The Dirichlet energy of mappings with values into the sphere. (English) Zbl 0678.49006

Summary: We discuss the relaxed functional of the Dirichlet energy. We also prove partial regularity of minimizers and concentration of the gradient on singular lines.


49J45 Methods involving semicontinuity and convergence; relaxation
49J99 Existence theories in calculus of variations and optimal control
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[1] BETHUEL F., BREZIS S., CORON J.M.,Relaxed energies for harmonic maps, preprint · Zbl 0793.58011
[2] BETHUEL F., ZHENG X.,Density of smooth functions between two manifolds in Sobolev spaces. J. Funct. Anal.80 (1988), p. 60–75 · Zbl 0657.46027
[3] BREZIS S.,S k -valued maps with singularities. In ”Topics in Calculus of Variations” Ed. M.Giaquinta, Lecture Notes in Math. n.1365, Springer-Verlag 1989
[4] BREZIS S., CORON J.M., LIEB E.H..Harmonic maps with defects. Comm. Math. Phys.107 (1986), p. 649–705 · Zbl 0608.58016
[5] FEDERER H.,Geometric measure theory. Springer-Verlag, New York, 1969 · Zbl 0176.00801
[6] GIAQUINTA M., MODICA G., SOUČEK J.,Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity. Archive for Rat. Mech. Anal.106 (1989) 97–159.Erratum and addendum, to appear in Archive for Rat. Mech. Anal. · Zbl 0677.73014
[7] GIAQUINTA M., MODICA G., SOUČEK J.,Cartesian currents and variational problems for mappings into spheres, to appear in Annali S.N.S. Pisa · Zbl 0713.49014
[8] HARDT R.,Point and line singularities in liquid crystals, preprint · Zbl 0752.49018
[9] SCHOEN R., UHLENBECK K.,A regularity theory for harmonic maps. J.Diff. Geom.17 (1982) 307–335 · Zbl 0521.58021
[10] SIMON L.,Lectures on Geometric Measure Theory. Proc. of the Centre for Math. Analysis vol.3 Australian National University, Canberra. 1983 · Zbl 0546.49019
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