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Interior gradient bounds for local minimizers of variational integrals under nonstandard growth conditions

Inspired by the work of Marcellini and Papi, we consider local minima u: ℝn ⊃ Ω → ℝM of variational integrals \( \int\limits_\Omega {h\left( {\left| {\nabla u} \right|} \right)} \) dx and prove interior gradient bounds under rather general assumptions on h provided that u is locally bounded. Our requirements on the density h do not involve the dimension n. Bibliography: 18 titles.

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Correspondence to D. Apushkinskaya.

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Translated from Problems in Mathematical Analysis 43, November 2009, pp. 35–50.

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Apushkinskaya, D., Bildhauer, M. & Fuchs, M. Interior gradient bounds for local minimizers of variational integrals under nonstandard growth conditions. J Math Sci 164, 345–363 (2010). https://doi.org/10.1007/s10958-009-9751-1

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Keywords

  • Strict Convexity
  • Young Inequality
  • Nonlinear Elliptic System
  • Local Boundedness
  • Nonstandard Growth