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

TL;DR

This paper investigates the existence and uniqueness of solutions to degenerate Monge--Ampère equations involving singular measures, using a variational framework and monotonicity techniques, and explores eigenvalue problems with complex measure singularities.

Contribution

It introduces a variational approach to analyze degenerate Monge--Ampère equations with singular measures, including eigenvalue problems, and provides new insights into solution existence, uniqueness, and approximation methods.

Findings

Solutions may not exist within the energy class for highly singular measures.

Eigenvalues can be approximated by truncated measure problems and iterative schemes.

Examples show nonuniqueness and symmetry breaking in solutions for certain singular measures.

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.