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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

49J45 Methods involving semicontinuity and convergence; relaxation
35D10 Regularity of generalized solutions of PDE (MSC2000)
49Q99 Manifolds and measure-geometric topics
Full Text: DOI
[1] Q-valued functions minimizing Dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension two, Princeton University Notes, 1983. · Zbl 0557.49021
[2] DiBenedetto, Nonlinear Analysis 7 pp 827– (1983)
[3] Federer, Ann. Math. 72 pp 480– (1960)
[4] and , Partial regularity for minimizers of certain functionals having non quadratic growth, preprint, CMA, Canberra, 1985.
[5] and , Remarks on the regularity of the minimizers of certain degenarate functionals, Preprint, Universita di Firenze.
[6] Homotopy Theory, Academic Press, New York, London, 1959.
[7] Hardt, Comm. Math. Physics 105 pp 547– (1986)
[8] , and , Stable defects of minimizers of constrained variational principles, I.M.A., Preprint 325. · Zbl 0657.49018
[9] Hardt, Manuscripta Math 56 pp 1– (1986)
[10] , and , Function Spaces, Noordhoff, Leyden, 1977.
[11] Partial Holder continuity of energy minimizing p-harmonic maps between Riemannian manifolds, preprint, CMA, Canberra, 1986.
[12] C1, {\(\epsilon\)}-Regularity for energy minimizing Hölder continuous p-harmonic maps between Riemannian manifolds, preprint, CMA, Canberra, 1986.
[13] Multiple Integrals in the Calculus of Variations, Springer-Verlag, Heidelberg and New York, 1966. · Zbl 0142.38701
[14] Schoen, J. Diff. Geom. 18 pp 307– (1982)
[15] Schoen, J. Diff. Geom. 18 pp 253– (1983)
[16] Tolksdorff, Annali di Math. Pura et Applicata 134 pp 241– (1983)
[17] Tolksdorff, J. Diff. Equations
[18] Uhlenbeck, Acta. Math. 138 pp 219– (1977)
[19] White, J. Diff. Geometry 23 pp 127– (1986)
[20] White, Bull, Amer. Math. Soc. 13 pp 166– (1985)
[21] Homotopy classes in Sobolev spaces and the existence of energy minimizing maps, Preprint.
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