Second order equations with prescribed boundary values and isolated singularities have received extensive study through the years. In 1965, Serrin studied the local behavior of the solutions of certain second-order, quasilinear equations $\text{div} A(x, u, u_x) = 0$ in the perforated domain $\Omega \setminus \{0\}$. Later, Kichenassamy-Veron continued the work of Serrin and studied the $p$-Laplace equation $\text{div}(| \nabla u|^{p−2} \nabla u) = 0$ in $\Omega \setminus \{0\}$. Weak solutions of various fully nonlinear equations with prescribed singularities have also been studied. For example, the $k$-Hessian equation has been investigated by Trudinger-Wang and Labutin. A very general family of fully nonlinear equations has been considered by Labutin and Harvey-Lawson.

In this talk, we will first explain what boundary value problems with prescribed singularities exactly are; then we will go through some related known results and explain their general ideas; finally, we will discuss the inspirations for our new ideas. These new ideas lead to the proof of the existence of a smooth solution of some fully nonlinear equations with prescribed density at their singularities. This is the first result showing the existence of smooth solutions for a fully nonlinear equation with prescribed density. Previously only continuous solutions were obtained.