The linear elasticity system under singular forces

Abstract

We study the linear elasticity system under singular forces. We show the existence and uniqueness of solutions in two frameworks: weighted Sobolev spaces $H^1(\varpi ,\Omega )$, where the weight belongs to the Muckenhoupt class $A_2$, and standard Sobolev spaces $W^{1,p}\left(\Omega \right)$, where the integrability index $p$ is less than $d/(d-1)$. We also propose a standard finite element scheme and provide optimal error estimates in the $L^2$–norm.