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A compactness theorem of $$n$$-harmonic maps. (English) Zbl 1229.58017
Summary: For $$n\geq 3$$, let $$\Omega\subset \mathbb R^n$$ be a bounded domain and $$N\subset\mathbb R^L$$ be a compact smooth Riemannian submanifold without boundary. Suppose that $$\{u_n\}\subset W^{1,n}(\Omega,N)$$ are weak solutions to a (perturbed) $$n$$-harmonic map equation satisfying a limit condition, and $$u_k\to u$$ weakly in $$W^{1,n}(\Omega,N)$$. Then $$u$$ is an $$n$$-harmonic map. In particular, the space of $$n$$-harmonic maps is sequentially compact for the weak-$$W^{1,n}$$ topology.

##### MSC:
 58E20 Harmonic maps, etc. 35J60 Nonlinear elliptic equations
##### Keywords:
Harmonic maps; Coulomb gauge frame; compensated-compactness
Full Text:
##### References:
 [1] Bethuel, F., Weak limits of palais – smale sequences for a class of critical functionals, Calc. var. partial differential equations, 1, 3, 267-310, (1993) · Zbl 0812.58018 [2] Bethuel, F., On the singular set of stationary harmonic maps, Manuscripta math., 78, 417-443, (1993) · Zbl 0792.53039 [3] Chen, Y.M., The weak solutions to the evolution problems of harmonic maps, Math. Z., 201, 1, 69-74, (1989) · Zbl 0685.58015 [4] Coifman, R.; Lions, P.; Meyer, Y.; Semmes, S., Compensated compactness and Hardy spaces, J. math. pures appl., 72, 247-286, (1993) · Zbl 0864.42009 [5] Evans, L.C., Partial regularity for stationary harmonic maps into spheres, Arch. rational mech. anal., 116, 101-113, (1991) · Zbl 0754.58007 [6] Evans, L.C., Weak convergence methods for nonlinear partial differential equations, CBMS regional conf. ser. in math., vol. 74, (1990) [7] Evans, L.C.; Gariepy, R., Measure theory and fine properties of functions, Stud. adv. math., (1992), CRC Press Boca Raton, FL · Zbl 0804.28001 [8] Fefferman, C.; Stein, E., $$H^p$$ spaces of several variables, Acta math., 129, 137-193, (1972) · Zbl 0257.46078 [9] Freire, A.; Müller, S.; Struwe, M., Weak convergence of wave maps from (1+2)-dimensional Minkowski space to Riemannian manifolds, Invent. math., 130, 3, 589-617, (1997) · Zbl 0906.35061 [10] Freire, A.; Müller, S.; Struwe, M., Weak compactness of wave maps and harmonic maps, Ann. inst. H. Poincaré anal. non linéaire, 15, 6, 725-754, (1998) · Zbl 0924.58011 [11] Fuchs, M., The blow-up of p-harmonic maps, Manuscripta math., 81, 1-2, 89-94, (1993) · Zbl 0794.58012 [12] Hélein, F., Regularite des applications faiblement harmoniques entre une surface et variete riemannienne, C. R. acad. sci. Paris, 312, 591-596, (1991) · Zbl 0728.35015 [13] Hardt, R.; Lin, F.H., Mappings minimizing the $$L^p$$ norm of the gradient, Comm. pure appl. math., 40, 5, 555-588, (1987) · Zbl 0646.49007 [14] Hardt, R.; Lin, F.H.; Mou, L., Strong convergence of p-harmonic mappings, (), 58-64 · Zbl 0833.35038 [15] Hélein, F., Harmonic maps, conservation laws and moving frames, Cambridge tracts in math., vol. 150, (2002), Cambridge Univ. Press Cambridge · Zbl 1010.58010 [16] Hungerbhler, N., m-harmonic flow, Ann. scuola norm. sup. Pisa cl. sci. (4), 24, 4, 593-631, (1997), (1998) [17] Iwaniec, T.; Martin, G., Quasiregular mappings in even dimensions, Acta math., 170, 1, 29-81, (1993) · Zbl 0785.30008 [18] John, F.; Nirenberg, L., On functions of bounded Mean oscillation, Comm. pure appl. math., 14, 415-426, (1961) · Zbl 0102.04302 [19] Lions, P.L., The concentration-compactness principle in the calculus of variations: the limit case, I, Rev. mat. iberoamericana, 1, 1, 145-201, (1985) · Zbl 0704.49005 [20] Lions, P.L., The concentration-compactness principle in the calculus of variations: the limit case, II, Rev. mat. iberoamericana, 1, 2, 45-121, (1985) · Zbl 0704.49006 [21] Luckhaus, S., Convergence of minimizers for the p-Dirichlet integral, Math. Z., 213, 3, 449-456, (1993) · Zbl 0798.58022 [22] Sacks, J.; Uhlenbeck, K., The existence of minimal immersions of 2-spheres, Ann. of math., 113, 1-24, (1981) · Zbl 0462.58014 [23] Schoen, R.; Uhlenbeck, K., A regularity theory for harmonic maps, J. differential geom., 17, 2, 307-335, (1982) · Zbl 0521.58021 [24] Shatah, J., Weak solutions and development of singularities of the $$\text{SU}(2)$$σ-model, Comm. pure appl. math., 41, 4, 459-469, (1988) · Zbl 0686.35081 [25] Strzelecki, P.; Zatorska-Goldstein, A., A compactness theorem for weak solutions of higher-dimensional H-systems, Duke math. J., 121, 2, 269-284, (2004) · Zbl 1054.58008 [26] Toro, T.; Wang, C.Y., Compactness properties of weakly p-harmonic maps into homogeneous spaces, Indiana univ. math. J., 44, 1, 87-113, (1995) · Zbl 0826.58014 [27] Uhlenbeck, K., Connections with $$L^p$$-bounds on curvature, Comm. math. phys., 83, 31-42, (1982) · Zbl 0499.58019 [28] Wang, C.Y., Bubble phenomena of certain palais – smale sequences from surfaces to general targets, Houston J. math., 22, 3, 559-590, (1996) · Zbl 0879.58019 [29] Wang, C.Y., Stationary biharmonic maps from $$\mathbf{R}^n$$ into a Riemannian manifold, Comm. pure appl. math., LVII, 0419-0444, (2004) [30] Wang, C.Y., Biharmonic maps from $$\mathbf{R}^4$$ into a Riemannian manifold, Math. Z., 247, 1, 65-87, (2004) · Zbl 1064.58016
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