Abstract
Recent results of the author concerning the minimizing properties of the cones\(C_n^\alpha = \left\{ {0 \leqslant x_{n + 1} \leqslant \sqrt {\frac{\alpha }{{n - 1}}[x_1^2 + ... + x_n^2 ]^{\frac{1}{2}} } } \right\}\) will be improved considerably. It is shown that the new results are optimal. Moreover the existence of “singular” minimizers of class C0,1/2 is established in any dimension n≥2.
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Bemelmans, J., Dierkes, U.: On a singular variational integral with linear growth, I: existence and regularity of minimizers. Arch. Rat Mech. Anal.100 (1987), 83–103
Dierkes, U.: Minimal hypercones and C0,1/2-minimizers for a singular variational problem. To appear in Indiana University Math. Journ.
Dierkes, U.: Boundary regularity for solutions of a singular variational problem with linear growth. To appear in Arch. Rat. Mech. Anal.
Simoes, P.A.Q.: A class of minimal cones in ℝn, n≥8 that minimize area. Ph.D. thesis, University of California, Berkeley, CA 1973
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I would like to thank Frank Morgan for directing my attention to the paper of Simoes: “On a class of minimal cones in ℝn”, Bull. A.M.S.80 (1974) 488–489.
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Dierkes, U. A classification of minimal cones in ℝn × ℝ+ and a counterexample to interior regularity of energy minimizing functions. Manuscripta Math 63, 173–192 (1989). https://doi.org/10.1007/BF01168870
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DOI: https://doi.org/10.1007/BF01168870