Previous |  Up |  Next


Title: A compactness result for polyharmonic maps in the critical dimension (English)
Author: Zheng, Shenzhou
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 66
Issue: 1
Year: 2016
Pages: 137-150
Summary lang: English
Category: math
Summary: For $n=2m\ge 4$, let $\Omega \in \mathbb {R}^n$ be a bounded smooth domain and ${\mathcal {N}\subset \mathbb {R}^L}$ a compact smooth Riemannian manifold without boundary. Suppose that $\{u_k\}\in W^{m,2}(\Omega ,\mathcal {N})$ is a sequence of weak solutions in the critical dimension to the perturbed $m$-polyharmonic maps $$\label {m-polyharmonic} \frac {\rm d}{{\rm d} t}\Big |_{t=0}E_m(\Pi (u+t\xi ))=0 $$ with $\Phi _k\rightarrow 0$ in $(W^{m,2}(\Omega ,\mathcal {N}))^*$ and $u_k\rightharpoonup u$ weakly in $W^{m,2}(\Omega ,\mathcal {N})$. Then $u$ is an $m$-polyharmonic map. In particular, the space of $m$-polyharmonic maps is sequentially compact for the weak-$W^{m,2}$ topology. (English)
Keyword: polyharmonic map
Keyword: compactness
Keyword: Coulomb moving frame
Keyword: Palais-Smale sequence
Keyword: removable singularity
MSC: 35J35
MSC: 35J48
MSC: 58J05
idZBL: Zbl 06587880
idMR: MR3483229
DOI: 10.1007/s10587-016-0246-1
Date available: 2016-04-07T15:02:14Z
Last updated: 2020-07-03
Stable URL:
Reference: [1] Angelsberg, G., Pumberger, D.: A regularity result for polyharmonic maps with higher integrability.Ann. Global Anal. Geom. 35 (2009), 63-81. Zbl 1172.58003, MR 2480664, 10.1007/s10455-008-9122-z
Reference: [2] Bethuel, F.: Weak limits of Palais-Smale sequences for a class of critical functionals.Calc. Var. Partial Differ. Equ. 1 (1993), 267-310. MR 1261547, 10.1007/BF01191297
Reference: [3] Freire, A., Müller, S., Struwe, M.: Weak compactness of wave maps and harmonic maps.Ann. Inst. Henri Poincaré, Anal. Non Linéaire 15 (1998), 725-754. MR 1650966, 10.1016/S0294-1449(99)80003-1
Reference: [4] Gastel, A.: The extrinsic polyharmonic map heat flow in the critical dimension.Adv. Geom. 6 (2006), 501-521. Zbl 1136.58010, MR 2267035, 10.1515/ADVGEOM.2006.031
Reference: [5] Gastel, A., Scheven, C.: Regularity of polyharmonic maps in the critical dimension.Commun. Anal. Geom. 17 (2009), 185-226. Zbl 1183.58010, MR 2520907, 10.4310/CAG.2009.v17.n2.a2
Reference: [6] Goldstein, P., Strzelecki, A., Zatorska-Goldstein, A.: On polyharmonic maps into spheres in the critical dimension.Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009), 1387-1405. Zbl 1188.35071, MR 2542730, 10.1016/j.anihpc.2008.10.008
Reference: [7] Hélein, F.: Regularity of weakly harmonic maps between a surface and a Riemannian manifold.C. R. Acad. Sci., Paris, Sér. (1) 312 French (1991), 591-596. MR 1101039
Reference: [8] Lamm, T., Rivière, T.: Conservation laws for fourth order systems in four dimensions.Commun. Partial Differ. Equations 33 (2008), 245-262. Zbl 1139.35328, MR 2398228, 10.1080/03605300701382381
Reference: [9] Laurain, P., Rivière, T.: Energy quantization for biharmonic maps.Adv. Calc. Var. 6 (2013), 191-216. Zbl 1275.35098, MR 3043576, 10.1515/acv-2012-0105
Reference: [10] Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The limit case. I.Rev. Mat. Iberoam. 1 (1985), 145-201. Zbl 0704.49005, MR 0834360, 10.4171/RMI/6
Reference: [11] Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The limit case. II.Rev. Mat. Iberoam. 1 (1985), 45-121. MR 0850686, 10.4171/RMI/12
Reference: [12] Mou, L., Wang, C.: Bubbling phenomena of Palais-Smale-like sequences of $m$-harmonic type systems.Calc. Var. Partial Differ. Equ. 4 (1996), 341-367. MR 1393269, 10.1007/BF01190823
Reference: [13] Rivière, T.: Conservation laws for conformally invariant variational problems.Invent. Math. 168 (2007), 1-22. Zbl 1128.58010, MR 2285745, 10.1007/s00222-006-0023-0
Reference: [14] Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of 2-spheres.Ann. Math. (2) 113 (1981), 1-24. MR 0604040
Reference: [15] Strzelecki, P.: On biharmonic maps and their generalizations.Calc. Var. Partial Differ. Equ. 18 (2003), 401-432. Zbl 1106.35021, MR 2020368, 10.1007/s00526-003-0210-4
Reference: [16] Strzelecki, P., Zatorska-Goldstein, A.: A compactness theorem for weak solutions of higher-dimensional $H$-systems.Duke Math. J. 121 (2004), 269-284. Zbl 1054.58008, MR 2034643, 10.1215/S0012-7094-04-12123-2
Reference: [17] Tartar, L.: Imbedding theorems of Sobolev spaces into Lorentz spaces.Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 1 (1998), 479-500. MR 1662313
Reference: [18] Uhlenbeck, K. K.: Connections with $L^p$ bounds on curvature.Commun. Math. Phys. 83 (1982), 31-42. MR 0648356, 10.1007/BF01947069
Reference: [19] Wang, C.: A compactness theorem of $n$-harmonic maps.Ann. Inst. Henri Poincaré, Anal. Non Linéaire 22 (2005), 509-519. Zbl 1229.58017, MR 2145723, 10.1016/j.anihpc.2004.10.007
Reference: [20] Zheng, S.: Weak compactness of biharmonic maps.Electron. J. Differ. Equ. (electronic only) 2012 (2012), Article No. 190, 7 pages. Zbl 1288.31012, MR 3001676


Files Size Format View
CzechMathJ_66-2016-1_14.pdf 288.6Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo