Regularity of weak solutions to the Monge-Ampère equation

Authors:
Cristian E. Gutiérrez and David Hartenstine

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

Published electronically:
January 14, 2003

MathSciNet review:
1973999

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Abstract: We study the properties of generalized solutions to the Monge-Ampère equation , where the Borel measure satisfies a condition, introduced by Jerison, that is weaker than the doubling property. When , this condition, which we call , admits the possibility of vanishing or becoming infinite. Our analysis extends the regularity theory (due to Caffarelli) available when , which implies that is doubling. The main difference between the case and the case when 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.

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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

DOI:
https://doi.org/10.1090/S0002-9947-03-03231-8

Keywords:
Aleksandrov solutions,
strict convexity,
Hausdorff metric,
doubling property,
H\"older estimates

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.

Article copyright:
© Copyright 2003
American Mathematical Society