Article

 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: http://hdl.handle.net/10338.dmlcz/144880 . 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. 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