Optimal regularity of solutions to fully nonlinear monotone operators

Problem's Description

The semilinear problem given by $$F(D^2 u, u):= \Delta u - f(u)=0,$$ where $f(u)$ is monotone-increasing and has a jump discontinuity for some values of $u$, can be seen as an unconstrained free boundary problem, where free boundary is given by the level surfaces for $u$ where $f(u)$ has a discontinuity. It was shown in [2] that solutions to this problem are $C^{1,1}$. It is tantalizing to analyze the case of fully nonlinear equations $F(D^2 u, u)=0$ with $F'_u \leq 0 $. Notice that in the particular case $F(D^2 u, u)=G(D^2u)-f(u)$ with $G$ convex and $f$ bounded, elliptic regularity gives that $D^2u$ belongs to BMO, so one only needs to understand the ``last step'' from BMO to $L^\infty$.

See [1] for further related problems.

1. ArticleIs an originAn overview of unconstrained free boundary problems

   • Alessio Figalli,
   • Henrik Shahgholian

   Philos. Trans. Roy. Soc. A 373 (2050), 2015arXiv

2. Article $C^{1,1}$-regularity in semilinear elliptic problems.

   • Henrik Shahgholian

   Comm. Pure Appl. Math. 56 (2), 278-281, 2003arXiv