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Partial Regularity of Strong Local Minimizers in the Multi-Dimensional Calculus of Variations

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Abstract.

Let Ω⊂ℝn be a bounded domain and F:𝕄→ℝ a given strongly quasiconvex integrand of class C 2 satisfying the growth condition \({{ |F(\xi)| \le c (1 + |\xi|^p)}}\) for some c>0 and 2≤p<∞. Consider the multiple integral \({{ I[u] = \int_{{\Omega}} \! F(\nabla u) }}\) where uW 1,p(Ω, ℝN). The main result of the paper is the proof that any strong local minimizer of I[·] is of class C 1,α loc for any α(0,1) on an open set of full n-dimensional measure. In the case of weak local minimizers we establish the same result under the extra assumption that the oscillations in the gradient of the minimizer are not too large. Without such an assumption weak local minimizers need not be partially regular as we show by a class of examples. We also briefly discuss the question of existence of strong local minimizers for I[·] and connections of our results to extensions of Weierstrass’ sufficiency theorem to the multi-dimensional setting.

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Authors and Affiliations

  1. School of Mathematical and Computer Sciences Scott Russell Building, Heriot-Watt University, Edinburgh, EH14 4AS, Scotland, U.K

    Jan Kristensen

  2. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, England, U.K

    Ali Taheri

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  1. Jan Kristensen
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  2. Ali Taheri
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Communicated by S. Müller

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Kristensen, J., Taheri, A. Partial Regularity of Strong Local Minimizers in the Multi-Dimensional Calculus of Variations. Arch. Rational Mech. Anal. 170, 63–89 (2003). https://doi.org/10.1007/s00205-003-0275-4

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