  We present a detailed analysis of the vibrational spectrum and heat capacity
of suspended mesoscopic dielectric plates, for various thickness-to-side ratios
at sub-Kelvin temperatures. The vibrational modes of the suspended cavity are
accurately obtained from the three-dimensional (3D) elastic equations in the
small strain limit and their frequencies assigned to the cavity phonon modes.
The calculations demonstrate that the heat capacity of realistic quasi-2D
phonon cavities approach the linear dependence on T at sub-Kelvin temperatures.
The behavior is more pronounced for the thinnest cavities, but takes place also
for moderately thick structures, with thickness-to-side ratios $\gamma$=0.1 to
0.2. It is also demonstrated that the heat capacity of the suspended phonon
cavities is invariant under the product of the temperature (T) with a
characteristic lateral dimension (L) of the sample. The present results
establish a lower bound for the heat capacity of suspended mesoscopic
structures and indicate the emergence of the quantum mechanical regime in the
dynamics of bounded phonon cavities.
