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Some counterexamples to the regularity of Monge-Ampère equations


Author: Xu Jia Wang
Journal: Proc. Amer. Math. Soc. 123 (1995), 841-845
MSC: Primary 35B65; Secondary 35J60
DOI: https://doi.org/10.1090/S0002-9939-1995-1223269-0
MathSciNet review: 1223269
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Abstract: We present examples to show that the solution u of the Monge-Ampère equation $ \det ({D^2}u) = f(x)$, with $ u = 0$ on the boundary, may not lie in $ {W^{2,p}}$ or in $ {C^{1,\alpha }}$ for noncontinuous and positive $ f(x)$ and for continuous and nonnegative $ f(x)$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1223269-0
Keywords: Monge-Ampère equations, elliptic regularity, convexity
Article copyright: © Copyright 1995 American Mathematical Society