Vector-valued Free Boundary Problems

Free Boundaries (FB) are mathematical challenges that arise in various fields such as Physics, Economics, and Biology, often involving phase transitions, like the change from ice to water. These problems are particularly difficult to analyze due to the uncertainty of initially unknown boundaries. This proposal aims to address this complexity by developing novel theoretical frameworks and numerical methodologies, leveraging machine learning and neural network approaches. Despite recent advancements in these technologies, their application to free boundary problems remains underexplored, leaving a significant gap in the field.