Some Properties of the Derivatives on Sierpinski Gasket Type Fractals
Tóm tắt
In this paper, we focus on Strichartz’s derivatives, a family of derivatives including the normal derivative, on post critically finite fractals, which are defined at vertices in the graphs that approximate the fractal. We obtain a weak continuity property of the derivatives for functions in the domain of the Laplacian. For a function with zero normal derivative at any fixed vertex, the derivatives, including the normal derivatives, of the neighboring vertices will decay to zero. The rates of approximations are described, and several nontrivial examples are provided to illustrate that our estimates are optimal. We also study the boundedness property of derivatives for functions in the domain of the Laplacian. A necessary condition for a function having a weak tangent of order one at a vertex is provided. Furthermore, we give a counterexample of a conjecture of Strichartz on the existence of higher-order weak tangents.
Tài liệu tham khảo
Barlow, M.T., Kigami, J.: Localized eigenfunctions of the Laplacian on p.c.f. self-similar sets. J. Lond. Math. Soc. 56, 320–332 (1997)
Ben-Bassat, O., Strichartz, R.S., Teplyaev, A.: What is not in the domain of the Laplacian on Sierpinski gasket type fractals. J. Funct. Anal. 166, 197–217 (1999)
Ben-Gal, N., Shaw-Krauss, A., Strichartz, R.S., Young, C.: Calculus on the Sierpinski gasket. II. Point singularities, eigenfunctions, and normal derivatives of the heat kernel. Trans. Am. Math. Soc. 358(9), 3883–3936 (2006)
Cipriani, F., Guido, D., Isola, T., Sauvageot, J.L.: Integrals and potentials of differential 1-forms on the Sierpinski gasket. Adv. Math. 239, 128–163 (2013)
Cipriani, F., Guido, D., Isola, T., Sauvageot, J.L.: Spectral triples for the Sierpinski gasket. J. Funct. Anal. 266, 4809–4869 (2014)
Cipriani, F., Sauvageot, J.L.: Derivations as square roots of Dirichlet forms. J. Funct. Anal. 201, 78–120 (2003)
Dalrymple, K., Strichartz, R.S., Vinson, J.P.: Fractal differential equations on the Sierpinski gasket. J. Fourier Anal. Appl. 5, 203–284 (1999)
DeGrado, J.L., Rogers, L.G., Strichartz, R.S.: Gradients of Laplacian eigenfunctions on the Sierpinski gasket. Proc. Am. Math. Soc. 137(2), 531–540 (2009)
Guo, Z., Kogan, R., Qiu, H., Strichartz, R.S.: Boundary value problems for a family of domains in the Sierpinski gasket. Ill. J. Math. 58(2), 497–519 (2014)
Hinz, M., Teplyaev, A.: Dirac and magnetic Schrödinger operators on fractals. J. Funct. Anal. 265(11), 2830–2854 (2013)
Hinz, M., Teplyaev, A.: Vector analysis on fractals and applications. Contemp. Math. 601, 147–163 (2013)
Hinz, M., Teplyaev, A.: Local Dirichlet forms, Hodge theory, and the Navier-Stokes equations on topologically one-dimensional fractals. Trans. Am. Math. Soc. 367(2), 1347–1380 (2015)
Ionescu, M., Pearse, E.P.J., Rogers, L.G., Ruan, H., Strichartz, R.S.: The resolvent kernel for p.c.f. self-similar fractals. Trans. Am. Math. Soc. 362(8), 4451–4479 (2010)
Ionescu, M., Rogers, L.G., Teplyaev, A.: Derivations and Dirichlet forms on fractals. J. Funct. Anal. 263, 2141–2169 (2012)
Kigami, J.: A harmonic calculus on the Sierpinski spaces. Jpn. J. Appl. Math. 6, 259–290 (1989)
Kigami, J.: Harmonic calculus on p.c.f. self-similar sets. Trans. Am. Math. Soc. 335, 721–755 (1993)
Kigami, J.: Harmonic metric and Dirichlet form on the Sierpinski gasket. In: asymptotic problems in probability theory: stochastic models and diffusions on fractals, Sanda/Kyoto, 1990. Pitman Research Notes in Mathematical Series, vol. 283, pp. 201–218, Longman, Harlow (1993)
Kigami, J.: Effective resistance for harmonic structures on p.c.f. self-similar sets. Math. Proc. Camb. Philos. Soc. 115, 291–303 (1994)
Kigami, J.: Harmonic calculus on limits of networks and its application to dendrites. J. Funct. Anal. 128, 48–86 (1995)
Kigami, J.: Distributions of localized eigenvalues of Laplacian on p.c.f. self-similar sets. J. Funct. Anal. 156, 170–198 (1998)
Kigami, J.: Analysis on Fractals. Cambridge University Press, Cambridge (2001)
Kusuoka, S.: Dirichlet forms on fractals and products of random matrices. Publ. Res. Inst. Math. Sci. 25, 659–680 (1989)
Kigami, J., Lapidus, M.L.: Weyl’s problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals. Commun. Math. Phys. 158, 93–125 (1993)
Kigami, J., Sheldon, D.R., Strichartz, R.S.: Green’s functions on fractals. Fractal 8(4), 385–402 (2000)
Lapidus, M.L.: Analysis on fractals, Laplacian on self-similar sets, noncommutative geometry and spectral dimensions. Topol. Methods Nonlinear Anal. 4, 137–195 (1994)
Malozemov, L., Teplyaev, A.: Pure point spectrum of the Laplacian on fractal graphs. J. Funct. Anal. 129, 390–405 (1995)
Needleman, J., Strichartz, R.S., Teplyaev, A., Yung, P.: Calculus on the Sierpinski gasket. I. Polynomials, exponentials and power series. J. Funct. Anal. 215(2), 290–340 (2004)
Qiu, H., Strichartz, R.S.: Mean value properties of harmonic functions on Sierpinski gasket type fractals. J. Fourier Anal. Appl. 5, 943–966 (2013)
Rogers, L.G., Strichartz, R.S.: Distribution theory on p.c.f. fractals. J. Anal. Math. 112, 137–191 (2010)
Strichartz, R.S.: Piecewise linear wavelets on Sierpinski gasket type fractals. J. Fourier. Anal. Appl. 3, 387–416 (1997)
Strichartz, R.S.: Fractals in the large. Can. J. Math. 50, 638–657 (1998)
Strichartz, R.S.: Isoperimetric estimates on Sierpinski gasket type fractals. Trans. Am. Math. Soc. 351, 1705–1752 (1999)
Strichartz, R.S.: Some properties of Laplacians on fractals. J. Funct. Anal. 164, 181–208 (1999)
Strichartz, R.S.: Taylor approximations on Sierpinski gasket type fractals. J. Funct. Anal. 174, 76–127 (2000)
Strichartz, R.S.: Function spaces on fractals. J. Funct. Anal. 198(1), 43–83 (2003)
Strichartz, R.S.: Solvability for differential equations on fractals. J. Anal. Math. 96, 247–267 (2005)
Strichartz, R.S.: Differential Equations on Fractals: A Tutorial. Princeton University Press, Princeton (2006)
Teplyaev, A.: Spectral analysis on infinite Sierpinski gaskets. J. Funct. Anal. 159, 537–567 (1998)
Teplyaev, A.: Gradients on fractals. J. Funct. Anal. 174, 128–154 (2000)