Regularity and relaxed problems of minimizing biharmonic maps into spheres

Hong, Min-Chun; Wang, Changyou
August 2005
Calculus of Variations & Partial Differential Equations;Aug2005, Vol. 23 Issue 4, p425
Academic Journal
For $n\ge 5$ and $k\ge 4$, we show that any minimizing biharmonic map from $\Omega\subset \Bbb R^n$ 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 $\Bbb H_{\lambda}$ of the Hessian energy for maps in $W^{2,2}(\Omega ; S^4)$ so that each minimizer $u_{\lambda}$ of $\Bbb H_{\lambda}$ is also a biharmonic map. We also establish the existence and partial regularity of a minimizer of $\Bbb H_{\lambda}$ for $\lambda \in [0,1)$.


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