| Abstract: |
| In this talk, we will discuss $\mathcal{C}^1$ regularity results for viscosity solutions of certain free boundary problems with gradient constraint of the form
$$
G(Dv,D^2v) \in L^\infty(\{|Dv|>\mu \}).
$$
Due to the lack of appropriate structure, solutions are to be understood throughout a limiting procedure and we prove gradient regularity results that remain uniform in the process. To do so, we first get compactness by the method of doubling the variables and we combine it with a Bernstein-type argument. We provide some applications and, in particular, we are able to study the behavior of the family of normalized solutions $\{v_q\}$ to
$$
|Dv_q|^q\Delta v_q = 1
$$
as $q\rightarrow \infty$. |
|