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We investigate lower hybrid wave trapping in cylindrically symmetric density depletions in the electrostatic approximation. Our investigation is inspired by previous observations of such trapping by spacecraft in the auroral region at altitudes up to about 2000km, and the recent discovery of this phenomenon at altitudes above 20000km in the inner magnetosphere. No particular shape is assumed for the density depletion, which need not be strictly zero outside some value of the radial coordinate <i>r</i>. Important previously known properties concerning parabolic density depletions extending to finite <i>r</i> are shown to hold also for arbitrary shapes and infinite extent: for a given parallel wave number <i>k<sub>z</sub></i>, modes below the ambient lower hybrid frequency <i>f</i><sub>LH</sub> are trapped in the density depletion (in the sense that they are evanescent outside the cavity), have a discrete spectrum and rotate in a left-handed sense, while there is a continuous spectrum of freely propagating right-handed rotating modes above <i>f</i><sub>LH</sub>. New results are such that even though the density depletion may go to zero slowly with increasing <i>r</i>, and thus be essentially infinite in extent, there is a maximum distance within which a trapped mode with given <i>k<sub>z</sub></i> and azimuthal mode number <i>m</i> may propagate. Furthermore, we find that for any monotonic density cavity and given <i>k<sub>z</sub></i>, there is a local relation between plasma density gradient and the lowest possible frequency that can be trapped. We combine our theoretical results with spacecraft observations to find an upper bound on <i>k<sub>z</sub></i>. Our examples indicate that the length of the cavities is larger than the width by a factor of at least 100.