We construct symmetric self-similar Dirichlet forms on unconstrained Sierpinski carpets, which are natural extension of planner Sierpinski carpets by allowing the small cells to live off the 1/k grids. The intersection of two cells can be a line segment of irrational length, and we also drop the non-diagonal assumption in this recurrent setting. The proof of the existence is purely analytic. A uniqueness theorem is also provided. Moreover, the additional freedom of unconstrained Sierpinski carpets allows us to slide the cells around. In this situation, we view unconstrained Sierpinski carpets as moving fractals, and we prove that the self-similar Dirichlet forms will vary continuously in a Γ-convergence sense.

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Abstract. In this paper, we focus on Strichartz's derivatives, a family of derivatives including the normal derivative, on p.c.f. (post critically finite) fractals, which are defined at vertex points 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 optimal rates of approximations are described and several non-trivial examples are provided to illustrate that our estimates are sharp. We also study the boundness 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 point is provided. Furthermore, we give a counter-example of a conjecture of Strichartz on the existence of higher order weak tangents.

We construct a Sierpinski carpet like fractal, on which a diffusion with sub-Gaussian heat kernel estimate does not exist, in contrast to previous researches on the existence of such diffusions, on the generalized Sierpinski carpets and recently introduced unconstrained Sierpinski carpets.

We study boundary value problems for harmonic functions on certain domains in the level-l Sierpinski gaskets SG l (l ≥ 2) whose boundaries are Cantor sets. We give explicit analogues of the Poisson integral formula to recover harmonic functions from their boundary values. Three types of domains, the left half domain of SG l and the upper and lower domains generated by horizontal cuts of SG l are considered at present. We characterize harmonic functions of finite energy and obtain their energy estimates in terms of their boundary values. This paper settles several open problems raised in previous work.