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

TL;DR

This paper establishes partial regularity results for minimizers of a variable exponent p(x)-growth functional, showing conditions under which solutions are partially Hölder or C^{1,α} regular.

Contribution

It proves partial regularity of minimizers for a p(x)-growth functional with non-standard growth conditions, extending regularity theory to variable exponent settings.

Findings

Partial Hölder regularity when coefficients are in VMO class.

Partial C^{1,α} regularity when coefficients are Hölder continuous.

Regularity results depend on the smoothness of the coefficient matrices.

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.