Iteration method for the Dirichlet problem of the elasticity theory
David Natroshvili · Georgian Mathematical Journal · 2025
Abstract We construct a convergent recurrence scheme for a solution of the three-dimensional Dirichlet boundary value problem of the elasticity theory. By the potential method, the Dirichlet problem is reduced to the uniquely solvable Fredholm integral equation of the first kind with a weakly singular boundary integral operator generated by the single layer potential. First, we construct a sequence of successive approximations which converges to the solution of the boundary integral equation in appropriate Bessel-potential spaces of functions defined on the boundary. Afterwards, using these approximations as densities of the single layer potential, we formulate another iteration which converges to the solution of the Dirichlet boundary value problem in the appropriate Sobolev–Slobodetskii spaces of functions defined in the region occupied by the elastic body.