A \(C^{1,\alpha}\) partial regularity result for integral functionals with \(p(x)\)-growth condition (Q323098)

The author proves a partial regularity result for the local minimizers of integral functionals of the type NEWLINE\[NEWLINE F(u,K)= \int_{\Omega} \left(1 + |Du| 2 ) \right)^{p(x)}\,dx NEWLINE\]NEWLINE where the gradient of the exponent function \(p(x)\geq 2\) is a continuous weakly differentiable function belonging to a suitable Orlicz-Zygmund class i.e. \( |Dp(x)| \in L ^{n} \log^{2n-1} L_{\mathrm{loc}} (\Omega)\).NEWLINENEWLINEUnder these assumptions, if \(u\in W_{\mathrm{loc}} ^{1,1}(\Omega,\mathbb{R}^N )\) is a local minimizer, there exists an open subset \(\Omega_0\) of \(\Omega\) such that \( | \Omega -\Omega_0| =0\) and \( u \in C^{ 1,\alpha }_{\mathrm{loc}}( \Omega -\Omega_0)\) for all \(\alpha<1\).NEWLINENEWLINEAn comparison method and a blow-up argument are the key tools to establish a decay estimate for the excess function of the minimizers. Moreover, by using an higher differentiability result proven in by the author and \textit{A. Passarelli di Napoli} [Math. Z. 280, No. 3--4, 873--892 (2015; Zbl 1320.49025)], a Caccioppoli-type inequality is established.