论文标题
Laplacian在整个偏移空间上的表述较弱
Weak Formulation of the Laplacian on the Full Shift Space
论文作者
论文摘要
我们考虑在有限符号集上的单方面全班空间上的Laplacian，该符号是有限差算子的重新规定限制的。我们通过选择测试函数作为具有有限能量并在各种边界集中消失的测试函数来提出对这种Laplacian的弱定义，类似于微积分中的laplacian。在移动空间的抽象设置中，选择边界集是定义有限差异操作员的集合。然后，我们定义了在这些边界集上的函数的诺伊曼衍生物，并在迄今为止分析中的三个重要概念之间建立了关系，即拉普拉斯式，双线性能量形式和函数的neumann衍生物。结果，我们获得了类似于经典情况下的高斯绿色公式。我们通过为偏移空间上的诺伊曼边界价值问题提供足够条件来结束本文。
We consider a Laplacian on the one-sided full shift space over a finite symbol set, which is constructed as a renormalized limit of finite difference operators. We propose a weak definition of this Laplacian, analogous to the one in calculus, by choosing test functions as those which have finite energy and vanish on various boundary sets. In the abstract setting of the shift space, the boundary sets are chosen to be the sets on which the finite difference operators are defined. We then define the Neumann derivative of functions on these boundary sets and establish a relation between three important concepts in analysis so far, namely, the Laplacian, the bilinear energy form and the Neumann derivative of a function. As a result, we obtain the Gauss-Green's formula analogous to the one in classical case. We conclude this paper by providing a sufficient condition for the Neumann boundary value problem on the shift space.
