The linear elasticity system under singular forces

We study the linear elasticity system subject to singular forces. We show existence and uniqueness of solutions in two frameworks: weighted Sobolev spaces, where the weight belongs to the Muckenhoupt class $A_2$; and standard Sobolev spaces where the integrability index is less than $d\mathrm{/}(d-1)$; $d$ is the spatial dimension. We propose a standard finite element scheme and provide optimal error estimates in the ${\mathbf{L}}^2$–norm. By proving well posedness, we clarify some issues concerning the study of generalized mixed problems in Banach spaces.