Spectroscopic observations of some metal-rich white dwarfs (WDs), believed to be polluted by planetary material, reveal the presence of compact gaseous metallic disks orbiting them. The observed variability of asymmetric, double-peaked emission line profiles in about half of such systems could be interpreted as the signature of precession of an eccentric gaseous debris disk. The variability timescales --- from decades down to $1.4$ yr (recently inferred for the debris disk around HE 1349--2305) --- are in rough agreement with the rate of general relativistic (GR) precession in the test particle limit. However, it has not been demonstrated that this mechanism can drive such a fast, coherent precession of a radially extended (out to $1 R_odot$) gaseous disk mediated by internal stresses (pressure). Here we use the linear theory of eccentricity evolution in hydrodynamic disks to determine several key properties of eccentric modes in gaseous debris disks around WDs. We find a critical dependence of both the precession period and radial eccentricity distribution of the modes on the inner disk radius, $r_mathrm{in}$. For small inner radii, $r_mathrm{in} lesssim (0.2 - 0.4) R_odot$, the modes are GR-driven, with periods of $approx 1 - 10$ yr. For $r_mathrm{in} gtrsim (0.2 - 0.4) R_odot$, the modes are pressure-dominated, with periods of $approx 3 - 20$ yr. Correspondence between the variability periods and inferred inner radii of the observed disks is in general agreement with this trend. In particular, the short period of HE 1349--2305 is consistent with its small $r_mathrm{in}$. Circum-WD debris disks may thus serve as natural laboratories for studying the evolution of eccentric gaseous disks.
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