Calculus of Variations and Geometric Measure Theory

V. Magnani - A. Minne

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

created by magnani on 26 Jul 2019
modified on 07 Jun 2024


Published Paper

Inserted: 26 jul 2019
Last Updated: 7 jun 2024

Journal: Anal. PDE
Volume: 15
Number: 6
Pages: 1429-1456
Year: 2022


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.

Keywords: subelliptic equations, stratified groups, obstacle problem, sub-Laplacian