A compactness result for polyharmonic maps in the critical dimension

Zheng, Shenzhou. "A compactness result for polyharmonic maps in the critical dimension." Czechoslovak Mathematical Journal 66.1 (2016): 137-150. <http://eudml.org/doc/276757>.

@article{Zheng2016,
abstract = {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 $\lbrace u_k\rbrace \in W^\{m,2\}(\Omega ,\mathcal \{N\})$ is a sequence of weak solutions in the critical dimension to the perturbed $m$-polyharmonic maps \[ \frac\{\rm d\}\{\{\rm d\} t\}\Big |\_\{t=0\}E\_m(\Pi (u+t\xi ))=0 \] 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.},
author = {Zheng, Shenzhou},
journal = {Czechoslovak Mathematical Journal},
keywords = {polyharmonic map; compactness; Coulomb moving frame; Palais-Smale sequence; removable singularity},
language = {eng},
number = {1},
pages = {137-150},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A compactness result for polyharmonic maps in the critical dimension},
url = {http://eudml.org/doc/276757},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Zheng, Shenzhou
TI - A compactness result for polyharmonic maps in the critical dimension
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 1
SP - 137
EP - 150
AB - 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 $\lbrace u_k\rbrace \in W^{m,2}(\Omega ,\mathcal {N})$ is a sequence of weak solutions in the critical dimension to the perturbed $m$-polyharmonic maps \[ \frac{\rm d}{{\rm d} t}\Big |_{t=0}E_m(\Pi (u+t\xi ))=0 \] 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.
LA - eng
KW - polyharmonic map; compactness; Coulomb moving frame; Palais-Smale sequence; removable singularity
UR - http://eudml.org/doc/276757
ER -