Dirichlet integrals and Gaffney-Friedrichs inequalities in convex domains

We study geometrical conditions guaranteeing the validity of the classical Gaffney-Friedrichs estimate (1) parallel tou parallel to (1,2)(H)((Omega)) less than or equal to C(parallel to du parallel to (2)(L)((Omega))+parallel to deltau parallel to (2)(L)((Omega))+parallel tou parallel to (2)(L)((Omega))) granted that the differential form u has a vanishing tangential or normal component on partial derivative Omega. Our main result is that (1) holds provided Omega satisfies a suitable convexity assumption. In the Euclidean setting, a uniform exterior ball condition suffices. As applications, certain regularity results of PDE's and eigenvalue inequalities in non-smooth domains are presented.