. Ericksen and D. Kinderlehrer (editors), Theory and applications of liquid crystals, IMA Series, Vol. 5, Springer, 1987. MR 88d:82007
- 26.
- L. C. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Rational Mech. Anal. 116 (1991), 101-113. MR 93m:58026
- 27.
- H. Federer, Geometric measure theory, Springer, 1969. MR 41:1976
- 28.
- M. Giaquinta, Multiple integrals in the calculus of variations and non-linear elliptic systems, Princeton Univ. Press, 1983. MR 86b:49003
- 29.
- M. Giaquinta and S. Hildebrandt, A priori estimates for harmonic mappings, J. Reine Angew. Math. 336 (1982), 124-164. MR 84b:58035
- 30.
- M. Giaquinta, G. Modica and J. Soucek, Cartesian Currents in the Calculus of Variations, Springer, 1998. MR 2000b:49001a, MR 2000b:49001b
- 31.
- E. Giusti (editor), Harmonic mappings and minimal immersions, Lecture Notes in Math. Vol. 1161, Springer, 1985; includes lectures by S. Hildebrandt, J. Jost and L. Simon. MR 86j:58037
- 32.
- R. Hamilton, Harmonic maps of manifolds with boundary, Lecture Notes in Math. 471, Springer, 1975. MR 58:2872
- 33.
- F. B. Hang and F. H. Lin, Topology of Sobolev mappings (to appear).
- 34.
- R. Hardt, D. Kinderlehrer and F. H. Lin, Existence and partial regularity of static liquid crystal configurations, Comm. Math. Phys. 105 (1986), 547-570. MR 88a:35207
- 35.
- R. Hardt and F. H. Lin, A remark on
mappings, Manuscripta Math. 56 (1986), 1-10. MR 87k:58068 - 36.
- -, Singularities of
-energy minimizing unit vector-fields on planar domains, Calc. of Variations and PDE 3 (1995), 311-342. MR 97d:58060 - 37.
- F. Hélein, Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne, C. R. Acad. Sci. Paris 312 (1991), 591-596. MR 92e:58055
- 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.
- 40.
- S. Hildebrandt, Nonlinear elliptic systems and harmonic mappings, Proc. 1980 Beijing Symp. Diff. Geom. Diff. Eq., Science Press, Beijing, 1982, 481-615. MR 85k:35078
- 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.
- 43.
- -, The Dirichlet problem for harmonic maps from a surface with boundary onto a
-sphere with non-constant boundary values, J. Diff. Geom. 19 (1984), 393-401. MR 86b:58031 - 44.
- S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), 1221-1268. MR 94h:35137
- 45.
- -, Smoothing estimates for null forms and applications, Duke Math. J. 81 (1995), 99-133. MR 97h:35022
- 46.
- M. Kléman, Points, lines and walls, John-Wiley, 1983. MR 85e:82058
- 47.
- F. H. Lin and T. Rivière, Complex Ginzburg-Landau equations in high dimensions and codimension-two area-minimizing currents, J. Eur. Math. Soc. 1 (1999), 237-311; Erratum 2 (2000), 87-91. MR 2000g:49048, MR 2001a:49041
- 48.
- C. Morrey, Multiple Integrals in the Calculus of Variations, Springer, 1966. MR 34:2380
- 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.
- 51.
- T. Rivière, Everywhere discontinuous harmonic maps into spheres, Acta Math. 175 (1995), 197-226. MR 96k:58059
- 52.
- -, Line vortices in the
- Higgs model, Control, Opt. and Calc. of Variations 1 (1996), 77-167. MR 97g:58043 - 53.
- J. Rubinstein and P. Sternberg, Homotopy classification of minimizers of the Ginzburg-Landau energy and the existence of permanent currents, Comm. Math. Phys. 179 (1996), 257-263. MR 97f:35208
- 54.
- J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. 113 (1981), 1-24. MR 82f:58035
- 55.
- R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Diff. Geom. 17 (1982), 307-335. MR 84b:58037a; correction in J. Diff. Geom. 18 (1983), 329. MR 84b:58037b
- 56.
- -, Boundary regularity and the Dirichlet problem for harmonic maps, J. Diff. Geom. 18 (1983), 253-268. MR 85b:58037
- 57.
- -, Regularity of minimizing harmonic maps into the sphere, Invent. Math. 78 (1984), 89-100. MR 86a:58024
- 58.
- R. Schoen and S. T. Yau, Lectures on Harmonic Maps, International Press, 1997. MR 98i:58072
- 59.
- H. Wente, An existence theorem for surfaces of constant mean curvature, J. Math. Annal. 26 (1969), 318-344. MR 39:4788
- 60.
- H. Weyl, The method of orthogonal projection in potential theory, Duke Math. J. 7 (1940), 411-440.
- 61.
- B. White, Homotopy classes in Sobolev spaces and the existence of energy minimizing maps, Acta Math. 160 (1988), 1-17. MR 89a:58031
Similar Articles:
Retrieve articles in Bulletin of the American Mathematical Society
with MSC
(2000):
35A15, 35A20, 35J50, 35J65, 35Qxx, 46Txx, 47Hxx, 47Jxx, 55Pxx, 58E15
Retrieve articles in all Journals with MSC
(2000):
35A15, 35A20, 35J50, 35J65, 35Qxx, 46Txx, 47Hxx, 47Jxx, 55Pxx, 58E15
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
Email:
brezis@math.rutgers.edu, brezis@ann.jussieu.fr, brezis@ccr.jussieu.fr
DOI:
10.1090/S0273-0979-03-00976-5
PII:
S 0273-0979(03)00976-5
Received by editor(s):
October 23, 2002
Posted:
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 of article:
Copyright
2003,
American Mathematical Society
|