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Positive solutions for non local elliptic problems
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 |
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. |
show all references
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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