Weighted composition operators as Daugavet centers

We investigate the norm identity $\mathrm{\parallel }u{C}_{\varphi }+T\mathrm{\parallel }=\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}+\mathrm{\parallel }T\mathrm{\parallel }$ for classes of operators on $C(S)$, where $S$ is a compact Hausdorff space without isolated point, and characterize those weighted composition operators which satisfy this equation for every weakly compact operator $T:C(S)\to C(S)$. We also give a characterization of such weighted composition operator acting on the disk algebra $A(D)\mathrm{.}$