A regularity theory for scalar local minimizers of splitting-type variational integrals

Bildhauer, Michael; Fuchs, Martin; Zhong, Xiao

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 3, pp. 385-404.

Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of $(p,q)$-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.

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