gradient constrained problems for fully nonlinear equations

Problem's Description

Define a general class of (double) gradient constrained problems for fully nonlinear equations as $$ \max\bigl\{\min \{ -F(D^2u,x) , |\nabla u -a|-h_1\}, |\nabla u-a|-h_2 \bigr\}=0\qquad \text{in $B_1$}, $$ where the equation is in the viscosity sense. Formally, the equation can be written as $$ F(D^2u,x)\chi_{\{|\nabla u-a| \neq h_1 \} \cup \{|\nabla u-a |\neq h_2\} }=0,\qquad h_1 \leq |\nabla u-a| \leq h_2. $$ Dropping the constraints, $h_1 \leq |\nabla u-a| \leq h_2$, one encounters an unconstrained problem.

A natural question is: How regular are solutions and the free boundary for such problems?

1. ArticleIs an originAn overview of unconstrained free boundary problems

   • Alessio Figalli,
   • Henrik Shahgholian

   Philos. Trans. Roy. Soc. A 373 (2015), no. 2050, 20140281, 11 pp. A373, 2015arXiv