We prove a global $ W^{2, p} $-estimate for the viscosity solution to fully nonlinear elliptic equations $ F(x, u, Du, D^{2}u) = f(x) $ with oblique boundary condition in a bounded $ C^{2, \alpha} $-domain for every $ \alpha\in (0, 1) $. Here, the nonlinearities $ F $ is assumed to be asymptotically $ \delta $-regular to an operator $ G $ that is $ (\delta, R) $-vanishing with respect to $ x $. We employ the approach of constructing a regular problem by an appropriate transformation. With a similar argument, we also obtain a global $ W^{2, p} $-estimate for the viscosity solution to fully nonlinear parabolic equations $ F(x, t, u, Du, D^{2}u)-u_{t} = f(x, t) $ with oblique boundary condition in a bounded $ C^{3} $-domain.

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$ W^{2, p} $-regularity 用于渐近正则完全非线性椭圆和抛物线方程的倾斜边界值

我们证明了一个全局 $ W^{2, p} $ - 对完全非线性椭圆方程 $ F(x, u, Du, D^{2}u) = f(x) $ 的粘性解的估计，具有倾斜边界条件在一个有界的 $ C^{2, \alpha} $-domain 中，对于每个 $ \alpha\in (0, 1) $。在这里，非线性$ F $ 被假定为渐近$ \delta $ - 正则于算子$ G $，即$ (\delta, R) $ - 相对于$ x $ 消失。我们采用通过适当的转换来构建正则问题的方法。通过类似的论证，我们还获得了全非线性抛物线方程的粘度解的全局 $W^{2, p} $-estimate $ F(x, t, u, Du, D^{2}u)-u_ {t} = f(x, t) $ 在有界 $ C^{3} $ 域中具有倾斜边界条件。