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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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$W^\{2,p\}$–estimates for the linearized Monge–Ampère equation
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by Cristian E. Gutiérrez and Federico Tournier PDF
Trans. Amer. Math. Soc. 358 (2006), 4843-4872 Request permission

Abstract:

Let $\Omega \subseteq \mathbb {R}^n$ be a strictly convex domain and let $\phi \in C^2(\Omega )$ be a convex function such that $\lambda \leq \text {det}D^2\phi \leq \Lambda$ in $\Omega$. The linearized Monge–Ampère equation is \begin{equation*} L_{\Phi }u=\textrm {trace}(\Phi D^2u)=f, \end{equation*} where $\Phi = (\text {det}D^2\phi )(D^2\phi )^{-1}$ is the matrix of cofactors of $D^2\phi$. We prove that there exist $p>0$ and $C>0$ depending only on $n,\lambda ,\Lambda$, and $\textrm {dist}(\Omega ^\prime ,\Omega )$ such that \begin{equation*} \|D^2u\|_{L^p(\Omega ^\prime )}\leq C(\|u\|_{L^\infty (\Omega )}+\|f\|_{L^n(\Omega )}) \end{equation*} for all solutions $u\in C^2(\Omega )$ to the equation $L_{\Phi }u=f$.
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Additional Information
  • Cristian E. Gutiérrez
  • Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
  • Email: gutierrez@math.temple.edu
  • Federico Tournier
  • Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-2067
  • Address at time of publication: Instituto Argentino de Matemática, Saavedra 15, 1038 Buenos Aires, Argentina
  • Email: fedeleti@aol.com
  • Received by editor(s): August 19, 2004
  • Published electronically: June 9, 2006
  • Additional Notes: The first author was partially supported by NSF grant DMS–0300004.
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 358 (2006), 4843-4872
  • MSC (2000): Primary 35B45, 35J60, 35J70
  • DOI: https://doi.org/10.1090/S0002-9947-06-04189-4
  • MathSciNet review: 2231875