# American Institute of Mathematical Sciences

February  2013, 33(2): 861-878. doi: 10.3934/dcds.2013.33.861

## Energy identity for a class of approximate biharmonic maps into sphere in dimension four

 1 Department of Mathematics, University of Kentucky, Lexington, KY 40506, United States 2 Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China

Received  January 2012 Revised  April 2012 Published  September 2012

We consider in dimension four weakly convergent sequences of approximate biharmonic maps into sphere with bi-tension fields bounded in $L^p$ for $p>1$. We prove an energy identity that accounts for the loss of hessian energies by the sum of hessian energies over finitely many nontrivial biharmonic maps on $\mathbb R^4$. As a corollary, we obtain an energy identity for the heat flow of biharmonic maps into sphere at time infinity.
Citation: Changyou Wang, Shenzhou Zheng. Energy identity for a class of approximate biharmonic maps into sphere in dimension four. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 861-878. doi: 10.3934/dcds.2013.33.861
##### References:
 [1] D. R. Adams, A note on Riesz potentials, Duke Math. J., 42 (1975), 765-778. doi: 10.1215/S0012-7094-75-04265-9. [2] A. Chang, L. Wang and P. Yang, Aregularity theory of biharmonic maps, Comm. Pure Appl. Math., 52 (1999), 1113-1137. doi: 10.1002/(SICI)1097-0312(199909)52:9<1113::AID-CPA4>3.0.CO;2-7. [3] W. Y. Ding and G. Tian, Energy identity for a class of approximate harmonic maps from surfaces, Comm. Anal. Geom., 3 (1995), 543-554. [4] A. Gastel, The extrinsic polyharmonic map heat flow in the critical dimension, Adv. Geom., 6 (2006), 501-521. doi: 10.1515/ADVGEOM.2006.031. [5] H. J. Gong, T. Lamm and C. Y. Wang, Boundary regularity for a class of biharmonicmaps, Calc. Var. Partial Differential Equations, 45 (2012), 165-191. doi: 10.1007/s00526-011-0455-2. [6] F. Hélein, "Harmonic Maps, Conservation Laws, and Moving Frames," Cambridge Tracts in Mathematics, 150, Cambridge: CambridgeUniversity Press 2002. [7] P. Hornung and R. Moser, Energy identity for instrinsically biharmonic maps in fourdimensions, Anal. PDE, to appear. [8] T. Iwaniec and G. Martin, Quasiregular mappings in even dimensions, Acta Math., 170 (1993), 29-81. doi: 10.1007/BF02392454. [9] Y. Ku, Interior and boundary regularity of intrinsic biharmonic maps to spheres, Pacific J. Math., 234 (2008), 43-67. [10] T. Lamm, Heat flow for extrinsic biharmonic maps with small initial energy, Ann. Global Anal. Geom., 26 (2004), 369-384. doi: 10.1023/B:AGAG.0000047526.21237.04. [11] T. Lamm and T. Rivière, Conservation laws for fourth order systems in four dimensions, Comm. PDE., 33 (2008), 245-262. [12] P. Laurain and T. Rivière, Angular energy quantization for linear elliptic systems with antisymmetric potentials and applications, Preprint, arXiv:1109.3599. [13] F. H. Lin and T. Rivière, Energy quantization for harmonic maps, Duke Math. J., 111 (2002), 177-193. doi: 10.1215/S0012-7094-02-11116-8. [14] F. H. Lin and T. Riviére, A quantization property for static Ginzburg-Landau vortices, Comm. Pure Appl. Math., 54 (2001), 206-228. doi: 10.1002/1097-0312(200102)54:2<206::AID-CPA3>3.0.CO;2-W. [15] F. H. Lin and C. Y. Wang, Harmonic and quasi-harmonic spheres II, Comm. Anal. Geom., 10 (2002), 341-375. [16] R. Moser, Weak solutions of a biharmonic map heat flow, Adv. Calc. Var., 2 (2009), 73-92. doi: 10.1515/ACV.2009.004. [17] C. Scheven, Dimension reduction for the singular set of biharmonic maps, Adv. Calc. Var., 1 (2008), 53-91. doi: 10.1515/ACV.2008.002. [18] C. Scheven, An optimal partial regularity result for minimizers of an intrinsically defined second-order functional, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1585-1605. [19] P. Strzelecki, On biharmonic maps and their generalizations, Calc. Var. Partial Differential Equations, 18 (2003), 401-432. [20] M. Struwe, Partial regularity for biharmonic maps, revisited, Calc. Var. Partial Differential Equations, 33 (2008), 249-262. [21] C. Y. Wang, Remarks on biharmonic maps into spheres, Calc. Var. Partial Differential Equations, 21 (2004), 221-242. [22] C. Y. Wang, Biharmonic maps from $\mathbb R^4$ into a Riemannian manifold, Math. Z., 247 (2004), 65-87. doi: 10.1007/s00209-003-0620-1. [23] C. Y. Wang, Stationray biharmonic maps from $\mathbb R^m$ into a Riemannian manifold, Comm. Pure Appl. Math., 57 (2004), 0419-0444. [24] C. Y. Wang, Heat flow of biharmonic maps in dimensions four and its application, Pure Appl. Math. Q., 3 (2007), 595-613. [25] C. Y. Wang and S. Z. Zheng, Energy identity of approximate biharmonic maps to Riemannian manifolds and its application, J. Funct. Anal., 263 (2012), 960-987. doi: 10.1016/j.jfa.2012.05.008.

