论文标题
标准化$ p $ - parabolic运算符的边界梯度连续性
Borderline gradient continuity for the normalized $p$-parabolic operator
论文作者
论文摘要
在本文中，我们证明了粘度解决方案的梯度连续性估算值为$δ_{p}^n u- u_t = f $，就缩放量表而言，关键$ l（n+2,1）$ quarm of $ f $，其中$δ_{p}^n $是游戏理论$ p-$ p-$ p-$ p-$ p-$ laplacian plaplacian Pertication nevernection nevernection p-$ p-$ laplacian pelector necnection（1.2）（1.2）。我们的主要结果，定理2.5构成了$ u $的边界梯度连续性估计，就修改后的抛物线riesz潜在$ \ mathbf {p}^{p}^{f} _ {n+1} $，如下（2.8）中所定义。此外，对于$ f \ in l^{m} $，带有$ m> n+2 $，我们还获得了解决方案$ u $的空间梯度的Hölder连续性，请参见下面的定理2.6。这改善了[3]梯度Hölder的连续性结果，它考虑了有限的$ f $。我们的主要结果定理2.5和定理2.6是[9]中的抛物线类似物。此外，在[3]中，我们的方法与ISHII-LIONS方法无关，该方法在[3]中至关重要地用于获得均匀摄取方程的Lipschitz估计值作为中间步骤。
In this paper, we prove gradient continuity estimates for viscosity solutions to $Δ_{p}^N u- u_t= f$ in terms of the scaling critical $L(n+2,1 )$ norm of $f$, where $Δ_{p}^N$ is the game theoretic normalized $p-$Laplacian operator defined in (1.2) below. Our main result, Theorem 2.5 constitutes borderline gradient continuity estimate for $u$ in terms of the modified parabolic Riesz potential $\mathbf{P}^{f}_{n+1}$ as defined in (2.8) below. Moreover, for $f \in L^{m}$ with $m>n+2$, we also obtain Hölder continuity of the spatial gradient of the solution $u$, see Theorem 2.6 below. This improves the gradient Hölder continuity result in [3] which considers bounded $f$. Our main results Theorem 2.5 and Theorem 2.6 are parabolic analogues of those in [9]. Moreover differently from that in [3], our approach is independent of the Ishii-Lions method which is crucially used in [3] to obtain Lipschitz estimates for homogeneous perturbed equations as an intermediate step.
