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 nonstandard growth conditions of $p(x)$-type. If $g^{\alpha \beta }(x)$ are in the class $VMO$, we have partial Hölder regularity. Moreover, if $g^{\alpha \beta }$ are Hölder continuous, we can show partial $C^{1,\alpha }$-regularity.