$ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values
$ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values
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DOI：
10.3934/dcdss.2021080
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发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Junjie Zhang;Shenzhou Zheng;Chunyan Zuo
中科院分区：
文献类型：
--
作者：
Junjie Zhang;Shenzhou Zheng;Chunyan Zuo
证明了具有斜边界条件的完全非线性椭圆型方程{Begin{Document}$F(x，u，Du，D^{2}u)=f(X)$\end{Document}在有界的\Begin{Document}$C^{2，α}$\end{Document}-区域上，对(0，1)$\end{Document}中的每个有界域，证明了其粘性解的全局估计.这里，假设非线性\Begin{Document}$F$\end{Document}渐近地\Begin{Document}$\Delta$\end{Document}-正则于算子\Begin{Document}$G$\end{Document}，即\Begin{Document}$(\Delta，R)$\end{Document}-相对于\Begin{Document}$x$\end{Document}消失。我们采用了通过适当的变换来构造正则问题的方法。在类似的情况下，我们还得到了在有界的{DOCUMENT}$C^C^{3}$END{DOCUMENT}-区域中具有斜边界条件的完全非线性抛物型方程的粘性解的全局BEGIN{DOCUMENT}$W^{2，p}$END{DOCUMENT}-估计。
We prove a global \begin{document}$ W^{2, p} $\end{document}-estimate for the viscosity solution to fully nonlinear elliptic equations \begin{document}$ F(x, u, Du, D^{2}u) = f(x) $\end{document} with oblique boundary condition in a bounded \begin{document}$ C^{2, \alpha} $\end{document}-domain for every \begin{document}$ \alpha\in (0, 1) $\end{document}. Here, the nonlinearities \begin{document}$ F $\end{document} is assumed to be asymptotically \begin{document}$ \delta $\end{document}-regular to an operator \begin{document}$ G $\end{document} that is \begin{document}$ (\delta, R) $\end{document}-vanishing with respect to \begin{document}$ x $\end{document}. We employ the approach of constructing a regular problem by an appropriate transformation. With a similar argument, we also obtain a global \begin{document}$ W^{2, p} $\end{document}-estimate for the viscosity solution to fully nonlinear parabolic equations \begin{document}$ F(x, t, u, Du, D^{2}u)-u_{t} = f(x, t) $\end{document} with oblique boundary condition in a bounded \begin{document}$ C^{3} $\end{document}-domain.