Optimal partial regularity for nonlinear elliptic systems of higher order (Q2780731)

The authors consider the regularity of weak solutions to the following homogeneous nonlinear elliptic systems of higher order in divergence form: NEWLINE\[NEWLINE\sum^N_{i=1} \sum_{|\alpha |=m_i}\int_\Omega A^\alpha_i (\cdot, du, D^{\mathbf m}u)D^\alpha \varphi^idx =0,\;\varphi\in C_0^\infty (\Omega, \mathbb{R}^N),\tag{P}NEWLINE\]NEWLINE where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \({\mathbf m}= (m_1,\dots,m_N)\) with \(m_i\geq 1\), \(D^{\mathbf m}u\) stands for \(\{D^\alpha u_i\}\) with \(i=1,\dots,N\) and \(|\alpha|=m_i\), and \(du\) stands for \(\{D^\alpha u_i\}\) with \(i=1,\dots,N\) and \(|\alpha|\leq m_i-1\). Under suitable assumptions on the coefficients \(A_i^\alpha(x,\xi,\nu)\) (namely \(\nu\)-differentiability with bounded continuous derivative, strong ellipticity condition and Hölder continuity of the function \((x,\xi)\mapsto (1+|\nu |)^{-1}A(x, \xi,\nu)\) with some exponent \(s\in(0,1))\), the authors prove the following result: If \(u\in H^{{\mathbf m},2} (\Omega,\mathbb{R}^N)\) is a weak solution of problem (P) in \(\Omega\), then there exists an open set \(\Omega_0\subset \Omega\) such that \(u\in C^{{\mathbf m},s} (\Omega_0,\mathbb{R}^N)\) and \({\mathcal L}^n (\Omega \backslash \Omega_0)=0\). The paper extends the results of [\textit{M. Giaquinta} and \textit{G. Modica}, J. Reine Angew. Math. 311-312, 145-169 (1979; Zbl 0409.35015)] and [\textit{M. Giaquinta} and \textit{G. Modica}, Manuscr. Math. 28, 109-158 (1979; Zbl 0411.35018)].