Regularity of weak solutions to the Monge–Ampère equation
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- by Cristian E. Gutiérrez and David Hartenstine PDF
- Trans. Amer. Math. Soc. 355 (2003), 2477-2500 Request permission
Abstract:
We study the properties of generalized solutions to the Monge–Ampère equation $\det D^2 u = \nu$, where the Borel measure $\nu$ satisfies a condition, introduced by Jerison, that is weaker than the doubling property. When $\nu = f dx$, this condition, which we call $D_{\epsilon }$, admits the possibility of $f$ vanishing or becoming infinite. Our analysis extends the regularity theory (due to Caffarelli) available when $0 < \lambda \leq f \leq \Lambda < \infty$, which implies that $\nu = f dx$ is doubling. The main difference between the $D_{\epsilon }$ case and the case when $f$ is bounded between two positive constants is the need to use a variant of the Aleksandrov maximum principle (due to Jerison) and some tools from convex geometry, in particular the Hausdorff metric.References
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Additional Information
- Cristian E. Gutiérrez
- Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
- Email: gutierrez@math.temple.edu
- David Hartenstine
- Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
- Email: hartenst@math.utah.edu
- Received by editor(s): March 3, 2002
- Received by editor(s) in revised form: October 7, 2002
- Published electronically: January 14, 2003
- Additional Notes: The first author was partially supported by NSF grant DMS–0070648.
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 2477-2500
- MSC (2000): Primary 35D10, 35J65, 35J60
- DOI: https://doi.org/10.1090/S0002-9947-03-03231-8
- MathSciNet review: 1973999