joint work with Daniel Steck
This talk deals with quasi-variational inequality problems (QVIs) in a generic Banach space setting. We provide a theoretical framework for the analysis of such problems based on the pseudomonotonicity (in the sense of Brezis) of the variational operator and a Mosco-type continuity of the feasible set mapping. These assumptions can be used to establish the existence of solutions and their computability via suitable approximation techniques. An augmented Lagrangian technique is then derived and shown to be convergent both theoretically and numerically.
In recent years, generalized Nash equilibrium problems in function spaces involving control of PDEs have gained increasing interest. One of the central issues arising is the question of existence, which requires the topological characterization of the set of minimizers for each player. In this talk, we propose conditions on the operator and the functional, that guarantee the reduced formulation to be a convex minimization problem. Subsequently, we generalize the results of convex analysis to derive optimality systems also for nonsmooth operators. Our findings are illustrated with examples.
joint work with Michael Hintermüller, Carlos Rautenberg
We consider quasi-variational inequalities with pointwise constraints on the gradient of the state, that arise in the modeling of sandpile growth, where the constraint depends on local properties of the supporting surface and the solution. We will demonstrate an example where multiple solutions exist in the unregularized problem but not after regularization. Solution algorithms in function space are derived, considering two approaches: a variable splitting method and a semismooth Newton method. The application can further be extended to regarding water-drainage as the limiting case.