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Transactions of the American Mathematical Society

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

 

 

Regularity of solutions of two-dimensional Monge-Ampère equations


Authors: Friedmar Schulz and Liang Yuan Liao
Journal: Trans. Amer. Math. Soc. 307 (1988), 271-277
MSC: Primary 35J60; Secondary 35B65
DOI: https://doi.org/10.1090/S0002-9947-1988-0936816-1
MathSciNet review: 936816
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Abstract: In the paper we investigate the regularity of solutions $ z(x,\,y) \in {C^{1,1}}(\Omega )$, resp. $ {C^{1,1}}(\overline \Omega )$ of elliptic Monge-Ampére equations of the form

$\displaystyle Ar + 2Bs + Ct + (rt - {s^2}) = E.$

It is shown that $ z(x,\,y) \in {C^{2,\alpha }}(\Omega )$, resp. $ {C^{2,\alpha }}(\overline \Omega )$, with corresponding a priori estimates, if $ A,\,B,\,C,\,E \in {C^\alpha }(\Omega \times {{\mathbf{R}}^3})$. The results are deduced via the Campanato technique for equations of variational structure invoking a Legendre-like transformation.

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DOI: https://doi.org/10.1090/S0002-9947-1988-0936816-1
Article copyright: © Copyright 1988 American Mathematical Society