Liouville theorem for degenerate optimal transport Monge–Ampère equation

Prove that any nonnegative solution ϕ: R_+^l × R^m → R_{≥0} of the Monge–Ampère equation det(D^2ϕ) = 1 in R_+^l × R^m with boundary condition ∑_{i=1}^l ∂ϕ/∂x_i = 0 on ∂(R_+^l) × R^m must be of the form ϕ(x) = c + ∑_{i=l+1}^{l+m} v_i x_i + ∑_{i,j=1}^l P_{ij} x_i x_j + p ϕ_{l,k}(x_1,...,x_l), where p > 0 and c, v_i are constants, P = (P_{ij}) is a positive-definite l × l matrix, and ϕ_{l,k} is the homogeneous solution on R_+^l of det(D^2ϕ_{l,k}) = 1 with ∑_{i=1}^l ∂ϕ_{l,k}/∂x_i = 0 on ∂R_+^l, as described in Corollary 1.1 with n + 1 = l.

Background

The paper studies a free-boundary Monge–Ampère equation and its Legendre-dual formulation motivated by constructing complete Calabi–Yau metrics on complements of simple normal crossing anticanonical divisors. Regularity near lower-dimensional faces and free-boundary behavior are connected to optimal transport with degenerate densities.

To advance regularity theory and boundary asymptotics, the authors propose a Liouville-type classification result for solutions to a degenerate optimal transport Monge–Ampère equation on R_+^l × R^m. This generalizes known results when l = 1 (or low-dimensional cases) and would yield strong control over blow-up limits and structure of solutions, crucial for understanding asymptotics needed in the Calabi–Yau setting.