Weak compactness of wave maps and harmonic maps
Annales de l'I.H.P. Analyse non linéaire, Volume 15 (1998) no. 6, pp. 725-754.
Freire, Alexandre 1; Müller, Stefan 2; Struwe, Michael 3

1 Dept. of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, USA
2 Max Planck Institute for Mathematics in the Sciences, D-04103 Leipzig
3 Mathematik, ETH-Zentrum, CH-8092 Zurich
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     title = {Weak compactness of wave maps and harmonic maps},
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     pages = {725--754},
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Freire, Alexandre; Müller, Stefan; Struwe, Michael. Weak compactness of wave maps and harmonic maps. Annales de l'I.H.P. Analyse non linéaire, Volume 15 (1998) no. 6, pp. 725-754. http://archive.numdam.org/item/AIHPC_1998__15_6_725_0/

[1] F. Bethuel and J.-M. Ghiadaglia, Weak limits of solutions to the steady incompressible

two-dimensional Euler equations in a bounded domain, Asympt. Anal., Vol. 8, 1994, pp. 277-291. | MR | Zbl

[2] F. Bethuel, Weak convergence of Palais-Smale sequences for some critical functionals, Calc. Var., Vol. 1, 1993, pp. 267-310. | MR | Zbl

[3] F. Bethuel, On the singular set of stationary harmonic maps, manusc. math., Vol. 78, 1993, pp. 417-443. | MR | Zbl

[4] S. Campanato, Sistemi ellittiche del secondo ordine e spazi L2.λ, Ann. Mat. Pura Appl., Vol. 69, 1965, pp. 321-380. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[8] L.C. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Rat. Mech. Anal., Vol. 116, 1991, 101-113. | MR | Zbl

[9] C. Fefferman and E.M. Stein, Hp spaces of several variables, Acta Math., Vol. 129, 1972, pp. 137-193. | MR | Zbl

[10] J. Frehse, Capacity methods in the theory of partial differential equations, Jahresb. Dt. Math. Ver., Vol. 84, 1982, pp. 1-44. | MR | Zbl

[11] A. Freire, Global weak solutions of the wave map system to compact homogeneous spaces, manuscripta math., Vol. 91, 1996, pp. 525-533. | MR | Zbl

[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. | MR | Zbl

[13] M. Giaquinta, Introduction to regularity theory for nonlinear elliptic systems, Lectures in Mathematics, Birkhäuser, 1993. | MR | Zbl

[14] D. Goldberg, A local version of the Hardy space, Duke Math. J., Vol. 46, 1979, pp. 27-42. | MR | Zbl

[15] E. Heinz, Elementare Bemerkung zur isoperimetrischen Ungleichung im R3, Math. Z., Vol. 132, 1973, pp. 319-322. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[18] S. Klainerman and M. Machedon, Smoothing estimates for null forms and applications, Int. Math. Res. Not., Vol. 4, 1994, pp. 383-389. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[22] S. Müller, M. Struwe and V. Šverák, Harmonic maps on planar lattices, Ann. SNS Pisa, to appear. | Numdam

[23] S. Müller and M. Struwe, Spatially discrete wave maps on (1+2)-dimensional space-time, Top. Methods Nonlinear Analysis, to appear. | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[27] E. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, 1970. | MR | Zbl

[28] E. Stein, Harmonic analysis, Princeton Univ. Press, 1993. | MR | Zbl

[29] M. Struwe, Wave maps, in: Nonlinear partial differential equations in geometry and physics-Barrett Lectures 1995 (Eds. G. Baker et al.), Birkhäuser, 1997, pp. 113-153. | MR | Zbl

[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. | MR | Zbl

[31] H.C. Wente, An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl., Vol. 26, 1969, pp. 318-344. | MR | Zbl

[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. | MR | Zbl

[33] Yi Zhou, Global weak solutions for the 2 + 1 dimensional wave maps with small energy, preprint, 1995.

[34] Yi Zhou, Global weak solutions for 1 + 2 dimensional wave maps into homogeneous spaces, Ann. IHP-Analyse non linéaire, to appear. | Numdam | Zbl