We present scattered light images of the TW Hya disk performed with SPHERE in PDI mode at 0.63, 0.79, 1.24 and 1.62 micron. We also present H2/H3-band ADI observations. Three distinct radial depressions in the polarized intensity distribution are seen, around 85, 21, and 6~au. The overall intensity distribution has a high degree of azimuthal symmetry; the disk is somewhat brighter than average towards the South and darker towards the North-West. The ADI observations yielded no signifiant detection of point sources in the disk. Our observations have a linear spatial resolution of 1 to 2au, similar to that of recent ALMA dust continuum observations. The sub-micron sized dust grains that dominate the light scattering in the disk surface are strongly coupled to the gas. We created a radiative transfer disk model with self-consistent temperature and vertical structure iteration and including grain size-dependent dust settling. This method may provide independent constraints on the gas distribution at higher spatial resolution than is feasible with ALMA gas line observations. We find that the gas surface density in the gaps is reduced by 50% to 80% relative to an unperturbed model. Should embedded planets be responsible for carving the gaps then their masses are at most a few 10 Mearth. The observed gaps are wider, with shallower flanks, than expected for planet-disk interaction with such low-mass planets. If forming planetary bodies have undergone collapse and are in the detachted phase then they may be directly observable with future facilities such as METIS at the E-ELT.
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