The Limiting Normal Cone to Pointwise Defined Sets in Lebesgue Spaces

We consider subsets of Lebesgue spaces which are defined by pointwise constraints. We provide formulas for corresponding variational objects (tangent and normal cones). Our main result shows that the limiting normal cone is always dense in the Clarke normal cone and contains the convex hull of the pointwise limiting normal cone. A crucial assumption for this result is that the underlying measure is non-atomic, and this is satisfied in many important applications (Lebesgue measure on subsets of \(\mathbb {R}^{d}\) or the surface measure on hypersurfaces in \(\mathbb {R}^{d}\)). Finally, we apply our findings to an optimization problem with complementarity constraints in Lebesgue spaces.