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Universal singular sets for one-dimensional variational problems

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Summary

A study is made of the regularity properties of minimizers u of the integralI(u)=∫ b a f(x, u, u′) dx subject to the boundary conditionsu(a)=α, u(b)=β as the interval (a, b) and boundary valuesα,β are varied. Under natural hypotheses onf it is shown that the set of points in the (x, u)-plane at which a minimizer u can have infinite derivative for some interval and boundary values is small in the sense of category.

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Authors and Affiliations

  1. Department of Mathematics, Heriot-Watt University, EH14 4AS, Edinburgh, UK

    J. M. Ball

  2. Schmidt Institute of Earth Physics, Academy of Sciences of Russia, Moscow, Russia

    N. S. Nadirashvili

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  1. J. M. Ball
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  2. N. S. Nadirashvili
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Ball, J.M., Nadirashvili, N.S. Universal singular sets for one-dimensional variational problems. Calc. Var 1, 429–438 (1993). https://doi.org/10.1007/BF01206961

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  • DOI: https://doi.org/10.1007/BF01206961

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