Geodesic
On the regularity of local minimizers of decomposable variational integrals on domains in $\Bbb R^2$
Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 2, pp. 321-341.

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We consider local minimizers $u : \Bbb R^2\supset \Omega \to \Bbb R^N$ of variational integrals like $\int_\Omega [(1+|\partial_1 u|^{2})^{p/2}+(1+|\partial_2 u|^{2})^{q/2}]\,dx$ or its degenerate variant $\int_\Omega [|\partial_1 u|^p+|\partial_2 u|^q]\,dx$ with exponents $2\leq p q \infty $ which do not fall completely in the category studied in Bildhauer M., Fuchs M., Calc. Var. {\bf 16} (2003), 177--186. We prove interior $C^{1,\alpha}$- respectively $C^{1}$-regularity of $u$ under the condition that $q 2p$. For decomposable variational integrals of arbitrary order a similar result is established by the way extending the work Bildhauer M., Fuchs M., Ann. Acad. Sci. Fenn. Math. {\bf 31} (2006), 349--362.
Classification : 35J35, 35J50, 49N60
Keywords: non-standard growth; vector case; local minimizers; interior regularity; problems of higher order 
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Bildhauer, M.; Fuchs, M. On the regularity of local minimizers of decomposable variational integrals on domains in $\Bbb R^2$. Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 2, pp. 321-341. http://geodesic.mathdoc.fr/item/CMUC_2007__48_2_a13/