Well-posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks

We prove a regularity result for the anisotropic linear elasticity equation Pu:= div(C. ∇u) = f, with mixed (displacement and traction) boundary conditions on a curved polyhedral domain Ω ⊂ ℝ³ in weighted Sobolev spaces K^m+1_a=1 (Ω), for which the weight is given by the distance to the set of edges. In particular, we show that there is no loss of K^m_a-regularity. Our curved polyhedral domains are allowed to have cracks. We establish a well-posedness result when there are no neighboring traction boundary conditions and {combining long vertical line overlay}a{combining long vertical line overlay} < η, for some small η > 0 that depends on P, on the boundary conditions, and on the domain Ω. Our results extend to other strongly elliptic systems and higher dimensions.