$C^{1,\al}$ regularity of solutions to parabolic Monge-Amp\'ere equations
$C^{1,\al}$ regularity of solutions to parabolic Monge-Amp\'ere equations
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$C^{1,al}$ 抛物线 Monge-Amper 方程解的正则性
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发表时间：
2009
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通讯作者：
O. Savin
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作者：
P. Daskalopoulos;O. Savin
研究了指数为$p >0$且系数为$b$的抛物型monge - ampsamre方程$$u_t = b(x,t) \ddua,$$黏度解的内部$C^{1, \al}$正则性。我们表明，当$p$小于临界功率$\frac{1}{n-2}$时，解立即在内部变为$C^{1, \al}$。同样，我们在解与初始数据分离或初始数据为$C^{1, \beta}$的点上，对任意次幂$p>0$证明了相同的结果。
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 $C^{1, \beta}$.