Geodesic
A compactness result for polyharmonic maps in the critical dimension
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 1, pp. 137-150
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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.
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.
DOI : 10.1007/s10587-016-0246-1
Classification : 35J35, 35J48, 58J05
Keywords: polyharmonic map; compactness; Coulomb moving frame; Palais-Smale sequence; removable singularity 
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Zheng, Shenzhou. A compactness result for polyharmonic maps in the critical dimension. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 1, pp. 137-150. doi: 10.1007/s10587-016-0246-1

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