Room P3.10, Mathematics Building

Edgard Pimentel, Universidade Federal de São Carlos
Geometric regularity theory for fully nonlinear elliptic equations

In this talk, we examine the regularity theory for fully nonlinear elliptic equations of the form \[ F(D^2u)=f(x) \qquad\text{ in } B_1, \] where $F$ is a $(\lambda,\Lambda)$-elliptic operator and $f:B_1\to\mathbb{R}$ is a continuous source term, in appropriate Lebesgue spaces. We recur to a set of tools known as geometric–tangential analysis to produce a priori estimates for the solutions in Sobolev spaces, under minimal (asymptotic) assumptions on the operator $F$. In addition, we discuss regularity in $p$-BMO spaces and the density of $W^{2,p}$- solutions in the class of continuous viscosity solutions. We conclude the talk with the study of a degenerate problem; in this case, we produce a result on the optimal regularity of solutions in Holder spaces.