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An $ L^p$ regularity theory for harmonic maps


Author: Roger Moser
Journal: Trans. Amer. Math. Soc. 367 (2015), 1-30
MSC (2010): Primary 53C44, 58E20; Secondary 35B65
DOI: https://doi.org/10.1090/S0002-9947-2014-06282-X
Published electronically: May 20, 2014
MathSciNet review: 3271251
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Abstract: Motivated by the harmonic map heat flow, we consider maps between Riemannian manifolds such that the tension field belongs to an $ L^p$-space. Under an appropriate smallness condition, a certain degree of regularity follows. For suitable solutions of the harmonic map heat flow, we have a partial regularity result as a consequence.


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  • [1] David R. Adams, A note on Riesz potentials, Duke Math. J. 42 (1975), no. 4, 765-778. MR 0458158 (56 #16361)
  • [2] David R. Adams and Michael Frazier, Composition operators on potential spaces, Proc. Amer. Math. Soc. 114 (1992), no. 1, 155-165. MR 1076570 (92e:46061), https://doi.org/10.2307/2159794
  • [3] Fabrice Bethuel, On the singular set of stationary harmonic maps, Manuscripta Math. 78 (1993), no. 4, 417-443. MR 1208652 (94a:58047), https://doi.org/10.1007/BF02599324
  • [4] A. P. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85-139. MR 0052553 (14,637f)
  • [5] Kung-Ching Chang, Heat flow and boundary value problem for harmonic maps, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989), no. 5, 363-395 (English, with French summary). MR 1030856 (90i:58037)
  • [6] Yun Mei Chen, Jiayu Li, and Fang-Hua Lin, Partial regularity for weak heat flows into spheres, Comm. Pure Appl. Math. 48 (1995), no. 4, 429-448. MR 1324408 (96e:58039), https://doi.org/10.1002/cpa.3160480403
  • [7] Yun Mei Chen, Dirichlet problems for heat flows of harmonic maps in higher dimensions, Math. Z. 208 (1991), no. 4, 557-565. MR 1136475 (92k:58063), https://doi.org/10.1007/BF02571545
  • [8] Yun Mei Chen and Michael Struwe, Existence and partial regularity results for the heat flow for harmonic maps, Math. Z. 201 (1989), no. 1, 83-103. MR 990191 (90i:58031), https://doi.org/10.1007/BF01161997
  • [9] R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9) 72 (1993), no. 3, 247-286 (English, with English and French summaries). MR 1225511 (95d:46033)
  • [10] R. R. Coifman, R. Rochberg, and Guido Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (1976), no. 3, 611-635. MR 0412721 (54 #843)
  • [11] G. Da Prato, Spazi $ {\mathfrak{L}}^{(p,\theta )}(\Omega ,\delta )$ e loro proprietà, Ann. Mat. Pura Appl. (4) 69 (1965), 383-392 (Italian). MR 0192330 (33 #556)
  • [12] Emmanuele DiBenedetto, Real analysis, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Boston Inc., Boston, MA, 2002. MR 1897317 (2003d:00001)
  • [13] Lawrence C. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Rational Mech. Anal. 116 (1991), no. 2, 101-113. MR 1143435 (93m:58026), https://doi.org/10.1007/BF00375587
  • [14] C. Fefferman and E. M. Stein, $ H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137-193. MR 0447953 (56 #6263)
  • [15] Mikhail Feldman, Partial regularity for harmonic maps of evolution into spheres, Comm. Partial Differential Equations 19 (1994), no. 5-6, 761-790. MR 1274539 (95i:58057), https://doi.org/10.1080/03605309408821034
  • [16] Alexandre Freire, Uniqueness for the harmonic map flow from surfaces to general targets, Comment. Math. Helv. 70 (1995), no. 2, 310-338. MR 1324632 (96f:58045), https://doi.org/10.1007/BF02566010
  • [17] Alexandre Freire, Uniqueness for the harmonic map flow in two dimensions, Calc. Var. Partial Differential Equations 3 (1995), no. 1, 95-105. MR 1384838 (97d:58058), https://doi.org/10.1007/BF01190893
  • [18] Emilio Gagliardo, Ulteriori proprietà di alcune classi di funzioni in più variabili, Ricerche Mat. 8 (1959), 24-51 (Italian). MR 0109295 (22 #181)
  • [19] F. W. Gehring, The $ L^{p}$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265-277. MR 0402038 (53 #5861)
  • [20] M. Giaquinta and G. Modica, Regularity results for some classes of higher order nonlinear elliptic systems, J. Reine Angew. Math. 311/312 (1979), 145-169. MR 549962 (81a:35035)
  • [21] Frédéric Hélein, Régularité des applications faiblement harmoniques entre une surface et une sphère, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 9, 519-524 (French, with English summary). MR 1078114 (92a:58034)
  • [22] Frédéric Hélein, Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 8, 591-596 (French, with English summary). MR 1101039 (92e:58055)
  • [23] Frédéric Hélein, Regularity of weakly harmonic maps from a surface into a manifold with symmetries, Manuscripta Math. 70 (1991), no. 2, 203-218. MR 1085633 (92a:58035), https://doi.org/10.1007/BF02568371
  • [24] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426. MR 0131498 (24 #A1348)
