Regularity and relaxed problems of minimizing biharmonic maps into spheres

Details Regularity and relaxed problems of minimizing biharmonic maps into spheres Regularity and relaxed problems of minimizing biharmonic maps into spheres

2004-05-04

arxiv.org/abs/math/0405059v1

Mathematics

Analysis of PDEs

28 pages

Scientific paper

For $n\ge 5$ and $k\ge 4$, we show that any minimizing biharmonic map from $\Omega\subset R^n$ to $S^k$ is smooth off a closed set whose Hausdorff dimension is at most $n-5$. When $n=5$ and $k=4$, for a parameter $\lambda\in [0,1]$ we introduce a $\lambda$-relaxed energy $\H_\lambda$ for the Hessian energy for maps in $W^{2,2}(\Omega,S^4)$ so that each minimizer $u_\lambda$ of $\H_\lambda$ is also a biharmonic map. We also estabilish the existence and partial regularity of a minimizer of $\H_\lambda$ for $\lambda\in [0,1)$.

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