Abstract
For higher order functionals \(\int_\Omega f(x, \delta u(x), {D^m}u(x))\,dx\) with p(x)-growth with respect to the variable containing D m u, we prove that D m u is Hölder continuous on an open subset \(\Omega_0 \subset \Omega\) of full Lebesgue-measure, provided that the exponent function \(p : \Omega \to (1, \infty)\) itself is Hölder continuous.
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Habermann, J. Partial regularity for minima of higher order functionals with p(x)-growth. manuscripta math. 126, 1–40 (2008). https://doi.org/10.1007/s00229-007-0147-6
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DOI: https://doi.org/10.1007/s00229-007-0147-6