The linear elasticity system under singular forces

Alejandro Allendes, Gilberto Campaña, Enrique Otárola, Abner J. Salgado

Published 2023-10-16Version 1

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 $d/(d-1)$; $d$ 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.