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##### References:
 [1] D. R. Adams, A note on Riesz potentials, Duke Math. J., 42 (1975), 765-778. doi: 10.1215/S0012-7094-75-04265-9. [2] A. Chang, L. Wang and P. Yang, Aregularity theory of biharmonic maps, Comm. Pure Appl. Math., 52 (1999), 1113-1137. doi: 10.1002/(SICI)1097-0312(199909)52:9<1113::AID-CPA4>3.0.CO;2-7. [3] W. Y. Ding and G. Tian, Energy identity for a class of approximate harmonic maps from surfaces, Comm. Anal. Geom., 3 (1995), 543-554. [4] A. Gastel, The extrinsic polyharmonic map heat flow in the critical dimension, Adv. Geom., 6 (2006), 501-521. doi: 10.1515/ADVGEOM.2006.031. [5] H. J. Gong, T. Lamm and C. Y. Wang, Boundary regularity for a class of biharmonicmaps, Calc. Var. Partial Differential Equations, 45 (2012), 165-191. doi: 10.1007/s00526-011-0455-2. [6] F. Hélein, "Harmonic Maps, Conservation Laws, and Moving Frames," Cambridge Tracts in Mathematics, 150, Cambridge: CambridgeUniversity Press 2002. [7] P. Hornung and R. Moser, Energy identity for instrinsically biharmonic maps in fourdimensions, Anal. PDE, to appear. [8] T. Iwaniec and G. Martin, Quasiregular mappings in even dimensions, Acta Math., 170 (1993), 29-81. doi: 10.1007/BF02392454. [9] Y. Ku, Interior and boundary regularity of intrinsic biharmonic maps to spheres, Pacific J. Math., 234 (2008), 43-67. [10] T. Lamm, Heat flow for extrinsic biharmonic maps with small initial energy, Ann. Global Anal. Geom., 26 (2004), 369-384. doi: 10.1023/B:AGAG.0000047526.21237.04. [11] T. Lamm and T. Rivière, Conservation laws for fourth order systems in four dimensions, Comm. PDE., 33 (2008), 245-262. [12] P. Laurain and T. Rivière, Angular energy quantization for linear elliptic systems with antisymmetric potentials and applications, Preprint, arXiv:1109.3599. [13] F. H. Lin and T. Rivière, Energy quantization for harmonic maps, Duke Math. J., 111 (2002), 177-193. doi: 10.1215/S0012-7094-02-11116-8. [14] F. H. Lin and T. Riviére, A quantization property for static Ginzburg-Landau vortices, Comm. Pure Appl. Math., 54 (2001), 206-228. doi: 10.1002/1097-0312(200102)54:2<206::AID-CPA3>3.0.CO;2-W. [15] F. H. Lin and C. Y. Wang, Harmonic and quasi-harmonic spheres II, Comm. Anal. Geom., 10 (2002), 341-375. [16] R. Moser, Weak solutions of a biharmonic map heat flow, Adv. Calc. Var., 2 (2009), 73-92. doi: 10.1515/ACV.2009.004. [17] C. Scheven, Dimension reduction for the singular set of biharmonic maps, Adv. Calc. Var., 1 (2008), 53-91. doi: 10.1515/ACV.2008.002. [18] C. Scheven, An optimal partial regularity result for minimizers of an intrinsically defined second-order functional, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1585-1605. [19] P. Strzelecki, On biharmonic maps and their generalizations, Calc. Var. Partial Differential Equations, 18 (2003), 401-432. [20] M. Struwe, Partial regularity for biharmonic maps, revisited, Calc. Var. Partial Differential Equations, 33 (2008), 249-262. [21] C. Y. Wang, Remarks on biharmonic maps into spheres, Calc. Var. Partial Differential Equations, 21 (2004), 221-242. [22] C. Y. Wang, Biharmonic maps from $\mathbb R^4$ into a Riemannian manifold, Math. Z., 247 (2004), 65-87. doi: 10.1007/s00209-003-0620-1. [23] C. Y. Wang, Stationray biharmonic maps from $\mathbb R^m$ into a Riemannian manifold, Comm. Pure Appl. Math., 57 (2004), 0419-0444. [24] C. Y. Wang, Heat flow of biharmonic maps in dimensions four and its application, Pure Appl. Math. Q., 3 (2007), 595-613. [25] C. Y. Wang and S. Z. Zheng, Energy identity of approximate biharmonic maps to Riemannian manifolds and its application, J. Funct. Anal., 263 (2012), 960-987. doi: 10.1016/j.jfa.2012.05.008.
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