It is well documented that subjecting perpendicular magnetic films which exhibit the interfacial Dzyaloshinskii-Moriya interaction (DMI) to an in-plane magnetic field results in a domain wall (DW) energy, $sigma$, that is highly anisotropic with respect to the orientation of the DW in the film plane, $Theta$. We demonstrate that this anisotropy has a profound impact on the elastic response of the DW as characterized by the surface stiffness, $tilde{sigma}(Theta) = sigma(Theta) + sigma(Theta)$, and evaluate its dependence on the length scale of deformation. The influence of the stiffness on DW mobility in the creep regime is assessed, with analytic and numerical calculations showing trends in $tilde{sigma}$ that better represent experimental measurements of domain wall velocity in magnetic thin films compared to $sigma$ alone. Our treatment provides experimental support for theoretical models of the mobility of anisotropic elastic manifolds and makes progress toward a more complete understanding of magnetic domain wall creep.
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