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Regularity theory for fully nonlinear elliptic operators By  ​Dr. Edgard Pimentel (Universidade Federal do Ceara, UFC, Brazil)

  • Class schedule:  Wednesday, March 11th, 2015   from  3:00 pm to 4:00 pm
  • Location: Building 1, Room 4214 
  • Refreshments: Available at 2:45 pm

In [Ann. of Math. (2) 130 (1989), no. 1, 189–213], Luis Caffarelli developed his famous $W^{2,p}$ regularity theory for convex fully nonlinear operators with continuous coefficients. The question on whether $W^{2,p}$ estimates could be established for general non-convex equations challenged the community for nearly thirty years when Nadirashvili and Vladut built up counterexamples. While it is not possible to develop a general $W^{2,p}$ regularity theory, in this talk we show how an analysis on the operator $F$  at the ends of $S(n)$, gives such estimates back to the original operator. This analysis rests upon the recession function of $F$, which is formally given by $F^{*}(M) = \infty^{-1} F(\infty M)$. The arguments are based on techniques from the so-called geometric tangential analysis. These build upon elementary results in harmonic analysis and measure theory yielding our main result. This is based on a joint work (in progress) with E. Teixeira (UFC and ICMC-USP).

Edgard Almeida Pimentel received his BA in Economics and his M.Sc. in Applied Mathematics (2005/2010) from Universidade de São Paulo (USP). In 2013, Dr Pimentel concluded his Ph.D. in Mathematics from the Instituto Superior Tecnico of Universidade de Lisboa (IST-UL), under the direction of Prof. Diogo Gomes. Since them, Dr. Pimentel has held post-doctoral position at IST (Portugal) and IMPA (Brazil). Currently, Dr. Pimentel is a post-doctoral fellow of the Department of Mathematics of the Universidade Federal do Ceara (UFC, Brazil). His research interests are in the Analysis of Partial Differential Equations, with emphasis on regularity theory for mean-field games and fully nonlinear elliptic equations.