A Density Result in Two-Dimensional Linearized Elasticity, and Applications

We show that in a two-dimensional bounded open set whose complement has a finite number of connected components, the vector fields uH ¹(Ωℝ²) are dense in the space of fields whose symmetrized gradient e(u) is in L ²(Ωℝ⁴). This allows us to show the continuity of some linearized elasticity problems with respect to variations of the set, with applications to shape optimization or the study of crack evolution.