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Everywhere Hölder continuity of vectorial local minimizers of special classes of integral functionals with rank one integrands

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Abstract

In this paper we study the everywhere Hölder continuity of the minima of the following class of vectorial integral functionals

$$\begin{aligned} \int \limits _{\Omega }\sum \limits _{\alpha =1}^{n}f_{\alpha }\left( x,u^{\alpha },\nabla u^{\alpha }\right) +G\left( x,u,\nabla u\right) \,dx \end{aligned}$$

The energy densities satisfy suitable structure assumptions and may have neither radial nor quasi-diagonal structure. The regularity of minimizers is obtained by proving that each component stays in a suitable De Giorgi class and, from this, we conclude about the Hölder continuity.

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Acknowledgements

I would like to thank all my family and friends for the support given to me over the years: Elisa Cirri, Caterina Granucci, Delia Granucci, Irene Granucci, Laura and Fiorenza Granucci, Massimo Masi and Monia Randolfi. Also I would like to thank my professors Luigi Barletti, Giorgio Busoni, Elvira Mascolo, Giorgio Talenti and Vincenzo Vepri.

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    Tiziano Granucci

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Communicated by Ansgar Jüngel.

Dedicated to the memory of Fiorella Pettini.

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Granucci, T. Everywhere Hölder continuity of vectorial local minimizers of special classes of integral functionals with rank one integrands. Monatsh Math 200, 271–300 (2023). https://doi.org/10.1007/s00605-022-01763-5

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