Nonvariational singular elliptic equations and their geometries

In this talk I will discuss fine geometric properties of
solutions to fully nonlinear elliptic equations with singular
potentials. Singularities occurs along the (unkown) free surface {u=0}
and the degree of the singularity determines optimal regularity
estimates for u though the singular level set. These classes of
equations are related to important free boundary problems that were
previously accessed only through variational considerations.