C^{1,{1}/{3}-}$$ very weak solutions to the two dimensional Monge–Ampère equation
For any $$\theta <\frac{1}{3}$$ θ < 1 3 , we show that very weak solutions to the two-dimensional Monge–Ampère equation with regularity $$C^{1,\theta }$$ C 1 , θ are dense in the space of continuous functions. This result is shown by a convex integration scheme involving a subtle decomposition...
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| Published in | Calculus of variations and partial differential equations Vol. 64; no. 5 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
01.06.2025
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| Online Access | Get full text |
| ISSN | 0944-2669 1432-0835 |
| DOI | 10.1007/s00526-025-03019-0 |
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| Abstract | For any $$\theta <\frac{1}{3}$$ θ < 1 3 , we show that very weak solutions to the two-dimensional Monge–Ampère equation with regularity $$C^{1,\theta }$$ C 1 , θ are dense in the space of continuous functions. This result is shown by a convex integration scheme involving a subtle decomposition of the defect at each stage. The decomposition diagonalizes the defect and, in addition, incorporates some of the leading-order error terms of the first perturbation, effectively reducing the required amount of perturbations to one. |
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| AbstractList | For any $$\theta <\frac{1}{3}$$ θ < 1 3 , we show that very weak solutions to the two-dimensional Monge–Ampère equation with regularity $$C^{1,\theta }$$ C 1 , θ are dense in the space of continuous functions. This result is shown by a convex integration scheme involving a subtle decomposition of the defect at each stage. The decomposition diagonalizes the defect and, in addition, incorporates some of the leading-order error terms of the first perturbation, effectively reducing the required amount of perturbations to one. |
| ArticleNumber | 160 |
| Author | Hirsch, Jonas Cao, Wentao Inauen, Dominik |
| Author_xml | – sequence: 1 givenname: Wentao surname: Cao fullname: Cao, Wentao – sequence: 2 givenname: Jonas surname: Hirsch fullname: Hirsch, Jonas – sequence: 3 givenname: Dominik orcidid: 0000-0003-3858-3291 surname: Inauen fullname: Inauen, Dominik |
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| Cites_doi | 10.4171/cmh/564 10.1016/j.jfa.2024.110616 10.1007/978-3-642-25361-4_5 10.1090/surv/130 10.4171/rmi/1019 10.2140/apde.2017.10.695 10.1090/S0002-9947-1934-1501735-3 10.4310/jdg/1668186787 10.1090/gsm/048 10.1007/BF02385981 10.1016/j.anihpc.2015.08.005 10.1016/j.aim.2020.106996 10.1007/s11425-018-9516-7 10.1090/bull/1713 10.2307/1969840 |
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| References | 3019_CR12 Wentao Cao (3019_CR4) 2019; 62 3019_CR13 3019_CR10 Anders Källén (3019_CR15) 1978; 16 3019_CR21 3019_CR9 3019_CR11 3019_CR22 3019_CR16 3019_CR17 3019_CR14 Wentao Cao (3019_CR3) 2024; 99 Hassler Whitney (3019_CR23) 1934; 36 John Nash (3019_CR20) 1954; 2 3019_CR2 3019_CR18 3019_CR1 3019_CR19 3019_CR8 3019_CR7 3019_CR6 3019_CR5 |
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