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Partial differential equations arising in the modeling of natural phenomena often exhibit qualitative changes in behavior, frequently in a solution-dependent manner. Two classical instances are free boundary problems -- where the region in which the equation holds is itself unknown -- and equations that degenerate at critical points, where the gradient vanishes.
In this talk, I will present a framework based on a novel class of (local) PDEs exhibiting nonlocal degeneracies. Within this unified setting, classical regularity estimates for locally degenerate models emerge naturally as limiting cases of nonlocal degeneracies, revealing a common underlying structure across seemingly distinct, classical local phenomena.
Beyond the specific results, the aim is to emphasize the structural ideas underlying this formulation and to illustrate how combining local and nonlocal viewpoints can provide new insights into well-established problems.
This is joint work with Damião Araújo, Aelson Sobral, and José Miguel Urbano.
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