Partial regularity of \(p(x)\)-harmonic maps (Q2838126)

A natural energy with non-standard growth condition for mappings \(u:{\mathbb R}^m\supset\Omega\to{\mathbb R}^n\) is the \(p(x)\)-energy \(\int_\Omega|Du(x)|^{p(x)}\,dx\). In the current paper, the authors consider the Riemannian version of it, NEWLINE\[NEWLINEF(u):=\int_\Omega(g^{\alpha\beta}(x) h_{ij}(x)D_\alpha u^iD_\beta u^j)^{p(x)/2}\,dx,NEWLINE\]NEWLINE where \(g^{\alpha\beta}(x)\) and \(h_{ij}(u)\) are uniformly elliptic matrix functions. While \(h_{ij}(u)\) and \(p(x):\Omega\to[2,\infty)\) are assumed to be sufficiently smooth, the assumptions on \(g^{\alpha\beta}(x)\) play a key role here.NEWLINENEWLINEAssuming that \(g^{\alpha\beta}(x)\) is bounded and in VMO, the authors prove that every local minimzer \(u\) of \(F\) is Hölder continuous on some open set \(\Omega_0\subseteq\Omega\) of vanishing \((m-\gamma)\)-dimensional Hausdorff measure, where \(\gamma\) is the infimum of the function \(p(x)\). Moreover, if \(g^{\alpha\beta}(x)\) is Hölder continuous, then also the \(D_\alpha u\) are Hölder continuous on \(\Omega_0\).