A Variational Approach to Degenerate Monge--Amp\`ere Equations with Mixed Measures and Monotonicity

Abstract

We study the solvability and uniqueness for several degenerate Monge--Amp\`ere equations including the Monge--Amp\`ere eigenvalue problem in real Euclidean spaces that involve singular Borel measures. Our approach systematically analyzes the Monge--Amp\`ere energy from the variational point of view and appropriately exploits monotonicity arguments. Our main tools consist of the mixed Monge--Amp\`ere measure, Aleksandrov--Blocki--Jerison-type maximum principles, integration by parts, convex envelope, and comparison principles for subcritical equations. For the Monge--Amp\`ere eigenvalue problem, we contrast the analysis within and without the energy class; even if it might not have solutions in the energy class, we show that the infimum of the Rayleigh quotient can be approximated from above by Monge--Amp\`ere eigenvalues of the truncated measures, and by Rayleigh quotients of an inverse iterative scheme. We give examples showing that for very singular Borel measures, the Monge--Amp\`ere eigenvalue problem has only solutions outside the energy class together with symmetry breaking and nonuniqueness.