Variational and quasi-variational problems with gradient constraints and applications to sandpile growth
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2021
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Abstract
In this dissertation, we study variational inequalities (VIs) and quasi-variational inequalities (QVIs) with gradient constraints in diffusive and non-diffusive settings together with several related problems. Specifically, we consider evolutionary, as well as stationary versions of the aforementioned problems, and address existence and uniqueness of solutions, differentiability properties in the context of optimal control, and rigorous Fenchel dualization approaches under low regularity of the data. Initially, we address features of the prototypical problem under study: The Prigozhin model of sandpile growth. In particular, we establish an illustrating example and show closed forms for its multiple solutions and prove that the elementary regularization of constraints leads to uniqueness. On the class of problems arising from the semi-discretization of the evolutionary version, we study existence of solutions under low regularity assumptions; we analyze the cases where the bounds of the gradient constraints are non-negative integrable functions, and also Borel measures. In the latter, we identify new mathematical tools for the application of the direct method. A complete characterization of the Fenchel pre-dual problem leads to the study of minimization problems in a non-standard state space given by vectorial Borel measures with square integrable divergences. The duality description is then exploited for the development of a primal-dual solution algorithm and numerical tests are shown. For a stationary problem formulation which includes a diffusive operator,we provide novel results on the Newton differentiability of the control-to-state map.
This is of interest in the investigation of sensitivity features and optimal control. In this framework, the control is the material source term and the state corresponds to the stationary shape of the material pile. The mathematical enabling tool here is a new implicit function theorem for Newton differentiable maps. In the evolutionary sandpile growth setting, an optimal control problem with a QVI as constraint, is considered. The main goal in this problem is to keep a part of the domain free of material accumulation by controlling the initial supporting surface. We consider fully discrete and semi-discrete approaches for this problem and provide an existence of solutions result.