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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Bibliographic Details
Published inJournal of mathematical analysis and applications Vol. 485; no. 2; p. 123865
Main Author Le, Nam Q.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.05.2020
Subjects
Global Hölder estimates
Green's function
Linearized Monge-Ampère equation
Linearized Monge-Ampère equation
Global Hölder estimates
Green's function
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ISSN0022-247X
1096-0813
DOI10.1016/j.jmaa.2020.123865

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Summary: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.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2020.123865