On Moreau-Yosida Approximation and on Stability of Second-Order Subdifferentials of Clarke's Type

For the Gateaux derivative of a C^1,1 function defined on a reflexive Banach space with Kadec norm $||\cdot||$, we use Moreau-Yosida regularization to show that Clarke's subdifferential of second-order can be weak$^*$-approximated from below. Moreover, in the convex case we can strengthen the inclusion to an equality in the limit. In another approach for C^1,1 functions, we establish a weak$^*$ stability result for second-order subdifferentials of Clarke's type. We apply the latter result to the continuous behaviour of the Lagrange multipliers in second-order necessary optimality conditions under epi-convergent perturbations and to stability of second-order subdifferentials of Clarke's type of integral functionals and also of the standard type of functionals in the calculus of variations