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

Let $(g^{αβ}(x))$ 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_Ω(g^{αβ}(x) h_{ij}(u) D_αu^iD_βu^j)^{p(x)/2} dx, \] under the non-standard growth conditions of $p(x)$-type. If $g^{αβ}(x)$ are in the class $VMO$, we have partial Hölder regularity. Moreover, if $g^{αβ}$ are Hölder continuous, we can show partial $C^{1,α}$-regularity.

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