A regularity theory for scalar local minimizers of splitting-type variational integrals
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 3, pp. 385-404.

Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of $\left(p,q\right)$-growth with exponents $p\le q<\infty$ and show for the scalar case that locally bounded local minimizers are of class ${C}^{1,\mu }$. Note that to our knowledge the only ${C}^{1,\mu }$-results without imposing a relation between $p$ and $q$ concern the case of two independent variables as it is outlined in Marcellini’s paper [15], Theorem A, and later on in the work of Fusco and Sbordone [10], Theorem 4.2.

Classification: 49N60
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Bildhauer, Michael; Fuchs, Martin; Zhong, Xiao. A regularity theory for scalar local minimizers of splitting-type variational integrals. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 3, pp. 385-404. http://www.numdam.org/item/ASNSP_2007_5_6_3_385_0/

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