Mappings minimizing the \(L^ p\) norm of the gradient.

*(English)*Zbl 0646.49007Let 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)].

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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\textit{R. Hardt} and \textit{F. Lin}, Commun. Pure Appl. Math. 40, No. 5, 555--588 (1987; Zbl 0646.49007)

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