Global Hölder estimates for 2D linearized Monge–Ampère equations with right-hand side in divergence form
We establish global Hölder estimates for solutions to inhomogeneous linearized Monge–Ampère equations in two dimensions with the right hand side being the divergence of a bounded vector field. These equations arise in the semi-geostrophic equations in meteorology and in the approximation of convex f...
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| Published in | Journal of mathematical analysis and applications Vol. 485; no. 2; p. 123865 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
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Elsevier Inc
15.05.2020
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| Online Access | Get full text |
| ISSN | 0022-247X 1096-0813 |
| DOI | 10.1016/j.jmaa.2020.123865 |
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| Abstract | We establish global Hölder estimates for solutions to inhomogeneous linearized Monge–Ampère equations in two dimensions with the right hand side being the divergence of a bounded vector field. These equations arise in the semi-geostrophic equations in meteorology and in the approximation of convex functionals subject to a convexity constraint using fourth order Abreu type equations. Our estimates hold under natural assumptions on the domain, boundary data and Monge-Ampère measure being bounded away from zero and infinity. They are an affine invariant and degenerate version of global Hölder estimates by Murthy-Stampacchia and Trudinger for second order elliptic equations in divergence form. |
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| AbstractList | We establish global Hölder estimates for solutions to inhomogeneous linearized Monge–Ampère equations in two dimensions with the right hand side being the divergence of a bounded vector field. These equations arise in the semi-geostrophic equations in meteorology and in the approximation of convex functionals subject to a convexity constraint using fourth order Abreu type equations. Our estimates hold under natural assumptions on the domain, boundary data and Monge-Ampère measure being bounded away from zero and infinity. They are an affine invariant and degenerate version of global Hölder estimates by Murthy-Stampacchia and Trudinger for second order elliptic equations in divergence form. |
| ArticleNumber | 123865 |
| Author | Le, Nam Q. |
| Author_xml | – sequence: 1 givenname: Nam Q. surname: Le fullname: Le, Nam Q. email: nqle@indiana.edu organization: Department of Mathematics, Indiana University, Bloomington, IN 47405, USA |
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| Cites_doi | 10.4171/170 10.1007/s00229-016-0874-7 10.1007/s002220000059 10.1007/s00220-018-3125-9 10.4310/MAA.2008.v15.n2.a10 10.1007/s00208-012-0895-9 10.1007/s12220-016-9739-2 10.3934/dcds.2014.34.1251 10.1080/03605302.2012.669443 10.1002/mana.19951730106 10.2307/2999574 10.1090/S0002-9939-2013-11748-X 10.1007/BF02413623 10.4007/annals.2008.167.993 10.1137/050629070 10.1007/s00526-004-0280-y 10.1090/S0894-0347-2012-00747-4 10.1007/BF01166225 10.1016/j.aim.2012.07.034 10.1007/s00229-015-0766-2 10.1090/tran/7220 10.1353/ajm.1997.0010 |
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| Title | Global Hölder estimates for 2D linearized Monge–Ampère equations with right-hand side in divergence form |
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