American Institute of Mathematical Sciences

Advanced
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

Changyou Wang 1, and  Shenzhou Zheng 2,

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.
Keywords: heat flow of biharmonic maps., energy identity, Approximate biharmonic maps.
Mathematics Subject Classification: Primary: 35J48; Secondary: 35G4.
Citation: Changyou Wang, Shenzhou Zheng. Energy identity for a class of approximate biharmonic maps into sphere in dimension four. Discrete & Continuous Dynamical Systems - A, 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.  doi: 10.1215/S0012-7094-75-04265-9.  Google Scholar

[2]

A. Chang, L. Wang and P. Yang, Aregularity theory of biharmonic maps,, Comm. Pure Appl. Math., 52 (1999), 1113.  doi: 10.1002/(SICI)1097-0312(199909)52:9<1113::AID-CPA4>3.0.CO;2-7.  Google Scholar

[3]

W. Y. Ding and G. Tian, Energy identity for a class of approximate harmonic maps from surfaces,, Comm. Anal. Geom., 3 (1995), 543.   Google Scholar

[4]

A. Gastel, The extrinsic polyharmonic map heat flow in the critical dimension,, Adv. Geom., 6 (2006), 501.  doi: 10.1515/ADVGEOM.2006.031.  Google Scholar

[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.  doi: 10.1007/s00526-011-0455-2.  Google Scholar

[6]

F. Hélein, "Harmonic Maps, Conservation Laws, and Moving Frames,", Cambridge Tracts in Mathematics, (2002).   Google Scholar

[7]

P. Hornung and R. Moser, Energy identity for instrinsically biharmonic maps in fourdimensions,, Anal. PDE, ().   Google Scholar

[8]

T. Iwaniec and G. Martin, Quasiregular mappings in even dimensions,, Acta Math., 170 (1993), 29.  doi: 10.1007/BF02392454.  Google Scholar

[9]

Y. Ku, Interior and boundary regularity of intrinsic biharmonic maps to spheres,, Pacific J. Math., 234 (2008), 43.   Google Scholar

[10]

T. Lamm, Heat flow for extrinsic biharmonic maps with small initial energy,, Ann. Global Anal. Geom., 26 (2004), 369.  doi: 10.1023/B:AGAG.0000047526.21237.04.  Google Scholar

[11]

T. Lamm and T. Rivière, Conservation laws for fourth order systems in four dimensions,, Comm. PDE., 33 (2008), 245.   Google Scholar

[12]

P. Laurain and T. Rivière, Angular energy quantization for linear elliptic systems with antisymmetric potentials and applications,, Preprint, ().   Google Scholar

[13]

F. H. Lin and T. Rivière, Energy quantization for harmonic maps,, Duke Math. J., 111 (2002), 177.  doi: 10.1215/S0012-7094-02-11116-8.  Google Scholar

[14]

F. H. Lin and T. Riviére, A quantization property for static Ginzburg-Landau vortices,, Comm. Pure Appl. Math., 54 (2001), 206.  doi: 10.1002/1097-0312(200102)54:2<206::AID-CPA3>3.0.CO;2-W.  Google Scholar

[15]

F. H. Lin and C. Y. Wang, Harmonic and quasi-harmonic spheres II,, Comm. Anal. Geom., 10 (2002), 341.   Google Scholar

[16]

R. Moser, Weak solutions of a biharmonic map heat flow,, Adv. Calc. Var., 2 (2009), 73.  doi: 10.1515/ACV.2009.004.  Google Scholar

[17]

C. Scheven, Dimension reduction for the singular set of biharmonic maps,, Adv. Calc. Var., 1 (2008), 53.  doi: 10.1515/ACV.2008.002.  Google Scholar

[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.   Google Scholar

[19]

P. Strzelecki, On biharmonic maps and their generalizations,, Calc. Var. Partial Differential Equations, 18 (2003), 401.   Google Scholar

[20]

M. Struwe, Partial regularity for biharmonic maps, revisited,, Calc. Var. Partial Differential Equations, 33 (2008), 249.   Google Scholar

[21]

C. Y. Wang, Remarks on biharmonic maps into spheres,, Calc. Var. Partial Differential Equations, 21 (2004), 221.   Google Scholar

[22]

C. Y. Wang, Biharmonic maps from $\mathbb R^4$ into a Riemannian manifold,, Math. Z., 247 (2004), 65.  doi: 10.1007/s00209-003-0620-1.  Google Scholar

[23]

C. Y. Wang, Stationray biharmonic maps from $\mathbb R^m$ into a Riemannian manifold,, Comm. Pure Appl. Math., 57 (2004), 0419.   Google Scholar

