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A Variational Approach to Degenerate Monge--Ampère Equations with Mixed Measures and Monotonicity

Nam Q. Le

Abstract

We study the solvability and uniqueness for several degenerate Monge--Ampère equations including the Monge--Ampère eigenvalue problem in real Euclidean spaces that involve singular Borel measures. Our approach systematically analyzes the Monge--Ampère energy from the variational point of view and appropriately exploits monotonicity arguments. Our main tools consist of the mixed Monge--Ampère measure, Aleksandrov--Blocki--Jerison-type maximum principles, integration by parts, convex envelope, and comparison principles for subcritical equations. For the Monge--Ampère 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--Ampère 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--Ampère eigenvalue problem has only solutions outside the energy class together with symmetry breaking and nonuniqueness.

A Variational Approach to Degenerate Monge--Ampère Equations with Mixed Measures and Monotonicity

Abstract

We study the solvability and uniqueness for several degenerate Monge--Ampère equations including the Monge--Ampère eigenvalue problem in real Euclidean spaces that involve singular Borel measures. Our approach systematically analyzes the Monge--Ampère energy from the variational point of view and appropriately exploits monotonicity arguments. Our main tools consist of the mixed Monge--Ampère measure, Aleksandrov--Blocki--Jerison-type maximum principles, integration by parts, convex envelope, and comparison principles for subcritical equations. For the Monge--Ampère 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--Ampère 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--Ampère eigenvalue problem has only solutions outside the energy class together with symmetry breaking and nonuniqueness.
Paper Structure (23 sections, 57 theorems, 344 equations)

This paper contains 23 sections, 57 theorems, 344 equations.

Table of Contents

  1. Introduction and statement of the main results
  2. Case of Lebesgue measure
  3. Case of general Borel measure
  4. Real Monge--Ampère variational method
  5. Main results
  6. Ingredients of the proofs
  7. Mixed Monge--Ampère measures and inequalities
  8. Mixed Monge--Ampère measures and energies
  9. Mixed Monge--Ampère inequalities
  10. Maximum and comparison principles and the Dirichlet problem
  11. Maximum principles
  12. The Dirichlet problem
  13. A comparison principle for degenerate Monge--Ampère equations
  14. Mixed Monge--Ampère energies
  15. Generalized Cauchy--Schwarz inequality
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb R^n$$(n\geq 1)$ be a bounded convex domain, and $p\in (-1, \infty)$. Let $\nu$ be a locally finite Borel measure on $\Omega$ satisfying $\nu(\Omega)>0$ and

Theorems & Definitions (123)

  • Theorem 1.1: Solvability, uniqueness, and variational characterization of degenerate Monge--Ampère equations
  • Theorem 1.2: Monge--Ampère eigenvalue problem, Rayleigh quotient, Poincaré inequality, and Inverse scheme
  • Theorem 1.3: Variational derivative of Monge--Ampère energy of convex envelopes
  • Theorem 1.4: Uniform Aleksandrov--Jerison maximum principle
  • Theorem 1.5: Comparison principle for degenerate subcritical Monge--Ampère equations
  • Definition 1.6: Mixed Monge--Ampère measure
  • Theorem 1.7: Blocki-type maximum principles for mixed Monge--Ampère measures
  • Theorem 1.8: Generalized Cauchy--Schwarz inequality and Integration by parts
  • Theorem 1.9: Mixed Monge--Ampère inequality
  • Remark 2.1
  • ...and 113 more