@misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {From resolvents to generalized equations and quasi-variational inequalities: existence and differentiability}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 27}, abstract = {We consider a generalized equation governed by a strongly monotone and Lipschitz single-valued mapping and a maximally monotone set-valued mapping in a Hilbert space. We are interested in the sensitivity of solutions w.r.t. perturbations of both mappings. We demonstrate that the directional differentiability of the solution map can be verified by using the directional differentiability of the single-valued operator and of the resolvent of the set-valued mapping. The result is applied to quasi-generalized equations in which we have an additional dependence of the solution within the set-valued part of the equation.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Maximal monotone operators with non-maximal graphical limit}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 4}, abstract = {We present a counterexample showing that the graphical limit of maximally monotone operators might not be maximally monotone. We also characterize the directional differentiability of the resolvent of an operator B in terms of existence and maximal monotonicity of the proto-derivative of B.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Elliptic quasi-variational inequalities under a smallness assumption: uniqueness, differential stability and optimal control}, series = {Calculus of Variations and Partial Differential Equations}, volume = {59}, journal = {Calculus of Variations and Partial Differential Equations}, number = {2}, issn = {1432-0835}, doi = {10.1007/s00526-020-01743-3}, pages = {15}, abstract = {We consider a quasi-variational inequality governed by a moving set. We employ the assumption that the movement of the set has a small Lipschitz constant. Under this requirement, we show that the quasi-variational inequality has a unique solution which depends Lipschitz-continuously on the source term. If the data of the problem is (directionally) differentiable, the solution map is directionally differentiable as well. We also study the optimal control of the quasi-variational inequality and provide necessary optimality conditions of strongly stationary type.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {A guided tour of polyhedric sets: basic properties, new results on intersections and applications}, series = {Journal of Convex Analysis}, volume = {26}, journal = {Journal of Convex Analysis}, number = {1}, issn = {0944-6532}, pages = {153 -- 188}, abstract = {The aim of this contribution is twofold. On the one hand, we give some new results concerning polyhedric sets. In particular, we show that sets with pointwise lower and upper bound are polyhedric in many important function spaces. Moreover, we show that the intersection of such a set with finitely many hyperplanes and half-spaces is polyhedric. We also provide counterexamples demonstrating that the intersection of polyhedric sets may fail to be polyhedric. On the other hand, we gather all important results from the literature concerning polyhedric sets in order to give a complete picture of the current knowledge. In particular, we illustrate the applications of polyhedricity.}, language = {en} }