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Geometric-tangential regularity for fully nonlinear equations Pimentel, Edgard


The regularity theory for fully nonlinear PDEsEpodes was inaugurated in the turn of 70s to the 80s, with the works of Krylov and Safonov. In the span of a decade, a number of important advances took place in the field; e.g., the Evans-Krylov theory and Caffarelli's estimates in Sobolev spaces. In face of those breakthroughs, a question on the possibility of a general theory for that class of problems naturally arose. Such a question was set in the negative only recently; in a series of papers, Nadirashvili and Vladut produced a number of counterexamples unveiling the subtleties of this theory. In this talk, we examine the regularity of the solutions to elliptic and parabolic fully nonlinear equations, in Sobolev spaces. We make use of a geometric-tangential approach, which enables us to work under asymptotic assumptions of the operator governing the problem. We also put forward a number of new estimates and applications, consequential to our results.

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