AI 中文总结
该研究针对带斜导数边界条件的完全非线性抛物方程，结合紧性与多项式逼近方法，在不同源项可积性条件下得到解的边界正则性结果，包括Log-Lipschitz连续性、最优边界估计及Schauder型估计。
AI 中文摘要
我们研究带有斜导数边界条件的完全非线性抛物方程的粘性解的边界附近的精确正则性，该方程为：在Q₁⁺内满足F(D²u,x,t) - uₜ = f(x,t)，在Q₁*上满足β(x,t)·Du = g(x,t)。正则性理论是根据源项的可积性、边界数据的光滑性以及算子F的系数的振荡，结合紧性方法与多项式逼近发展起来的。在临界边界情形f∈L^{n+2}时，我们得到解在边界附近的Log-Lipschitz连续性；在更强的可积性下，即当f属于L^p（p>n+2）时，我们建立最优的C^{1+α', (1+α')/2}边界估计；在更高的正则性层面，我们在算子F和边界数据的适当假设下证明Schauder型估计。作为副产品，在源项属于BMO空间的临界边界情形，我们得到抛物型C^{1, Log-Lip}正则性。
英文摘要
We investigate the sharp regularity up to the boundary for viscosity solutions of fully nonlinear parabolic equations with oblique derivative boundary conditions given by \begin{equation*} \left\{ \begin{array}{rclcl} F(D^2u,x,t) - u_{t} &=& f(x,t) & \text{in} & Q_{1}^{+}, \\ β(x,t) \cdot Du &=& g(x,t) & \text{on} & Q_{1}^{*}. \end{array} \right. \end{equation*} The regularity theory is developed according to the integrability of the source term, the smoothness of the boundary data, and the oscillation of the coefficients of the operator \(F\), using a compactness method combined with polynomial approximation. In the borderline case \(f\in L^{n+2}\), we obtain Log-Lipschitz continuity of solutions up to the boundary. Under stronger integrability, specifically when \(f\in L^{p}\) for some \(p>n+2\), we establish optimal \(C^{1+α',\,\frac{1+α'}{2}}\) boundary estimates. At a higher regularity level, we prove Schauder-type estimates under appropriate assumptions on the operator \(F\) and the boundary data. As a byproduct, we obtain parabolic \(C^{1,\mathrm{Log\text{-}Lip}}\) regularity in the critical borderline case where the source term belongs to BMO space.
Comments30 pages
