The Minkowski Content Of Some Curve

We say that the set is Minkowski measurable if the Minkowski content of the set exists. The Minkowski contents of the sets may be finite or infi-nite(possibly zero), which is concerned by many scholars. Especially, in the paper that Lapidus studied the Weyl-Berry conjecture of the fractal drum ,he transformed the problem of estimating eigenvalue numbers of the Laplace's equation into the problem of calculating the Minkowski content on the fractal boundary. So the study of the Minkowski contents of some sets becomes the work to be paid closely attention to. Now many good results have been obtained for the Minkowski contents of some sets in R. For example, Associate Professor Shirong Chen in CCNU studied the countable sets in the interval [0, 1] and obtained its Minkowski content and dimension. Besides, Falconer and Lapidus also made some good researches on them. But little work has been carried out for Minkowski contents of the sets in Rn(n > 2).For example, Von Koch curve, Cantor dust and Sierpinski gasket are the famous fractal sets. But their Minkowski contents are unsolved yet. In this paper, we study Von Koch curve and Cantor dust firstly. We give some properties of Von Koch curve and Cantor dust and then get estimation formulae of upper and lower Minkowski contents of them respectively by some skill. Thus we easily obtain that the Minkowski contents of them don't exist. Besides, through another definition and method, we calculate the upper Minkowski content M*(d,E) and the lower Minkowski content M*(d,E) of Sierpinski gasket E respectively. That is, M*(d,E) ≈ 1.814 and M*(d,E) ≈ 1.811. Therefore we obtain that Sierpinski gasket is not Minkowski measurable by a new method.