Abstract
We consider rotationally symmetric 1-harmonic maps from D 2 to S 2 subject to Dirichlet boundary conditions. We prove that the corresponding energy—a degenerate non-convex functional with linear growth—admits a unique minimizer, and that the minimizer is smooth in the bulk and continuously differentiable up to the boundary. We also show that, in contrast with 2-harmonic maps, a range of boundary data exists such that the energy admits more than one smooth critical point: more precisely, we prove that the corresponding Euler–Lagrange equation admits a unique (up to scaling and symmetries) global solution, which turns out to be oscillating, and we characterize the minimizer and the smooth critical points of the energy as the monotone, respectively non-monotone, branches of such solution.
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R. Dal Passo passed away on 8th August 2007. Endowed with great strength, creativity and humanity, Roberta has been an outstanding mathematician, an extraordinary teacher and a wonderful friend. Farewell, Roberta.
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Dal Passo, R., Giacomelli, L. & Moll, S. Rotationally symmetric 1-harmonic maps from D 2 to S 2 . Calc. Var. 32, 533–554 (2008). https://doi.org/10.1007/s00526-007-0153-2
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DOI: https://doi.org/10.1007/s00526-007-0153-2