A Variational Approach to Degenerate Monge--Ampère Equations with Mixed Measures and Monotonicity
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915875843473408 |
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| author | Le, Nam Q. |
| author_facet | Le, Nam Q. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_19114 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Variational Approach to Degenerate Monge--Ampère Equations with Mixed Measures and Monotonicity Le, Nam Q. Analysis of PDEs 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. |
| title | A Variational Approach to Degenerate Monge--Ampère Equations with Mixed Measures and Monotonicity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.19114 |