The obstacle problem for the Monge-Ampère equation with the lower obstacle
The authors consider the existence and optimal regularity of the solution and the regularity of the free boundary of the obstacle problem for the following equation involving the log-concave Monge-Ampère operator \[ \begin{cases}{c}\max \left\{ \varphi -u,\det D^{2}u-f(x,u,Du)\right\} =0\quad&\text{in }\Omega,\\ u=g&\text{in }\partial \Omega, \end{cases} \] with a strictly convex lower obstacle function \(\varphi \in C^{2,1}\left( \overline{\Omega }\right) \). Here \(\Omega \) is a strictly convex bounded domain in \(\mathbb{R}^{n}\) with \(\partial \Omega \in C^{3,1}\). About the forcing term \(f\) it is assumed that \(f\in C^{2,1}\left( \overline{\Omega }\times\mathbb{R}\times\mathbb{R}^{n}\right) \), \(f(\overline{\Omega }\times \lbrack -M,M]\times \lbrack-M,M]^{n})\subset [\sigma _{1}(M),\sigma _{2}(M)]\) (\(-\infty <\sigma _{1}\leq \sigma _{2}<+\infty \)), and the the boundary data \(g\in C^{3,1}\left( \partial \Omega \right) \) is such that \(\varphi <g\) on \(\partial\Omega\). The obstacle problem for the Monge-Ampère equation arises in the study of the prescribed Gaussian curvature equation, optimal transportation, and affine geometry with an obstacle. The incongruousness of the operator and the location of the obstacle below the solution cause various difficulties in the penalization method, the classification of the global solutions, and the directional monotonicity.
scientific article; zbMATH DE number 7358557
scientific article