A property gradient coating to be applied by additive manufacture to an inner wall of a casing mounted on the periphery of moving blades of a turbomachine rotor, the coating including in superimposed layers from an outer surface of the coating to this inner wall of the casing, on the one hand, a first layer consisting of a three-dimensional scaffolding of filaments of an abradable material forming an ordered network of channels or microchannels whose pore sizes are between 50 and 250 microns and whose porosity is greater than 85%, and, on the other hand, a second layer having a function of dissipating energy from acoustic waves striking the outer surface of the coating and consisting of a three-dimensional scaffolding of filaments of a first thermosetting material forming an ordered network of channels or microchannels whose pore sizes are between 50 and 400 microns and whose porosity is greater than 60%.