A variational inequality approach to free boundary problems with applications in mould filling

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Since the early 1960s, the mathematical theory of variational inequalities has rapidly evolved, influenced by real-life applications and complex analysis. Many moving free boundary problems from engineering and economics can be formulated as variational inequalities, either directly or through transformations. This work investigates an evolutionary variational inequality with a memory term, resulting from applying such a transformation to a degenerate moving free boundary problem. The study encompasses mathematical modeling, existence, uniqueness, and regularity results, alongside numerical analysis of finite element and finite volume approximations, and numerical simulations for applications in polymer processing. Key aspects of this research were developed during my time at the Chair of Applied Mathematics (LAM) at the Technical University of Munich. I am deeply grateful to K.-H. Hoffmann, head of the chair and current scientific director of the Center of Advanced European Studies and Research (caesar), for his encouragement and support. This work reflects a broader concept of Applied Mathematics that he inspired in me, integrating mathematical modeling, analysis, computational methods, and visualization of simulation outcomes.