一类分形集上的Dirichlet形式研究
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摘要
本文首先讨论了在研究分形集时我们用到的一个新的重要的方法——Dirichlet分析方法，其是最近在研究“分形分析”中提出来的一种研究分形集的很有用的工具，它本身类似于我们在研究经典数学分析中能量的定义形式。我们首先给出分形集上Dirichlet形式的定义以及在本论文中将要用到的一些定理和主要结论。通过上述这些有用的方法和工具，我们研究了定义在自相似分形空间上的调和函数的Dirichlet形式的性质，我们得到的主要结论是将定义在自相似分形集上的Dirichlet形式进行分解，分别研究部分的分形空间上的方向Dirichlet形式的性质，得到了一个更进一步的结果，即其上的Dirichlet形式划分具有加权平均值的性质，这和在经典的数学分析中多元函数的积分与各个自变量之间的积分具有本质的区别。由于分形上的分析研究的基本内容是在分形集上建立微分方程，并且在分形集上建立导数的概念是有困难的，因而我们转而考虑在分形集上定义Laplace算子，这其中也将用到我们给出的Dirichlet形式定义，所以我们在本文中给出的Dirichlet形式的一些性质在分形分析中是很有必要的。
In this paper, we firstly discuss a new and important method during studying of fractal set—Dirichlet analysis method,Which is a more useful tool intrduced on analysis of fractal recently and is similar to energy form on mathematics analysis. We firstly give the definition of Dirichlet form on fractal set and some therom which is useful to our paper.Through this useful methods and tools, we study the characteristics Dirichlet form of harmonic function defined on self-similar fractal space. The main result of this paper is that the Dirichlet form defined on self-similar fractal sets can be resolved into several part Dirichlet forms.The relation of the total Dirichlet form and part Dirichlet form is that the former can be noted by laters with lines. This result is very different from the relationship in classic mathematics analysis. Though main content on fractal analysis is contruction of differential equations on fractal sets,and the contruction of definition of differential is more difficult, we consider the the definition of Laplace operator, which will use the the definition and characteristics of Dirichlet form. So some characteristics of Dirichlet form is more necessary on fractal analysis.
In this paper, we firstly discuss a new and important method during studying of fractal set—Dirichlet analysis method,Which is a more useful tool intrduced on analysis of fractal recently and is similar to energy form on mathematics analysis. We firstly give the definition of Dirichlet form on fractal set and some therom which is useful to our paper.Through this useful methods and tools, we study the characteristics Dirichlet form of harmonic function defined on self-similar fractal space. The main result of this paper is that the Dirichlet form defined on self-similar fractal sets can be resolved into several part Dirichlet forms.The relation of the total Dirichlet form and part Dirichlet form is that the former can be noted by laters with lines. This result is very different from the relationship in classic mathematics analysis. Though main content on fractal analysis is contruction of differential equations on fractal sets,and the contruction of definition of differential is more difficult, we consider the the definition of Laplace operator, which will use the the definition and characteristics of Dirichlet form. So some characteristics of Dirichlet form is more necessary on fractal analysis.
引文
[1] B. B. Mandelbrot. Fractals: form, chance, and dimension[M]. California: Freeman, 1997.
[2] B. B. Mandelbrot. The Fractal Geometry of Nature[M]. California: Freeman, 1982.
[3] 肯尼思．法尔科内(英)．曾文曲等译 分形几何—数学基础及其应用[M]．沈阳：东北大学出版社 1991
[4] J.Hutchinson. Fractals and self-similarity[J] Indiana Univ. Math. J. 30 (1981) 713-747.
[5] K. Falconer. Technique in Fractal geometry[M].New York: wiley 1995
[6] C. Bandt and Graf. Self-similar sets Ⅶ. A characterization of self-Similar fractals with positive Hausdorff measure[]]. Proc. Amer. Math. Soc. 114(1999) 319-344.
