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Volume 22, Issue 3
The Regularity of a Class of Degenerate Elliptic Monge-Ampere Equations

Jiaxing Hong & Weiye Wang

DOI: 10.4208/jpde.v22.n3.4

J. Part. Diff. Eq., 22 (2009), pp. 234-265.

Published online: 2009-08

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  • Abstract

In the present paper the regularity of solutions to Dirichlet problem of degenerate elliptic Monge-Ampère equations is studied. Let Ω⊂R^2 be smooth and convex. Suppose that u∈C^2(Ω) is a solution to the following problem: det(u_{ij}) = K(x) f (x,u,Du) in Ω with u = 0 on ∂Ω. Then u∈C^∞(\bar{Ω}) provided that f (x,u,p) is smooth and positive in \bar{Ω}×R×R^2, K > 0 in Ω and near ∂Ω, K=d^m\tilde{K}, where d is the distance to ∂Ω, m some integer bigger than 1 and \tilde{K} smooth and positive on \bar{Ω}.

  • Keywords

Degenerate Monge-Ampère equation regularity

  • AMS Subject Headings

35B65 35B45

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-22-234, author = {Jiaxing Hong and Weiye Wang }, title = {The Regularity of a Class of Degenerate Elliptic Monge-Ampere Equations}, journal = {Journal of Partial Differential Equations}, year = {2009}, volume = {22}, number = {3}, pages = {234--265}, abstract = {

In the present paper the regularity of solutions to Dirichlet problem of degenerate elliptic Monge-Ampère equations is studied. Let Ω⊂R^2 be smooth and convex. Suppose that u∈C^2(Ω) is a solution to the following problem: det(u_{ij}) = K(x) f (x,u,Du) in Ω with u = 0 on ∂Ω. Then u∈C^∞(\bar{Ω}) provided that f (x,u,p) is smooth and positive in \bar{Ω}×R×R^2, K > 0 in Ω and near ∂Ω, K=d^m\tilde{K}, where d is the distance to ∂Ω, m some integer bigger than 1 and \tilde{K} smooth and positive on \bar{Ω}.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v22.n3.4}, url = {http://global-sci.org/intro/article_detail/jpde/5256.html} }
TY - JOUR T1 - The Regularity of a Class of Degenerate Elliptic Monge-Ampere Equations AU - Jiaxing Hong & Weiye Wang JO - Journal of Partial Differential Equations VL - 3 SP - 234 EP - 265 PY - 2009 DA - 2009/08 SN - 22 DO - http://doi.org/10.4208/jpde.v22.n3.4 UR - https://global-sci.org/intro/article_detail/jpde/5256.html KW - Degenerate Monge-Ampère equation KW - regularity AB -

In the present paper the regularity of solutions to Dirichlet problem of degenerate elliptic Monge-Ampère equations is studied. Let Ω⊂R^2 be smooth and convex. Suppose that u∈C^2(Ω) is a solution to the following problem: det(u_{ij}) = K(x) f (x,u,Du) in Ω with u = 0 on ∂Ω. Then u∈C^∞(\bar{Ω}) provided that f (x,u,p) is smooth and positive in \bar{Ω}×R×R^2, K > 0 in Ω and near ∂Ω, K=d^m\tilde{K}, where d is the distance to ∂Ω, m some integer bigger than 1 and \tilde{K} smooth and positive on \bar{Ω}.

Jiaxing Hong and Weiye Wang . (2009). The Regularity of a Class of Degenerate Elliptic Monge-Ampere Equations. Journal of Partial Differential Equations. 22 (3). 234-265. doi:10.4208/jpde.v22.n3.4
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