The zero level set for a certain weak solution, with applications to the Bellman equations (Q2838066)

The questions in this work are related to a Bellman equation. The zero level set of the weak solution to the problem NEWLINE\[NEWLINE\begin{aligned} \Delta u= 0\quad &\text{when }u> 0,\\ \nabla\cdot({\mathbf A}\nabla u)= 0\quad &\text{when }u< 0,\end{aligned}NEWLINE\]NEWLINE is studied in a domain \(\Omega\) in \(\mathbb{R}^n\). Here, \({\mathbf A}\) is a constant matrix with strictly positive eigenvalues. The zero set \(\{u=0\}\) is treated as a free boundary.NEWLINENEWLINE The available regularity of the solution is not strong enough for the conventional proofs of establishing ``flatness'' of the zero set. Instead, a method from geometric measure theory is developed to prove that the free boundary is flat a.e.~with respect to the measure NEWLINE\[NEWLINE\mu= \Delta\max\{u,0\}.NEWLINE\]NEWLINE An interesting consequence is that the free boundary is \(\sigma\)-finite with respect to the \((n-1)\)-dimensional Hausdorff measure.NEWLINENEWLINE As always, blow-up arguments and monotonicity formulas are expedient.