Abstract
We consider local minimizers \(u: {\mathbb{R}}^n \supset \Omega \to {\mathbb{R}}^N\) of variational integrals \(I[u] := \int_{\Omega} F(\nabla u)\,{\rm dx}\) , where F is of anisotropic (p, q)-growth with exponents \(1 < p \leq q < \infty\). If F is in a certain sense decomposable, we show that the dimensionless restriction \(q \leq 2p+2\) together with the local boundedness of u implies local integrability of \(\nabla u\) for all exponents \(t \leq p+2\). More precisely, the initial exponents for the integrability of the partial derivatives can be increased by two, at least locally. If n = 2, then we use these facts to prove \(C^{1,\alpha}\)-regularity of u for any exponents \(2\leq p \leq q\).
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Bildhauer, M., Fuchs, M. Higher integrability of the gradient for vectorial minimizers of decomposable variational integrals. manuscripta math. 123, 269–283 (2007). https://doi.org/10.1007/s00229-007-0096-0
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DOI: https://doi.org/10.1007/s00229-007-0096-0