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A regularity theory of biharmonic maps. (English) Zbl 0953.58013

The authors consider mappings \(u:\Omega\to S^k=\{z\in \mathbb{R}^{k +1}: |z|=1\}\) from an \(n\)-dimensional domain \(\Omega\) into the unit \(k\)-sphere which are of Sobolev class \(W^2_2\), and in addition critical points of the energy \(\int_\Omega\sum^{k+1}_{\alpha=1} (\Delta_gu^\alpha)^2dV_g\) where \(\Delta_g\) and \(dV_g\) denote the Laplacian and the volume element w.r.t. the given metric \(g\) on \(\Omega\).
The following regularity results are established for this class of biharmonic mapping: any biharmonic map in \(W^2_2\) defined on a four-disk is Hölder continuous. If in addition a biharmonic map defined on a \(n\)-dimensional domain \((n\geq 5)\) is also stationary, then the \((n-4)\)-dimensional Hausdorff measure of the singular set is zero. Moreover, any continuous biharmonic map is smooth.

MSC:

58E20 Harmonic maps, etc.
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
58E35 Variational inequalities (global problems) in infinite-dimensional spaces
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