Free boundary problems (FBP) lie at the heart of mathematical models describing phase transitions, moving interfaces, and competitive dynamics in physics, biology, and economics. They typically involve phase transitions (such as ice melting into water), moving obstacles, or competition for resources. From a mathematical perspective, they arise when a partial differential equation (PDEs) must be solved together with the unknown location of part of the boundary. This "free boundary" significantly complicates the analysis. Over the past few decades, this field has expanded rapidly due to its relevance in modeling tumor growth, crystal formation, environmental changes, and option pricing in finance.
The research in this project focuses on systems of variational inequalities and free boundaries arising in vectorial settings. These settings are essential for modeling complex interactions like optimal switching, mean-field game theory, and multi-phase or multi-membrane problems. Unlike scalar problems, the vectorial setting addresses multiple interacting components, often represented as weakly or strongly coupled systems, where solutions may fail to be continuous. Our primary focus will be to explore the existence and qualitative behavior of both the solutions and the free boundaries. Beyond existence theory, we will investigate uniqueness, geometric properties, stability, and optimal smoothness, which form the core of a general mathematical approach to these systems. Additionally, we will study time-dependent versions of these problems, where the standard variational framework often fails, introducing further analytical challenges. While the immediate practical applications of such general mathematical theories can seem distant, the framework developed here is expected to significantly influence the exploration of models prevalent in current literature. A compelling application lies in optimal switching problems involving multiple choices. For instance, this model can address the transition between diverse energy sources—such as fossil fuels, solar, wind, and nuclear energy. Within the context of sustainability, this allows decision-makers to determine the most cost-effective and environmentally sound energy mix. By providing a rigorous framework for energy planning, this project aligns with Qatar key national priorities in energy and sustainability.