Constantin Christof — Page 2

arXiv2017

On the Non-Polyhedricity of Sets with Upper and Lower Bounds in Dual Spaces

We demonstrate that the set $L^\infty(X, [-1,1])$ of all measurable functions over a Borel measure space $(X, \mathcal B, μ)$ with values in the unit interval is typically non-polyhedric when interpreted as a subset of a dual space. Our findings contrast the classical result that subsets of Dirichlet spaces with pointwise upper and lower bounds are polyhedric. In particular, additional structural assumptions are unavoidable when the concept of polyhedricity is used to study the differentiability properties of solution maps to variational inequalities of the second kind in, e.g., the spaces $H^{1/2}(\partial Ω)$ or $H_0^1(Ω)$.

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arXiv2017

Differential Sensitivity Analysis of Variational Inequalities with Locally Lipschitz Continuous Solution Operators

This paper is concerned with the differential sensitivity analysis of variational inequalities in Banach spaces whose solution operators satisfy a generalized Lipschitz condition. We prove a sufficient criterion for the directional differentiability of the solution map that turns out to be also necessary for elliptic variational inequalities in Hilbert spaces (even in the presence of asymmetric bilinear forms, nonlinear operators and nonconvex functionals). In contrast to classical results, our method of proof does not rely on Attouch's theorem on the characterization of Mosco convergence but is fully elementary. Moreover, our technique allows us to also study those cases where the variational inequality at hand is not uniquely solvable and where directional differentiability can only be obtained w.r.t. the weak or the weak-$\star$ topology of the underlying space. As tangible examples, we consider a variational inequality arising in elastoplasticity, the projection onto prox-regular sets, and a bang-bang optimal control problem.

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