The zero level set for a certain weak solution, with applications to the Bellman equations

We will prove a partial regularity result for the zero level set of weak solutions to

$$\textrm {div}(B\nabla u)=0,$$

where $B=B(u)=I+(A-I)\chi _{\{u<0\}}$, where $I$ is the identity matrix and the eigenvalues of $A$ are strictly positive and bounded.

We will apply this to describe the regularity of solutions to the Bellman equations.