[24]

C. Y. Wang, Heat flow of biharmonic maps in dimensions four and its application,, Pure Appl. Math. Q., 3 (2007), 595.   Google Scholar

[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.  doi: 10.1016/j.jfa.2012.05.008.  Google Scholar

show all references

References:
[1]

D. R. Adams, A note on Riesz potentials,, Duke Math. J., 42 (1975), 765.  doi: 10.1215/S0012-7094-75-04265-9.  Google Scholar

[2]

A. Chang, L. Wang and P. Yang, Aregularity theory of biharmonic maps,, Comm. Pure Appl. Math., 52 (1999), 1113.  doi: 10.1002/(SICI)1097-0312(199909)52:9<1113::AID-CPA4>3.0.CO;2-7.  Google Scholar

[3]

W. Y. Ding and G. Tian, Energy identity for a class of approximate harmonic maps from surfaces,, Comm. Anal. Geom., 3 (1995), 543.   Google Scholar

[4]

A. Gastel, The extrinsic polyharmonic map heat flow in the critical dimension,, Adv. Geom., 6 (2006), 501.  doi: 10.1515/ADVGEOM.2006.031.  Google Scholar

[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.  doi: 10.1007/s00526-011-0455-2.  Google Scholar

[6]

F. Hélein, "Harmonic Maps, Conservation Laws, and Moving Frames,", Cambridge Tracts in Mathematics, (2002).   Google Scholar

[7]

P. Hornung and R. Moser, Energy identity for instrinsically biharmonic maps in fourdimensions,, Anal. PDE, ().   Google Scholar

[8]

T. Iwaniec and G. Martin, Quasiregular mappings in even dimensions,, Acta Math., 170 (1993), 29.  doi: 10.1007/BF02392454.  Google Scholar

[9]

Y. Ku, Interior and boundary regularity of intrinsic biharmonic maps to spheres,, Pacific J. Math., 234 (2008), 43.   Google Scholar

[10]

T. Lamm, Heat flow for extrinsic biharmonic maps with small initial energy,, Ann. Global Anal. Geom., 26 (2004), 369.  doi: 10.1023/B:AGAG.0000047526.21237.04.  Google Scholar

[11]

T. Lamm and T. Rivière, Conservation laws for fourth order systems in four dimensions,, Comm. PDE., 33 (2008), 245.   Google Scholar

[12]

P. Laurain and T. Rivière, Angular energy quantization for linear elliptic systems with antisymmetric potentials and applications,, Preprint, ().   Google Scholar

[13]

F. H. Lin and T. Rivière, Energy quantization for harmonic maps,, Duke Math. J., 111 (2002), 177.  doi: 10.1215/S0012-7094-02-11116-8.  Google Scholar

[14]

F. H. Lin and T. Riviére, A quantization property for static Ginzburg-Landau vortices,, Comm. Pure Appl. Math., 54 (2001), 206.  doi: 10.1002/1097-0312(200102)54:2<206::AID-CPA3>3.0.CO;2-W.  Google Scholar

[15]

F. H. Lin and C. Y. Wang, Harmonic and quasi-harmonic spheres II,, Comm. Anal. Geom., 10 (2002), 341.   Google Scholar

[16]

R. Moser, Weak solutions of a biharmonic map heat flow,, Adv. Calc. Var., 2 (2009), 73.  doi: 10.1515/ACV.2009.004.  Google Scholar

[17]

C. Scheven, Dimension reduction for the singular set of biharmonic maps,, Adv. Calc. Var., 1 (2008), 53.  doi: 10.1515/ACV.2008.002.  Google Scholar

[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.   Google Scholar

[19]

P. Strzelecki, On biharmonic maps and their generalizations,, Calc. Var. Partial Differential Equations, 18 (2003), 401.   Google Scholar

[20]

M. Struwe, Partial regularity for biharmonic maps, revisited,, Calc. Var. Partial Differential Equations, 33 (2008), 249.   Google Scholar

[21]

C. Y. Wang, Remarks on biharmonic maps into spheres,, Calc. Var. Partial Differential Equations, 21 (2004), 221.   Google Scholar

[22]

C. Y. Wang, Biharmonic maps from $\mathbb R^4$ into a Riemannian manifold,, Math. Z., 247 (2004), 65.  doi: 10.1007/s00209-003-0620-1.  Google Scholar

[23]

C. Y. Wang, Stationray biharmonic maps from $\mathbb R^m$ into a Riemannian manifold,, Comm. Pure Appl. Math., 57 (2004), 0419.   Google Scholar

[24]

