Some counterexamples to the regularity of Monge-Ampère equations
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- by Xu Jia Wang PDF
- Proc. Amer. Math. Soc. 123 (1995), 841-845 Request permission
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 $f(x)$.References
- L. A. Caffarelli, A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. of Math. (2) 131 (1990), no. 1, 129–134. MR 1038359, DOI 10.2307/1971509
- Luis A. Caffarelli, Interior $W^{2,p}$ estimates for solutions of the Monge-Ampère equation, Ann. of Math. (2) 131 (1990), no. 1, 135–150. MR 1038360, DOI 10.2307/1971510
- Luis A. Caffarelli, Some regularity properties of solutions of Monge Ampère equation, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 965–969. MR 1127042, DOI 10.1002/cpa.3160440809 X. J. Wang, Remarks on the regularity of Monge-Ampère equations, Proceedings of the International Conference on Nonlinear P.D.E (Hangzhou, 1992) (F. H. Lin and G. C. Dong, eds.), Academic Press, Beijing, 1992.
Additional Information
- © Copyright 1995 American Mathematical Society
- 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