Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



$ W^{2,p}$-estimates for the linearized Monge-Ampère equation

Authors: Cristian E. Gutiérrez and Federico Tournier
Journal: Trans. Amer. Math. Soc. 358 (2006), 4843-4872
MSC (2000): Primary 35B45, 35J60, 35J70
Published electronically: June 9, 2006
MathSciNet review: 2231875
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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$   det$ D^2\phi \leq\Lambda$ in $ \Omega$. The linearized Monge-Ampère equation is

$\displaystyle L_{\Phi}u=\textrm{trace}(\Phi D^2u)=f, $

where $ \Phi = ($det$ 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

$\displaystyle \Vert D^2u\Vert _{L^p(\Omega^\prime)}\leq C(\Vert u\Vert _{L^\infty(\Omega)}+\Vert f\Vert _{L^n(\Omega)}) $

for all solutions $ u\in C^2(\Omega)$ to the equation $ L_{\Phi}u=f$.

References [Enhancements On Off] (What's this?)

  • [C] Caffarelli, L. A. 1990. Interior $ W^{2,p}$ estimates for solutions to the Monge-Ampère equation. Ann. of Math. 131, 135-150. MR 1038360 (91f:35059)
  • [C90] Caffarelli, L. A. 1990. A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity. Ann. of Math. 131, 129-134. MR 1038359 (91f:35058)
  • [C91] Caffarelli, L. A. 1991. Some regularity properties of solutions of Monge-Ampère equation. Comm. Pure Appl. Math. 44, 965-969. MR 1127042 (92h:35088)
  • [CC95] Cafarelli, L. A. and Cabré, X. 1995. Fully nonlinear elliptic equations American Mathematical Society Colloquium Publications, volume 43. MR 1351007 (96h:35046)
  • [CG96] Caffarelli, L. A. and Gutiérrez, C. E. 1996. Real analysis related to the Monge-Ampère equation. Trans. A. M. S. 348(3), 1075-1092. MR 1321570 (96h:35047)
  • [CG97] Caffarelli L. A. and Gutiérrez C. E. 1997. Properties of the solutions of the linearized Monge-Ampère equation. Amer. J. Math. 119(2), 423-465. MR 1439555 (98e:35060)
  • [E85] Evans, L. C. 1985. Some estimates for nondivergence structure, second order elliptic equations. Trans. A. M. S. 287(2), 701-712. MR 0768735 (86g:35056)
  • [GT83] D. Gilbarg and N. S. Trudinger 1983. Elliptic Partial Differential Equations of Second Order. Springer-Verlag, New York. MR 0737190 (86c:35035)
  • [G01] C. E. Gutiérrez 2001. The Monge-Ampère Equation. Birkhaüser, Boston. MR 1829162 (2002e:35075)
  • [GH00] C. E. Gutiérrez and Qingbo Huang 2000. Geometric properties of the sections of solutions to the Monge-Ampère equation. Trans. A. M. S. 352, 4381-4396. MR 1665332 (2000m:35060)
  • [L86] Fang-Hua Lin 1986. Second derivative $ L^p$-estimates for elliptic equations of nondivergent type. Proc. Amer. Math. Soc. 96(3), 447-451. MR 0822437 (88b:35058)
  • [P78] Pogorelov, A.V. 1978. The Minkowski Multidimensional Problem. V.H. Winston and Sons, Washington. MR 0478079 (57:17572)
  • [Sc93] Schneider, R. 1993. Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge. MR 1216521 (94d:52007)
  • [TW00] Trudinger, N. S. and Xu-Jia Wang 2000. The Bernstein problem for affine maximal hypersurfaces. Invent. Math. 140(2), 399-422. MR 1757001 (2001h:53016)
  • [W95] Xu-Jia Wang 1995. Some counterexamples to the regularity of Monge-Ampère equations. Proc. Amer. Math. Soc. 123(3), 841-845. MR 1223269 (95d:35025)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35B45, 35J60, 35J70

Retrieve articles in all journals with MSC (2000): 35B45, 35J60, 35J70

Additional Information

Cristian E. Gutiérrez
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122

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

Keywords: A priori estimates of second derivatives, cross sections of solutions, viscosity solutions, nonuniformly elliptic equations
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.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society