  In this paper we investigate continuity properties of first and second order
shape derivatives of functionals depending on second order elliptic PDE's
around nonsmooth domains, essentially either Lipschitz or convex, or satisfying
a uniform exterior ball condition. We prove rather sharp continuity results for
these shape derivatives with respect to Sobolev norms of the boundary-traces of
the displacements. With respect to previous results of this kind, the approach
is quite different and is valid in any dimension $N\geq 2$. It is based on
sharp regularity results for Poisson-type equations in such nonsmooth domains.
We also enlarge the class of functionals and PDEs for which these estimates
apply. Applications are given to qualitative properties of shape optimization
problems under convexity constraints for the variable domains or their
complement.
