臺灣博碩士論文加值系統

大家都知道若 E 是一個局部有限周長的集合 (set with locally
finite perimeter)，則 E 的本質邊界 (essential boundary) 和縮減邊
界(reduced boundary) 的差，其 (n-1) 維的 Hausdorff 測度是零。兩
個問題自然的產生： 1.上述的式子是否可
以改善到E的邊界和縮減邊界的差，其 (n-1)維的 Hausdorff 測度 是
零？
2.對於哪些函數的等高線集，會滿足 1 的敘述？
這篇論文的第一部份 (第一章和第二章) 致力於解決上述的兩個問題。
這篇論文的第二部份 (附錄A和附錄B) 中，提供了另一種方式去證明等周
長不等式 (isoperimetric inequality) 以及重新定義了縮減邊界，並證
明此新定義下的縮減邊界對於 (n-1) 維 Hausdorff 測度而言是可測的
(measurable)，此解決了原定義下的縮減邊 界對於 (n-1) 維 Hausdorff
測度而言是可測的問題。

It is well known that if E is a set with locally finite
perimeter (or a Caccioppoli set), then the (n-1)-dimensional
Hausdorff measure of the difference between essential
boundary of E and reduced boundary of E is zero. Two questions
rise naturally:
1.Can we have the refined equality that the (n-1)-dimensional
Hausdorff measure of the difference between the boundary
of E and the reduced boundary of E is zero?
2.For what kind of functions f can we have the level sets of f
satisfy the statement in 1?
This thesis will deal with these problems in chapter 1 and 2.
In appendix A and B, we give another proof of isoperimetric
inequality, and redefine the reduced boundary of a set. We
prove that the reduced boundaryunder new definition is (n-1)-
dimensional Hausdorff measure measurable. This solves the
measurability of the reduced boundary under original definition
with respect to (n-1)-dimensional Hausdorff measure.