Abstract
Let and be smooth Riemannian manifolds, of the dimension n≥2 with nonempty boundary, and compact without boundary. We consider stationary harmonic maps u ∈ H 1(, ) with a free boundary condition of the type u(∂) ⊂ Γ, given a submanifold Γ⊂. We prove partial boundary regularity, namely (sing(u))=0, a result that was until now only known in the interior of the domain (see [B]). The key of the proof is a new lemma that allows an extension of u by a reflection construction. Once the partial regularity theorem is known, it is possible to reduce the dimension of the singular set further under additional assumptions on the target manifold and the submanifold Γ.
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References
Bethuel, F.: On the singular set of stationary harmonic maps. Manuscripta Math. 78, 417–443 (1993)
Coifman, R., Lions, P.-L., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. IX, 72,(3), 247–286 (1993)
Duzaar, F., Steffen, K.: A partial regularity theorem for harmonic maps at a free boundary. Asymptotic Analysis 2, 299–343 (1989)
Federer, H., Ziemer, W.: The Lebesgue set of a function whose distribution derivatives are p-th power summable. Indiana Univ. Math. J. 22, 139–158 (1972)
Gulliver, R., Jost, J.: Harmonic maps which solve a free boundary problem. J. reine angew. Math. 381, 61–89 (1987)
Hélein, F.: Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne. C.R. Acad. Sci. Paris 312, 591–596 (1991)
Hardt, R., Lin, F.H.: Partially constrained boundary conditions with energy minimizing mappings. Comm. Pure Appl. Math. 42, 309–334 (1989)
Iwaniec, T., Martin, G.: Quasiregular mappings in even dimensions. Acta Math. 170(1), 29–81 (1993)
Lin, F.H.: Gradient estimates and blow-up analysis for stationary harmonic maps. Annals of Math. 149, 785–829 (1999)
Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces - Fractals and rectifiability. Cambridge Studies in advanced mathematics 44, Cambridge University Press, 1995
Moser, R.: Regularity for the approximated harmonic map equation and application to the heat flow for harmonic maps, Math. Zeit. 243, No. 2, 263–289 (2003)
Preiss, D.: Geometry of measures in ℝn: Distribution, rectifiability, and densities. Annals of Math. 125, 537–643 (1987)
Rivière, T.: Everywhere discontinuous harmonic maps into spheres. Acta Math. 175, 197–226 (1995)
Schoen, R.: Analytic aspects of the harmonic map problem. In: Chern, S.S. (ed.), Seminar on Nonlinear Partial Differential Equations, Springer-Verlag, 321–358, 1984
Simon, L.: Lectures on Geometric Measure Theory. Proc. of Centre for Math. Anal. 3, Australian National Univ. 1983
Simon, L.: Theorems on regularity and singularity of harmonic maps. ETH Lecture notes, Birkhäuser, Zürich, 1996
Scheven, C.: Zur Regularitätstheorie stationärer harmonischer Abbildungen mit freier Randbedingung, Heinrich-Heine-Universität Düsseldorf, Mathematisch-Naturwissenschaftliche Fakultät, 2004 (download under http://diss.ub.uni-duesseldorf.de/ebib/diss/show?dissid=886).
Scheven, C.: Variationally harmonic maps with general boundary conditions: Boundary regularity. Calc. Var. 25, 409–429 (2006)
Schoen, R., Uhlenbeck, K.: A regularity theory for harmonic maps. J. Differential Geom. 17, 307–335 (1982) Differential Equations, Springer-Verlag, 321–358, 1984
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Scheven, C. Partial regularity for stationary harmonic maps at a free boundary. Math. Z. 253, 135–157 (2006). https://doi.org/10.1007/s00209-005-0891-9
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DOI: https://doi.org/10.1007/s00209-005-0891-9