[7] 文志英．分形几何的数学基础[M]．上海：上海科技教育出版社 1992
[8] J.Kigami. A Harmonic Calculus on the Sierpinski Spaces[J].Japan J. Appl. Math 6(1989).259-290
[9] J. Kigami. Harmonic Calculus on p.c.f, self-similar sets[J].Trans. Amer. Math. Soc. 1993,355:721-755
[10] J. Kigami. Analysis on fractals[M]. London: Cambridge Univ. Press. 2001
[11] Strichartz R. Some Properties of Laplacian on Fractals[J]. J. Funct. Anal. 1999,164: 181-208
[12] Strichartz R. Taylor Approximations on Sierpinski Gasket Type Fractals[J]. J. Funct. Anal. 2000,174(1): 76-127
[13] Strichartz R. The Laplacian on the Sierpinski Gasket via the method of averages[J]. Pracific J. Math. 2001,201(1): 241-256
[14] Strichartz R. and M. Usher. Splines on fractals[J]. Math. Proc. Camb. Phil. Soc. 129 (2000), 331-360
[15] K. S. Lau, S. M. Ngai and H. Rao. Iterated function systems with overlaps and self-similar mmeasures[J]. J. London. Math. Soc. (2) 63(2001), no. 1,99-116
[16] K. S. Lau and X. Y. Wang. Iterated function systems with a weak separation condition[J]. Studia Math. 161 (2004), no. 3, 249-268
[17] S. M. Ngai and Y. Wang. Hausdorff dimension of Self-similar sets with Overlaps [J]. J. London Math. Soc. (2) 63 (2001), no. 3, 655-672
[18] 谢和平，薛秀谦，分形应用中的数学基础与方法[M]，北京：科学出版社1998
[19] 金以文，鲁世杰，分形几何原理及其应用[M]，杭州：浙江大学出版社1998
[20] 文志英，井竹君，分形几何和分维数[J]，数学的实践与认识，第四期1995
[21] 苏维宜，分形与局部紧群上的调和分析，非线性论文集1990
[22] M. Yamaguti and J. Kigami, Some remarks on Dirichlet problem of Poisson Equation. Analysis Mathematics and Application, Gauthier-Villars, Paris,1988,465-471
[23] P. A. P. Moran, Additive functions of interval and Hausdroff measure,Proc. Camb. Philos. Soc, 42(1946),15-23
[24] 李文侠，一类准自相似集的研究[J]，数学年刊，15A，6(1994)
[25] V. Metz, The Laplacian of the Hany fractal, Arab, J. Sci. Eng. Sect.C Theme Issues 28 (IC), 199-211
[26] J. Kigami, A Harmonic calculus on the Sierpinski spaces, Japan J. Appl.Math. 6, 259-290, 1989
[27] B. M. Hambly, Heat kernels and spectral asymototics for some random Sierpinski gaskets, Fractal Geometry and Stochastics Ⅱ (C. Bandt et al, eds),Progress in Probability, vol. 46, Birkhauser, pp. 239-267,2000
[28] 吴敏，关于自相似集的一个维数定理[J]，数学学报，第38卷，第3期1995
[29] J. Kigami, Effective resistances for harmonic structures on p.c.f.Semi-similar sets, Math. Proc. Cambridge Phil. Soc. 115, 291-303,1994
[30] A. Teplyaev, Energy and Lapacian on the Sierpinski gasket, Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Part 1, Proc.Sympos. Pure Math. 72, Amer. Math. Soc. 131-154 2004
[31] 江惠坤，R~n上分形集的多重维数[J]，数学学报，第38卷，第3期，1995
[32] C. Sabot, Existence and uniqueness of diffusions on finitely ramified self-Similar fractals, Ann. Scient. Ec. Norm. Sup. 4eme serie 30, 605-673, 1997
[33] R. Strichartz, Mock Fouriers series and transforms associated with certain Cantor measures, Journal d' Analyse Mathematicsc 81, 209-238, 2000
[34] 郑维行，王声望，实变函数与泛函分析概要，北京：人民教育出版社，1980
[35] R. Meyers, R. Strichartz and A. Teplyaev, Dirichlet forms on the Sierpinski Gasket, Pacific Journal of Mathematics 217, 149-174. 2004
[36] Lau and Ngai,Multifractal measures and a weak separation condition,Adv. Math.141 pp 45-96 1999
[37] M. T. Barlow and E. A. Perkins, Brownian Motion on the Sierpinski gasket, Probab. Th.Rel. Fields 79, 543-623 1988
[2] B. B. Mandelbrot. The Fractal Geometry of Nature[M]. California: Freeman, 1982.
[3] 肯尼思．法尔科内(英)．曾文曲等译 分形几何—数学基础及其应用[M]．沈阳：东北大学出版社 1991
[4] J.Hutchinson. Fractals and self-similarity[J] Indiana Univ. Math. J. 30 (1981) 713-747.
[5] K. Falconer. Technique in Fractal geometry[M].New York: wiley 1995
[6] C. Bandt and Graf. Self-similar sets Ⅶ. A characterization of self-Similar fractals with positive Hausdorff measure[]]. Proc. Amer. Math. Soc. 114(1999) 319-344.
