  Interactions between rigid inclusions in continuous three-dimensional
linearly elastic solids and low-Reynolds-number viscous fluids have largely
been quantified during the past. Prime example systems are given by
functionalized elastic composite materials or fluid colloidal suspensions.
Here, we address the significantly less frequently studied situation of rigid
inclusions in two-dimensional elastic or low-Reynolds-number fluid films. We
concentrate on the situation in which disk-like inclusions remain well
separated from each other and do not get into contact. Specifically, we
demonstrate and explain that the logarithmic divergence of the associated
Green's function is removed in the absence of net external forces on the
inclusions, in line with physical intuition. For instance, this situation
applies when only pairwise mutual interactions between the inclusions prevail.
Our results will support, for example, investigations on membranes
functionalized by appropriate inclusions, both of technical or biological
origin, or the dynamics of active microswimmers in appropriately prepared thin
films.
