Weak compactness of wave maps and harmonic maps

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1. [1] F. Bethuel and J.-M. Ghiadaglia, Weak limits of solutions to the steady incompressible 
2. two-dimensional Euler equations in a bounded domain, Asympt. Anal., Vol. 8, 1994, pp. 277-291. Zbl0808.76014MR1288610
3. [2] F. Bethuel, Weak convergence of Palais-Smale sequences for some critical functionals, Calc. Var., Vol. 1, 1993, pp. 267-310. Zbl0812.58018MR1261547
4. [3] F. Bethuel, On the singular set of stationary harmonic maps, manusc. math., Vol. 78, 1993, pp. 417-443. Zbl0792.53039MR1208652
5. [4] S. Campanato, Sistemi ellittiche del secondo ordine e spazi L2.λ, Ann. Mat. Pura Appl., Vol. 69, 1965, pp. 321-380. Zbl0145.36603MR192168
6. [5] D. Christodoulou and A.S. Tahvildar-Zadeh, On the regularity of spherically symmetric wave maps, Comm. Pure Appl. Math., Vol. 46, 1993, pp. 1041-1091. Zbl0744.58071MR1223662
7. [6] R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., Vol. 72, 1993, pp. 247-286. Zbl0864.42009MR1225511
8. [7] R. Diperna and A. Majda, Reduced Hausdorff dimension and concentration cancellation for two-dimensional incompressible flows, J. Amer. Math. Soc., Vol. 1. 1988, pp. 59-95. Zbl0707.76026MR924702
9. [8] L.C. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Rat. Mech. Anal., Vol. 116, 1991, 101-113. Zbl0754.58007MR1143435
10. [9] C. Fefferman and E.M. Stein, Hp spaces of several variables, Acta Math., Vol. 129, 1972, pp. 137-193. Zbl0257.46078MR447953
11. [10] J. Frehse, Capacity methods in the theory of partial differential equations, Jahresb. Dt. Math. Ver., Vol. 84, 1982, pp. 1-44. Zbl0486.35002MR644068
12. [11] A. Freire, Global weak solutions of the wave map system to compact homogeneous spaces, manuscripta math., Vol. 91, 1996, pp. 525-533. Zbl0867.58019MR1421290
13. [12] A. Freire, S. Müller and M. Struwe, Weak convergence of wave maps from (2+1) dimensional Minkowski space to Riemannian manifolds, Invent. Math., Vol. 130, 1997, pp. 589-617. Zbl0906.35061MR1483995
14. [13] M. Giaquinta, Introduction to regularity theory for nonlinear elliptic systems, Lectures in Mathematics, Birkhäuser, 1993. Zbl0786.35001MR1239172
15. [14] D. Goldberg, A local version of the Hardy space, Duke Math. J., Vol. 46, 1979, pp. 27-42. Zbl0409.46060MR523600
16. [15] E. Heinz, Elementare Bemerkung zur isoperimetrischen Ungleichung im R3, Math. Z., Vol. 132, 1973, pp. 319-322. Zbl0251.52014MR323976
17. [16] F. Hélein, Régularité des applications faiblement harmoniques entre une surface et une variété Riemannienne, C. R. Acad. Sci. Paris Ser. I Math., Vol. 312, 1991, pp. 591-596. Zbl0728.35015MR1101039
18. [17] S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math., Vol. 46, 1993, pp. 1221-1268. Zbl0803.35095MR1231427
19. [18] S. Klainerman and M. Machedon, Smoothing estimates for null forms and applications, Int. Math. Res. Not., Vol. 4, 1994, pp. 383-389. Zbl0832.35096MR1301438
20. [19] P.-L. Lions, The concentration compactness principle in the calculus of variations, the limit case, Part 2 , Rev. Mat. Iberoam., Vol. 1/2 , 1985, pp. 45-121. Zbl0704.49006MR850686
21. [20] E.J. Nussenzweig Lopes, An estimate on the Hausdorff dimension of a concentration set for the incompressible 2-D Euler equations, Indiana Univ. Math. J., Vol. 43, 1994, pp. 521-534. Zbl0807.35113MR1291527
22. [21] S. Müller and M. Struwe, Global existence of wave maps in 1+2 dimensions for finite energy data, Top. Methods Nonlinear Analysis, Vol. 7, 1996, pp. 245-259. Zbl0896.35086MR1481698
23. [22] S. Müller, M. Struwe and V. Šverák, Harmonic maps on planar lattices, Ann. SNS Pisa, to appear. Zbl1004.58007
24. [23] S. Müller and M. Struwe, Spatially discrete wave maps on (1+2)-dimensional space-time, Top. Methods Nonlinear Analysis, to appear. Zbl0933.58020
25. [24] S. Semmes, A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller, Conun. P.D.E., Vol. 19, 1994, 277-319. Zbl0836.35030MR1257006
26. [25] J. Shatah, Weak solutions and development of singularities in the SU (2) σ-model, Comm. Pure Appl. Math., Vol. 41, 1988, pp. 459-469. Zbl0686.35081MR933231
27. [26] J. Shatah and A.S. Tahvildar-Zadeh, On the Cauchy problem for equivariant wave maps, Comm. Pure Appl. Math., Vol. 45, 1992, pp. 947-971. Zbl0769.58015MR1168115
28. [27] E. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, 1970. Zbl0207.13501MR290095
29. [28] E. Stein, Harmonic analysis, Princeton Univ. Press, 1993. Zbl0821.42001MR1232192
30. [29] M. Struwe, Wave maps, in: Nonlinear partial differential equations in geometry and physics-Barrett Lectures1995 (Eds. G. Baker et al.), Birkhäuser, 1997, pp. 113-153. Zbl0869.53017MR1437153
31. [30] L. Tartar, Remarks on oscillations and Stokes's equation, in: Macroscopic modelling of turbulent flows (Nice, 1984), Lect. Notes Phys., Vol. 230, Springer, 1985, pp. 24-31. Zbl0611.76042MR815930
32. [31] H.C. Wente, An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl., Vol. 26, 1969, pp. 318-344. Zbl0181.11501MR243467
33. [32] Y. Zheng, Regularity of solutions to a two dimensional modified Dirac-Klein-Gordon system of equations, Comm. Math. Phys., Vol. 151, 1993, pp. 67-87. Zbl0765.35046MR1201656
34. [33] Yi Zhou, Global weak solutions for the 2 + 1 dimensional wave maps with small energy, preprint, 1995. 
35. [34] Yi Zhou, Global weak solutions for 1 + 2 dimensional wave maps into homogeneous spaces, Ann. IHP-Analyse non linéaire, to appear. Zbl0997.58012

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