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Description: |
Operators with solution-dependent discontinuities give rise to interfaces or free boundaries. In this context, two fundamental questions appear. First, one is interested in the regularity properties of the solutions across the free boundary. Then the analysis focuses on the interfaces and examines their geometric properties. The interdependence of those structures leads to genuine difficulties, which require involved methods and techniques from several disciplines. We study free boundary problems driven by fully nonlinear elliptic operators, including degenerate models and Hessian-dependent functionals. Our findings comprise optimal regularity results for the solutions and information on the associated free boundaries. An application to the theory of fully nonlinear mean-field games concludes the talk.
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