Boundary value problems in elastostatics with singular data (Q829822)

The present paper deals with well-posedness of three-dimensional BVPs of elasticity, in the isotropic case, under weak regularity assumptions on the boundary data, e.g., Borel measurability of boundary data or the presence of concentrated loads. The paper consists of six sections and references with 21 sources. Section 1 is intended to motivate investigations carried out in the paper, it contains notation and a brief survey of classical results as well. Section 2 provides a detailed exposition of the single-layer and double-layer potentials with densities from \(L^1\). Section 3 is devoted to the Dirichlet-Neumann-Robin problem for bounded or exterior domains of class \(C^k\), \(k\ge 2\). Section 4 deals with mixed problems, and the existence of a unique very weak solution is established. Section 5 is devoted to the Dirichlet-Neumann-Robin problem in bounded or exterior domains of class \(C^{1,\alpha}\). Existence of a unique solution expressed by a simple-layer potential with density from \(L^q\) is proved. For the exterior domain the existence of a unique solution expressed by a simple-layer potential with density from \(W^{-1,q}\) is also proved. Section 6 deals with displacement problem in a bounded domain of class \(C^1\) and in exterior domains either of class \(C^1\) or of class \(C^2\).