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.

翻译：研究奇异力作用下的线性弹性系统。我们在两个框架中证明了解的存在唯一性：加权Sobolev空间（权函数属于Muckenhoupt类$A_2$）和标准Sobolev空间（可积指数小于$d/(d-1)$，其中$d$为空间维数）。我们提出了一种标准有限元格式，并给出了$\mathbf{L}^2$范数下的最优误差估计。通过证明适定性，我们澄清了Banach空间中广义混合问题研究中的若干问题。