### Regularity of p-harmonic maps from the p-dimensional ball into a sphere.

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We prove that minimizers $u\in {W}^{1,n}$ of the functional ${E}_{}\left(u\right)=1/n{\int}_{|}{\nabla u|}^{n}dx+1/\left({4}^{n}\right){\int}_{(}1-{\left|u\right|}^{2}{)}^{2}dx$, ⊂ ${\mathbb{R}}^{n}$, n ≥ 3, which satisfy the Dirichlet boundary condition ${u}_{}=g$ on for g: → ${S}^{n-1}$ with zero topological degree, converge in ${W}^{1,n}$ and ${C}_{loc}^{\alpha}$ for any α<1 - upon passing to a subsequence ${}_{k}\to 0$ - to some minimizing n-harmonic map. This is a generalization of an earlier result obtained for n=2 by Bethuel, Brezis, and Hélein. An example of nonunique asymptotic behaviour (which cannot occur in two dimensions if deg g = 0) is presented.

We consider a class of fourth order elliptic systems which include the Euler-Lagrange equations of biharmonic mappings in dimension 4 and we prove that a weak limit of weak solutions to such systems is again a weak solution to a limit system.

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