Optimal Regularity of Solutions to No-Sign Obstacle-Type Problems for the Sub-Laplacian

We establish the optimal $C_{H}^{1,1}$ interior regularity of solutions to $$\Delta_{H}u=f\chi_{\{u\ne0\}},$$ where $\Delta_{H}$ denotes the sub-Laplacian operator in a stratified group. We assume the weakest regularity condition on $f$, namely $f*\Gamma$ is $C_{H}^{1,1}$, where $\Gamma$ is the fundamental solution of $\Delta_{H}$. The $C_{H}^{1,1}$ regularity is understood in the sense of Folland and Stein. In the classical Euclidean setting, the first seeds of the above problem are already present in the 1991 paper of Sakai and are also related to quadrature domains. As a special instance of our results, when $u$ is nonnegative and satisfies the above equation we recover the $C_{H}^{1,1}$ regularity of solutions to the obstacle problem in stratified groups, that was previously established by Danielli, Garofalo and Salsa. Our regularity result is sharp: it can be seen as the subelliptic counterpart of the $C^{1,1}$ regularity result due to Andersson, Lindgren and Shahgholian.