$C^{1,\al}$ regularity of solutions to parabolic Monge-Ampére equations

Details $C^{1,\al}$ regularity of solutions to parabolic Monge-Ampére equations $C^{1,\al}$ regularity of solutions to parabolic Monge-Ampére equations

2009-05-11

arxiv.org/abs/0905.1685v1

Mathematics

Analysis of PDEs

Scientific paper

We study interior $C^{1, \al}$ regularity of viscosity solutions of the parabolic Monge-Amp\'ere equation $$u_t = b(x,t) \ddua,$$ with exponent $p >0$ and with coefficients $b$ which are bounded and measurable. We show that when $p$ is less than the critical power $\frac{1}{n-2}$ then solutions become instantly $C^{1, \al}$ 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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