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Potential Analysis

, Volume 24, Issue 3, pp 267–301 | Cite as

Regularity of Degenerate Monge–Ampère and Prescribed Gaussian Curvature Equations in Two Dimensions

  • Eric T. SawyerEmail author
  • Richard L. Wheeden
Article

Abstract

We use a priori inequalities for quasilinear equations to obtain a regularity theorem for the Dirichlet problem for the Monge–Ampère equation,
$$u_{xx}u_{yy}-(u_{xy})^{2}=k(x,y),$$
and the prescribed Gaussian curvature equation,
$$u_{xx}u_{yy}-(u_{xy})^{2}=k(x,y)(1+u_{x}^{2}+u_{y}^{2})^{2},$$
where k(x,y) is close to a function of one variable alone when k is small, but permitted to vanish to infinite order.

Keywords

Monge–Ampère quasilinear regularity 

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Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.McMaster UniversityHamilton
  2. 2.Rutgers UniversityNew Brunswick

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