Abstract

We study interior $C^{1, \alpha}$ regularity of viscosity solutions of the parabolic Monge-Amp\'ere equation $$ u_t = b(x,t)\,\big(\!\det D^2 u\big)^p, $$ with exponent $p >0$ and with coefficients $b$ which are bounded and measurable. We show that when $p$ is less than the critical power $1\over{n-2}$ then solutions become instantly $C^{1,\alpha}$ in the interior. Also, we prove the same result for any power $p>0$ at those points where either the solution separates from the initial data, or where the initial data is $C^{1,\beta}$.

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Additional Information

ISSN
1080-6377
Print ISSN
0002-9327
Pages
pp. 1051-1087
Launched on MUSE
2012-07-25
Open Access
No
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