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Mappings minimizing the $$L^ p$$ norm of the gradient. (English) Zbl 0646.49007
Let M be a compact m-dimensional $$C^ 2$$ Riemannian manifold with boundary $$\partial M$$, and let N be a compact connected n-dimensional $$C^ 2$$ Riemannian manifold without boundary. The authors consider maps $$u: M\to N$$ which minimize$$\int_{M}| \nabla u|^ pdM$$ among $$L^{1,p}$$ mappings having a given trace on $$\partial M$$, where $$1<p<2$$. Let Z be the set of points $$a\in M$$ for which the normalized p-energy on the ball $${\mathbb{B}}_ r(a)$$, $r^{p-n}\int_{M\cap {\mathbb{B}}_ r(a)}| \nabla u|^ p dM,$ fails to approach zero; it is shown that Z has Hausdorff dimension at most m-[p]-1 and is finite in case $$m=[p]+1$$. The authors prove that u is locally Hölder continuous on $$M\sim Z$$ and the gradient of u is also locally Hölder continuous on $$M\sim (Z\cup \partial M).$$
The first part of the proof of these results is to establish that, near points $$a\in M\sim (Z\cup \partial M)$$, the normalized p-energy decays like a positive power of r as r approaches zero. This is done by an argument by contradiction based on a sequence $$u_ i$$ of p-minimizers on $${\mathbb{B}}_ 1$$ which do not exhibit energy decay, but which have total p- energies $$\int_{{\mathbb{B}}}| \nabla u|^ pdM$$ approaching zero as $$i\to \infty$$. Hölder continuity follows by Morrey’s lemma. The proof that a Hölder continuous minimizer has a Hölder continuous gradient is obtained by modifying E. DiBenedetto’s proof [Nonlinear Anal., Theory Methods Appl. 7, 927-950 (1983; Zbl 0539.35027)] for a single equation. Regularity near the boundary, in the case of smooth boundary values on smooth domains, is based on the non-existence of nonconstant boundary tangent maps, as in the work of R. Schoen and K. Uhlenbeck [J. Differ. Geom. 17, 307-335, Correction ibid. 19, 329 (1982; Zbl 0521.58021)].
Reviewer: H.Parks

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 35D10 Regularity of generalized solutions of PDE (MSC2000) 49Q99 Manifolds and measure-geometric topics
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