[7] 文志英．分形几何的数学基础[M]．上海：上海科技教育出版社 1992
[8] J.Kigami. A Harmonic Calculus on the Sierpinski Spaces[J].Japan J. Appl. Math 6(1989).259-290
[9] J. Kigami. Harmonic Calculus on p.c.f, self-similar sets[J].Trans. Amer. Math. Soc. 1993,355:721-755
[10] J. Kigami. Analysis on fractals[M]. London: Cambridge Univ. Press. 2001
[11] Strichartz R. Some Properties of Laplacian on Fractals[J]. J. Funct. Anal. 1999,164: 181-208
[12] Strichartz R. Taylor Approximations on Sierpinski Gasket Type Fractals[J]. J. Funct. Anal. 2000,174(1): 76-127
[13] Strichartz R. The Laplacian on the Sierpinski Gasket via the method of averages[J]. Pracific J. Math. 2001,201(1): 241-256
[14] Strichartz R. and M. Usher. Splines on fractals[J]. Math. Proc. Camb. Phil. Soc. 129 (2000), 331-360
[15] K. S. Lau, S. M. Ngai and H. Rao. Iterated function systems with overlaps and self-similar mmeasures[J]. J. London. Math. Soc. (2) 63(2001), no. 1,99-116
[16] K. S. Lau and X. Y. Wang. Iterated function systems with a weak separation condition[J]. Studia Math. 161 (2004), no. 3, 249-268
[17] S. M. Ngai and Y. Wang. Hausdorff dimension of Self-similar sets with Overlaps [J]. J. London Math. Soc. (2) 63 (2001), no. 3, 655-672
[18] 谢和平，薛秀谦，分形应用中的数学基础与方法[M]，北京：科学出版社1998
[19] 金以文，鲁世杰，分形几何原理及其应用[M]，杭州：浙江大学出版社1998
[20] 文志英，井竹君，分形几何和分维数[J]，数学的实践与认识，第四期1995
[21] 苏维宜，分形与局部紧群上的调和分析，非线性论文集1990
[22] M. Yamaguti and J. Kigami, Some remarks on Dirichlet problem of Poisson Equation. Analysis Mathematics and Application, Gauthier-Villars, Paris,1988,465-471
[23] P. A. P. Moran, Additive functions of interval and Hausdroff measure,Proc. Camb. Philos. Soc, 42(1946),15-23
[24] 李文侠，一类准自相似集的研究[J]，数学年刊，15A，6(1994)
[25] V. Metz, The Laplacian of the Hany fractal, Arab, J. Sci. Eng. Sect.C Theme Issues 28 (IC), 199-211
[26] J. Kigami, A Harmonic calculus on the Sierpinski spaces, Japan J. Appl.Math. 6, 259-290, 1989
[27] B. M. Hambly, Heat kernels and spectral asymototics for some random Sierpinski gaskets, Fractal Geometry and Stochastics Ⅱ (C. Bandt et al, eds),Progress in Probability, vol. 46, Birkhauser, pp. 239-267,2000
[28] 吴敏，关于自相似集的一个维数定理[J]，数学学报，第38卷，第3期1995
[29] J. Kigami, Effective resistances for harmonic structures on p.c.f.Semi-similar sets, Math. Proc. Cambridge Phil. Soc. 115, 291-303,1994
[30] A. Teplyaev, Energy and Lapacian on the Sierpinski gasket, Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Part 1, Proc.Sympos. Pure Math. 72, Amer. Math. Soc. 131-154 2004
[31] 江惠坤，R~n上分形集的多重维数[J]，数学学报，第38卷，第3期，1995
[32] C. Sabot, Existence and uniqueness of diffusions on finitely ramified self-Similar fractals, Ann. Scient. Ec. Norm. Sup. 4eme serie 30, 605-673, 1997
[33] R. Strichartz, Mock Fouriers series and transforms associated with certain Cantor measures, Journal d' Analyse Mathematicsc 81, 209-238, 2000
[34] 郑维行，王声望，实变函数与泛函分析概要，北京：人民教育出版社，1980
[35] R. Meyers, R. Strichartz and A. Teplyaev, Dirichlet forms on the Sierpinski Gasket, Pacific Journal of Mathematics 217, 149-174. 2004
[36] Lau and Ngai,Multifractal measures and a weak separation condition,Adv. Math.141 pp 45-96 1999
[37] M. T. Barlow and E. A. Perkins, Brownian Motion on the Sierpinski gasket, Probab. Th.Rel. Fields 79, 543-623 1988