The Dirichlet problem for the Monge-Ampere equation in convex (but not strictly convex) domains
Texas State University-San Marcos, Department of Mathematics
It is well-known that the Dirichlet problem for the Monge-Ampère equation det D2u = μ in a bounded strictly convex domain Ω in ℝn has a weak solution (in the sense of Aleksandrov) for any finite Borel measure μ on Ω and for any continuous boundary data. We consider the Dirichlet problem when Ω is only assumed to be convex, and give a necessary and sufficient condition on the boundary data for solvability.
Aleksandrov solutions, Perron method, Viscosity solutions
Hartenstine, D. (2006). The Dirichlet problem for the Monge-Ampere equation in convex (but not strictly convex) domains. <i>Electronic Journal of Differential Equations, 2006</i>(138), pp. 1-9.