满移位空间上Laplacian算子的弱形式
Weak Formulation of the Laplacian on the Full Shift Space
摘要：
考虑有限符号集上单侧全移位空间上的拉普拉斯算子，它被构造为有限差分算子的重整化极限。我们提出了一个弱定义的拉普拉斯算子，类似于一个在微积分，通过选择检验函数作为那些具有有限的能量和消失的各种边界集。在移位空间的抽象设置中，边界集被选择为有限差分算子定义于其上的集合。然后，我们定义的诺依曼导数的功能，这些边界集，并建立了三个重要的概念之间的关系分析到目前为止，即拉普拉斯算子，双线性能量形式和诺依曼导数的功能。结果得到了类似于经典情形的高斯-格林公式。最后给出了移位空间上Neumann边值问题的一个充分条件。
英文摘要：
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.
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