A compactness result for polyharmonic maps in the critical dimension

Zheng, Shenzhou

Czechoslovak Mathematical Journal, Tome 66 (2016) no. 1, pp. 137-150.

Résumé

For $n=2m\ge 4$, let $\mathrm{\Omega }\in {\mathbb{R}}^n$ be a bounded smooth domain and $\mathcal{N}\subset {\mathbb{R}}^L$ a compact smooth Riemannian manifold without boundary. Suppose that $\{u_k\}\in W^{m,2}(\mathrm{\Omega },\mathcal{N})$ 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 ${\mathrm{\Phi }}_k\to 0$ in $(W^{m,2}(\mathrm{\Omega },\mathcal{N}){)}^{\ast }$ and $u_k\rightharpoonup u$ weakly in $W^{m,2}(\mathrm{\Omega },\mathcal{N})$. 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.

Détail

DOI : 10.1007/s10587-016-0246-1

Classification : 35J35, 35J48, 58J05
Mots-clés : polyharmonic map; compactness; Coulomb moving frame; Palais-Smale sequence; removable singularity