Abstract: Diffusion processes span across various scientific disciplines including physics, chemistry, biology, finance, and mathematical sciences. At the core of these models lies a unified regularity theory, providing essential information for their respective analyses. When the domain in which the system is driven by diffusion depends on the solution itself, the problem is significantly more complex and over the last 50 years, robust mathematical tools have been developed to tackle these challenging free boundary problems.
In this talk I will revise key milestones of these successful theories, drawing a comparative parallel between them. The goal is illustrate how powerful geometric insights originally developed for free boundary problems can be reshaped to tackle regularity questions in classical PDE theory that would otherwise be difficult to access or even conceptualize.
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