Hessian-dependent Functionals: Mean Field Games, Free Boundaries and Improved Regularity


Hessian-dependent functionals play a pivotal role in a wide latitude of problems in mathematics. Arising in the context of differential geometry and probability theory, this class of problems find applications in the mechanics of deformable media (mostly in elasticity theory) and the modelling of slow viscous fluids. We study such functionals from three distinct perspectives. First, we relate their Euler-Lagrange equations with the mean-field games theory and study the existence and improved regularity of the solutions. Motivated by a Bernoulli problem, we then embed those models in the context of free boundary problems and examine their geometric properties. Finally, we examine semi-linear variants of the bi-Laplacian equation and establish improved regularity results. We close the talk with a discussion of fascinating directions for further research.

Brief Biography

Edgard Pimentel is a research mathematician who works as an Assistant Professor at the Department of Mathematics of the University of Coimbra. His research interests are in partial differential equations, with emphasis on regularity theory, free boundary problems and geometric analysis. He is an associate fellow of the International Centre for Theoretical Physics and an affiliate member of the Brazilian Academy of Sciences. In 2024, he became an LMS Distinguished Visiting Fellow and joined the Young Scientists Seminar of the Academy of Sciences of Lisbon.

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