Singular Solutions to the Monge-Ampere Equation
This thesis contains the author's results on singular solutions to the Monge-Ampere equation \det D^2u = 1. We first prove that solutions are smooth away from a small closed singular set of Hausdorff (n-1)-dimensional measure zero. We also construct solutions with a singular set of Hausdorff di...
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2015
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| Online Access: | https://doi.org/10.7916/D89K4955 |
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ndltd-columbia.edu-oai-academiccommons.columbia.edu-10.7916-D89K49552019-05-09T15:14:43ZSingular Solutions to the Monge-Ampere EquationMooney, Connor R.2015ThesesMathematicsThis thesis contains the author's results on singular solutions to the Monge-Ampere equation \det D^2u = 1. We first prove that solutions are smooth away from a small closed singular set of Hausdorff (n-1)-dimensional measure zero. We also construct solutions with a singular set of Hausdorff dimension n-1, showing that this result is optimal. As a consequence we obtain unique continuation for the Monge-Ampere equation. Finally, we prove an interior W^{2,1} estimate for singular solutions, and we construct an example to show that this estimate is optimal.Englishhttps://doi.org/10.7916/D89K4955 |
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NDLTD |
| language |
English |
| sources |
NDLTD |
| topic |
Mathematics |
| spellingShingle |
Mathematics Mooney, Connor R. Singular Solutions to the Monge-Ampere Equation |
| description |
This thesis contains the author's results on singular solutions to the Monge-Ampere equation \det D^2u = 1. We first prove that solutions are smooth away from a small closed singular set of Hausdorff (n-1)-dimensional measure zero. We also construct solutions with a singular set of Hausdorff dimension n-1, showing that this result is optimal. As a consequence we obtain unique continuation for the Monge-Ampere equation. Finally, we prove an interior W^{2,1} estimate for singular solutions, and we construct an example to show that this estimate is optimal. |
| author |
Mooney, Connor R. |
| author_facet |
Mooney, Connor R. |
| author_sort |
Mooney, Connor R. |
| title |
Singular Solutions to the Monge-Ampere Equation |
| title_short |
Singular Solutions to the Monge-Ampere Equation |
| title_full |
Singular Solutions to the Monge-Ampere Equation |
| title_fullStr |
Singular Solutions to the Monge-Ampere Equation |
| title_full_unstemmed |
Singular Solutions to the Monge-Ampere Equation |
| title_sort |
singular solutions to the monge-ampere equation |
| publishDate |
2015 |
| url |
https://doi.org/10.7916/D89K4955 |
| work_keys_str_mv |
AT mooneyconnorr singularsolutionstothemongeampereequation |
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1719046172816965632 |