  For higher order integral functionals with $p(x)$ growth with respect to the
highest order derivative $D^m u$, we prove that $D^m u$ is H\"older 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 H\"older
continuous.
