在离散盒的边界处的离散调和函数的finite differences上的$L^p$比较，$p\in (1,\infty)$
An $L^p$-comparison, $p\in (1,\infty)$, on the finite differences of a discrete harmonic function at the boundary of a discrete box
摘要：
众所周知，对于定义在$d$维欧氏空间$d\geq 2$的单位球上的调和函数$u$，球上梯度$\nabla u$的切向分量和法向分量可通过$L^p$范数$p\in(1,\infty)$进行比较，直到乘法常数仅取决于$d,p$ .本文制定并证明了一个离散模拟这一结果的离散调和函数定义在一个离散框上的$d$ -维晶格与乘法常数，不依赖于大小的盒子。
英文摘要：
It is well-known that for a harmonic function $u$ defined on the unit ball of the $d$-dimensional Euclidean space, $d\geq 2$, the tangential and normal component of the gradient $\nabla u$ on the sphere are comparable by means of the $L^p$-norms, $p\in(1,\infty)$, up to multiplicative constants that depend only on $d,p$. This paper formulates and proves a discrete analogue of this result for discrete harmonic functions defined on a discrete box on the $d$-dimensional lattice with multiplicative constants that do not depend on the size of the box.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
