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 elasticity equation Pu := div ` C · ∇u) = f , with mixed (displacement and traction) boundary conditions Lk on a curved polyhedral domain Ω ⊂ R 3 in weighted Sobolev spaces Ka (Ω), for which the weight given by the distance to the set of edges. In particular, we show that there is no loss of Ka –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 |a| < η, for some small η > 0 that depends on P and Lk and the domain Ω. Our results extend to other strongly elliptic systems.