Approximation and Characterization of Elastic Relaxed Dirichlet Problems and Shape Optimization

In this study, we consider a relaxed Dirichlet problem in linear elasticity, which is a generalized Dirichlet problem for homogeneous linear elastic materials involving a potential in the form of a symmetric and positive semi-definite matrix of Borel measures that do not charge polar sets. We present an explicit approximation procedure by means of sequences of classical Dirichlet problems in strongly perturbed domains. Then, we give a characterization of these measures, when the components of the data are nonnegative, in terms of solutions in closed convex sets of particular relaxed Dirichlet problems. Finally, we give some applications to the shape optimization for Dirichlet problems in the linear elasticity framework.