Coupling and Harnack Inequalities for Sierpinski Carpets

Uniform Harnack inequalities for harmonic functions on the preand graphical Sierpinski carpets are proved using a probabilistic coupling argument. Various results follow from this, including the construction of Brownian motion on Sierpinski carpets embedded in Md , d > 3, estimates on the fundamental solution of the heat equation, and Sobolev and Poincaré inequalities. The Sierpinski carpets (SCs) we will study are generalizations of the Cantor set. Let F0 = [0, l]d be the unit cube in Rd, d > 2, centered at z0 = (1/2, ... , 1/2). Let k, a be integers with 1 < a < k and a + k even. Divide Fo into kd equal subcubes, remove a central block of ad subcubes, and let F\ be what remains: thus F\ = Fq ((k a)/2k, (k + a)/2k)d. Now repeat this operation on each of the kd ad remaining subcubes to obtain F2. Iterating, we obtain a decreasing sequence of closed sets Fn ; then F = f)TM=0Fn is a Sierpinski carpet and has Hausdorff dimension df = df(F) — log(kd-ad)/ log(Ä:). (When d = 2, k = 3, and a = 1, we get the usual Sierpinski carpet.) Let F„ = knFn c [0, oo)d , and define the pre-Sierpinski carpet by F U~ 1 Fn (see [10]). The graphical Sierpinski carpetis the graph G = (V, E) with vertex set V = (zn + Zd ) n F and edge set E { {x, y} e V :\x-y\ = 1}. Thus int(.F) is a domain in Rd with a large-scale structure which mimics the small-scale structure of F . We are interested in the behavior of solutions of the Laplace and heat equations on F, F, and G. One reason for this is applications to "transport phenomena" in disordered media (see [6]); another is the new type of behavior of the heat kernel on these spaces. Let W be Brownian motion on F with normal reflection on dF, and let q(t, x, y) be the transition density of W, so that q solves the heat equation on F with Neumann boundary conditions on dF . Received by the editors November 9, 1992. 1991 Mathematics Subject Classification. Primary 60B99; Secondary 60J35.