Partial regularity of $p(x)$-harmonic maps

Abstract

Let $(g^{\alpha\beta}(x))$ and $(h_{ij}(u))$ be uniformly elliptic symmetric matrices, and assume that $h_{ij}(u)$ and $p(x) \, (\, \geq 2)$ are sufficiently smooth. We prove partial regularity of minimizers for the functional [ {\mathcal F}(u) = \int_\Omega (g^{\alpha \beta}(x) h_{ij}(u) D_\alpha u^iD_\beta u^j)^{p(x)/2} dx, \] under the non-standard growth conditions of $p(x)$-type. If $g^{\alpha\beta}(x)$ are in the class $VMO$, we have partial H\"older regularity. Moreover, if $g^{\alpha\beta}$ are H\"older continuous, we can show partial $C^{1,\alpha}$-regularity.