Partial regularity result for non-autonomous elliptic systems with general growth

<p style='text-indent:20px;'>In this paper we prove a partial Hölder regularity result for weak solutions <inline-formula><tex-math id="M1">\begin{document}$ u:\Omega\to \mathbb{R}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ N\geq 2 $\end{document}</tex-math></inline-formula>, to non-autonomous elliptic systems with general growth of the type:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} -{\rm{div}} a(x, u, Du) = b(x, u, Du) \quad \;{\rm{ in }}\; \Omega. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>The crucial point is that the operator <inline-formula><tex-math id="M3">\begin{document}$ a $\end{document}</tex-math></inline-formula> satisfies very weak regularity properties and a general growth, while the inhomogeneity <inline-formula><tex-math id="M4">\begin{document}$ b $\end{document}</tex-math></inline-formula> has a controllable growth.</p>

A C1,α$C^{1,\alpha}$ partial regularity result for integral functionals with p⁢(x)$p(x)$-growth condition

We establish${C^{1,\alpha}}$partial regularity for the local minimizers of integral functionals of the type$\mathcal{F}(u;\Omega):=\int_{\Omega}(1+|Du|^{2})^{\frac{p(x)}{2}}\,dx,$where the gradient of the exponent function${p(\,\cdot\,)\geq 2}$belongs to a suitable Orlicz–Zygmund class.