Nonlinear bounds in Hölder spaces for the Monge-Ampère equation

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• 作者：Alessio Figalli ^{figalli@math.utexas.edu" class="auth_mail" title="E-mail the corresponding author} ; Yash Jhaveri ^{yjhaveri@math.utexas.edu" class="auth_mail" title="E-mail the corresponding author} ; Connor Mooney ; ^{cmooney@math.utexas.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Monge&ndash ; Ampè ; re ; Schauder estimates
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：15 May 2016
• 年：2016
• 卷：270
• 期：10
• 页码：3808-3827
• 全文大小：376 K

文摘

We demonstrate that C^2,α$C^{2,\alpha }$ estimates for the Monge–Ampère equation depend in a highly nonlinear way both on the C^α$C^{\alpha }$ norm of the right-hand side and 1/α$1/\alpha$. First, we show that if a solution is strictly convex, then the C^2,α$C^{2,\alpha }$ norm of the solution depends polynomially on the C^α$C^{\alpha }$ norm of the right-hand side. Second, we show that the C^2,α$C^{2,\alpha }$ norm of the solution is controlled by exp⁡((C/α)log⁡(1/α))$\exp ((C/\alpha )\log (1/\alpha ))$ as α→0$\alpha \to 0$. Finally, we construct a family of solutions in two dimensions to show the sharpness of our results.