2 results
Search Results
Now showing 1 - 2 of 2
- ItemEssential boundedness for solutions of the Neumann problem on general domains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) ter Elst, A.F.M.; Meinlschmidt, Hannes; Rehberg, JoachimLet the domain under consideration be bounded. Under the suppositions of very weak Sobolev embeddings we prove that the solutions of the Neumann problem for an elliptic, second order divergence operator are essentially bounded, if the right hand sides are taken from the dual of a Sobolev space which is adapted to the above embedding.
- ItemRegularization for optimal control problems associated to nonlinear evolution equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Meinlschmidt, Hannes; Meyer, Christian; Rehberg, JoachimIt is well-known that in the case of a sufficiently nonlinear general optimal control problem there is very frequently the necessity for a compactness argument in order to pass to the limit in the state equation in the standard ``calculus of variations'' proof for the existence of optimal controls. For time-dependent state equations, i.e., evolution equations, this is in particular unfortunate due to the difficult structure of compact sets in Bochner-type spaces. In this paper, we propose an abstract function space and a suitable regularization- or Tychonov term for the objective functional which allows for the usual standard reasoning in the proof of existence of optimal controls and which admits a reasonably favorable structure in the characterization of optimal solutions via first order necessary conditions in, generally, the form of a variational inequality of obstacle-type in time. We establish the necessary properties of the function space and the Tychonov term and derive the aforementioned variational inequality. The variational inequality can then be reformulated as a projection identity for the optimal control under additional assumptions. We give sufficient conditions on when these are satisfied. The considerations are complemented with a series of practical examples of possible constellations and choices in dependence on the varying control spaces required for the evolution equations at hand.