  • [25] Tobias Lamm and Tristan Rivière, Conservation laws for fourth order systems in four dimensions, Comm. Partial Differential Equations 33 (2008), no. 1-3, 245-262. MR 2398228 (2009h:35095), https://doi.org/10.1080/03605300701382381
  • [26] Xian-Gao Liu, Partial regularity for weak heat flows into a general compact Riemannian manifold, Arch. Ration. Mech. Anal. 168 (2003), no. 2, 131-163. MR 1991990 (2004d:53085), https://doi.org/10.1007/s00205-003-0250-0
  • [27] Yves Meyer and Tristan Rivière, A partial regularity result for a class of stationary Yang-Mills fields in high dimension, Rev. Mat. Iberoamericana 19 (2003), no. 1, 195-219. MR 1993420 (2004h:35032), https://doi.org/10.4171/RMI/343
  • [28] Charles B. Morrey Jr., The problem of Plateau on a Riemannian manifold, Ann. of Math. (2) 49 (1948), 807-851. MR 0027137 (10,259f)
  • [29] Roger Moser, Regularity for the approximated harmonic map equation and application to the heat flow for harmonic maps, Math. Z. 243 (2003), no. 2, 263-289. MR 1961867 (2003k:53081), https://doi.org/10.1007/s00209-002-0463-1
  • [30] Roger Moser, Partial regularity for harmonic maps and related problems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. MR 2155901 (2006d:58016)
  • [31] Roger Moser, Remarks on the regularity of biharmonic maps in four dimensions, Comm. Pure Appl. Math. 59 (2006), no. 3, 317-329. MR 2200257 (2007b:58026), https://doi.org/10.1002/cpa.20117
  • [32] Roger Moser, A Trudinger type inequality for maps into a Riemannian manifold, Ann. Global Anal. Geom. 35 (2009), no. 1, 83-90. MR 2480665 (2010f:47140), https://doi.org/10.1007/s10455-008-9123-y
  • [33] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 115-162. MR 0109940 (22 #823)
  • [34] L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa (3) 20 (1966), 733-737. MR 0208360 (34 #8170)
  • [35] Peter Price, A monotonicity formula for Yang-Mills fields, Manuscripta Math. 43 (1983), no. 2-3, 131-166. MR 707042 (84m:58033), https://doi.org/10.1007/BF01165828
  • [36] Tristan Rivière, Everywhere discontinuous harmonic maps into spheres, Acta Math. 175 (1995), no. 2, 197-226. MR 1368247 (96k:58059), https://doi.org/10.1007/BF02393305
  • [37] Tristan Rivière, Conservation laws for conformally invariant variational problems, Invent. Math. 168 (2007), no. 1, 1-22. MR 2285745 (2008d:58010), https://doi.org/10.1007/s00222-006-0023-0
  • [38] Tristan Rivière and Michael Struwe, Partial regularity for harmonic maps and related problems, Comm. Pure Appl. Math. 61 (2008), no. 4, 451-463. MR 2383929 (2009a:58016), https://doi.org/10.1002/cpa.20205
  • [39] Melanie Rupflin, An improved uniqueness result for the harmonic map flow in two dimensions, Calc. Var. Partial Differential Equations 33 (2008), no. 3, 329-341. MR 2429534 (2009g:53106), https://doi.org/10.1007/s00526-008-0164-7
  • [40] Richard Schoen and Karen Uhlenbeck, A regularity theory for harmonic maps, J. Differential Geom. 17 (1982), no. 2, 307-335. MR 664498 (84b:58037a)
  • [41] Richard Schoen and Karen Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps, J. Differential Geom. 18 (1983), no. 2, 253-268. MR 710054 (85b:58037)
  • [42] B. Sharp, Higher integrability for solutions to a system of critical elliptic PDE, arXiv:1112.1127v1 [math.AP], 2011.
  • [43] Ben Sharp and Peter Topping, Decay estimates for Rivière's equation, with applications to regularity and compactness, Trans. Amer. Math. Soc. 365 (2013), no. 5, 2317-2339. MR 3020100, https://doi.org/10.1090/S0002-9947-2012-05671-6
  • [44] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192 (95c:42002)
  • [45] Michael Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helv. 60 (1985), no. 4, 558-581. MR 826871 (87e:58056), https://doi.org/10.1007/BF02567432
  • [46] Michael Struwe, On the evolution of harmonic maps in higher dimensions, J. Differential Geom. 28 (1988), no. 3, 485-502. MR 965226 (90j:58037)
  • [47] Michael Struwe, Partial regularity for biharmonic maps, revisited, Calc. Var. Partial Differential Equations 33 (2008), no. 2, 249-262. MR 2413109 (2009b:35068), https://doi.org/10.1007/s00526-008-0175-4
  • [48] P. Strzelecki, Gagliardo-Nirenberg inequalities with a BMO term, Bull. London Math. Soc. 38 (2006), no. 2, 294-300. MR 2215922 (2007a:46038), https://doi.org/10.1112/S0024609306018169
  • [49] Terence Tao and Gang Tian, A singularity removal theorem for Yang-Mills fields in higher dimensions, J. Amer. Math. Soc. 17 (2004), no. 3, 557-593. MR 2053951 (2005f:58013), https://doi.org/10.1090/S0894-0347-04-00457-6
  • [50] Karen K. Uhlenbeck, Connections with $ L^{p}$ bounds on curvature, Comm. Math. Phys. 83 (1982), no. 1, 31-42. MR 648356 (83e:53035)

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Additional Information

Roger Moser
Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
Email: r.moser@bath.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-2014-06282-X
Received by editor(s): April 23, 2012
Published electronically: May 20, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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