The study of the properties of disks around young brown dwarfs can provide important clues on the formation of these very low-mass objects and on the possibility of forming planetary systems around them. The presence of warm dusty disks around brown dwarfs is well known, based on near- and mid-infrared studies. High angular resolution observations of the cold outer disk are limited; we used ALMA to attempt a first survey of young brown dwarfs in the $rho$-Oph star-forming region. All 17 young brown dwarfs in our sample were observed at 890 $mu $m in the continuum at $sim0.!^{primeprime}5$ angular resolution. The sensitivity of our observations was chosen to detect $sim0.5$ M$_oplus$ of dust. We detect continuum emission in 11 disks ($sim65$% of the total), and the estimated mass of dust in the detected disks ranges from $sim0.5$ to $sim6$ M$_oplus$. These disk masses imply that planet formation around brown dwarfs may be relatively rare and that the supra-Jupiter mass companions found around some brown dwarfs are probably the result of a binary system formation. We find evidence that the two brightest disks in $rho$-Oph have sharp outer edges at R<~25 AU, in contrast to disks around Taurus brown dwarfs. This difference may suggest that the different environment in $rho$-Oph may lead to significant differences in disk properties. A comparison of the M$_{disk}$/M$_ast$ ratio for brown dwarf and solar-mass systems also shows a possible deficit of mass in brown dwarfs, which could support the evidence for dynamical truncation of disks in the substellar regime. These findings are still tentative and need to be put on firmer grounds by studying the gaseous disks around brown dwarfs and by performing a more systematic and unbiased survey of the disk population around the more massive stars.
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