C. Y. Wang, Heat flow of biharmonic maps in dimensions four and its application,, Pure Appl. Math. Q., 3 (2007), 595.   Google Scholar

[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.  doi: 10.1016/j.jfa.2012.05.008.  Google Scholar

[1]

Filippo Gazzola, Hans-Christoph Grunau. Eventual local positivity for a biharmonic heat equation in RN. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 83-87. doi: 10.3934/dcdss.2008.1.83
[2]

Marco Abate, Francesca Tovena. Formal normal forms for holomorphic maps tangent to the identity. Conference Publications, 2005, 2005 (Special) : 1-10. doi: 10.3934/proc.2005.2005.1
[3]

Sikhar Patranabis, Debdeep Mukhopadhyay. Identity-based key aggregate cryptosystem from multilinear maps. Advances in Mathematics of Communications, 2019, 13 (4) : 759-778. doi: 10.3934/amc.2019044
[4]

Lam Quoc Anh, Pham Thanh Duoc, Tran Ngoc Tam. Continuity of approximate solution maps to vector equilibrium problems. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1685-1699. doi: 10.3934/jimo.2017013
[5]

Paulo Cesar Carrião, R. Demarque, Olímpio H. Miyagaki. Nonlinear Biharmonic Problems with Singular Potentials. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2141-2154. doi: 10.3934/cpaa.2014.13.2141
[6]

Roman Chapko, B. Tomas Johansson. Integral equations for biharmonic data completion. Inverse Problems & Imaging, 2019, 13 (5) : 1095-1111. doi: 10.3934/ipi.2019049
[7]

Zhongliang Wang. Nonradial positive solutions for a biharmonic critical growth problem. Communications on Pure & Applied Analysis, 2012, 11 (2) : 517-545. doi: 10.3934/cpaa.2012.11.517
[8]

Teemu Tyni, Valery Serov. Scattering problems for perturbations of the multidimensional biharmonic operator. Inverse Problems & Imaging, 2018, 12 (1) : 205-227. doi: 10.3934/ipi.2018008
[9]

Guillaume Warnault. Regularity of the extremal solution for a biharmonic problem with general nonlinearity. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1709-1723. doi: 10.3934/cpaa.2009.8.1709
[10]

Jiří Benedikt. Continuous dependence of eigenvalues of $p$-biharmonic problems on $p$. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1469-1486. doi: 10.3934/cpaa.2013.12.1469
[11]

Alberto Ferrero, Filippo Gazzola, Hans-Christoph Grunau. Decay and local eventual positivity for biharmonic parabolic equations. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1129-1157. doi: 10.3934/dcds.2008.21.1129
[12]

Tran Ngoc Thach, Nguyen Huy Tuan, Donal O'Regan. Regularized solution for a biharmonic equation with discrete data. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020008
[13]

Ying Gao, Xinmin Yang, Jin Yang, Hong Yan. Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps. Journal of Industrial & Management Optimization, 2015, 11 (2) : 673-683. doi: 10.3934/jimo.2015.11.673
[14]

Chérif Amrouche, Yves Raudin. Singular boundary conditions and regularity for the biharmonic problem in the half-space. Communications on Pure & Applied Analysis, 2007, 6 (4) : 957-982. doi: 10.3934/cpaa.2007.6.957
[15]

Federica Sani. A biharmonic equation in $\mathbb{R}^4$ involving nonlinearities with critical exponential growth. Communications on Pure & Applied Analysis, 2013, 12 (1) : 405-428. doi: 10.3934/cpaa.2013.12.405
[16]

Marie-Françoise Bidaut-Véron, Marta García-Huidobro, Cecilia Yarur. Large solutions of elliptic systems of second order and applications to the biharmonic equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 411-432. doi: 10.3934/dcds.2012.32.411
[17]

Filippo Gazzola. On the moments of solutions to linear parabolic equations involving the biharmonic operator. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3583-3597. doi: 10.3934/dcds.2013.33.3583
[18]

Elvise Berchio, Filippo Gazzola. Positive solutions to a linearly perturbed critical growth biharmonic problem. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 809-823. doi: 10.3934/dcdss.2011.4.809
[19]

Luiz F. O. Faria. Existence and uniqueness of positive solutions for singular biharmonic elliptic systems. Conference Publications, 2015, 2015 (special) : 400-408. doi: 10.3934/proc.2015.0400
[20]

Angelo Favini, Rabah Labbas, Keddour Lemrabet, Stéphane Maingot, Hassan D. Sidibé. Resolution and optimal regularity for a biharmonic equation with impedance boundary conditions and some generalizations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4991-5014. doi: 10.3934/dcds.2013.33.4991

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences