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Boundary regularity for the second boundary-value problem of Monge-Ampère equations in dimension two

Journal Article


Abstract


  • In this paper, we introduce an iteration argument to prove that a convex

    solution to the Monge-Amp\`ere equation $\mbox{det } D^2 u =f $ in dimension

    two subject to the natural boundary condition $Du(\Omega) = \Omega^*$ is

    $C^{2,\alpha}$ smooth up to the boundary. We establish the estimate under the

    sharp conditions that the inhomogeneous term $f\in C^{\alpha}$ and the domains

    are convex and $C^{1,\alpha}$ smooth. When $f\in C^0$ (resp. $1/C

    positive constant $C$), we also obtain the global $W^{2,p}$ (resp.

    $W^{2,1+\epsilon}$) regularity.

Publication Date


  • 2018

Web Of Science Accession Number


Abstract


  • In this paper, we introduce an iteration argument to prove that a convex

    solution to the Monge-Amp\`ere equation $\mbox{det } D^2 u =f $ in dimension

    two subject to the natural boundary condition $Du(\Omega) = \Omega^*$ is

    $C^{2,\alpha}$ smooth up to the boundary. We establish the estimate under the

    sharp conditions that the inhomogeneous term $f\in C^{\alpha}$ and the domains

    are convex and $C^{1,\alpha}$ smooth. When $f\in C^0$ (resp. $1/C

    positive constant $C$), we also obtain the global $W^{2,p}$ (resp.

    $W^{2,1+\epsilon}$) regularity.

Publication Date


  • 2018

Web Of Science Accession Number