We study boundary value problems for harmonic functions on certain domains in the level-$l$ Sierpinski gaskets $\mathcal{SG}_l$($l\geq 2$) 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 $\mathcal{SG}_l$ and the upper and lower domains generated by horizontal cuts of $\mathcal{SG}_l$ 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.
We study the Laplace operator on the Sierpinski gasket with nonlinear Robin boundary conditions. We show that for certain Robin boundary conditions the Laplace operator generates a positive, order preserving, $L^\infty$-contractive semigroup which is sandwiched (in the sense of domination) between the semigroups generated by the Dirichlet-Laplace operator and the Neumann-Laplace operator. We also characterise all local semigroups which are sandwiched between these two extremal semigroups by showing that their generators are Robin-Laplace operators.
We prove versions of the strong Szëgo limit theorem for certain classes of pseudodifferential operators defined on the Sierpiński gasket. Our results used in a fundamental way the existence of localized eigenfunctions for the Laplacian on this fractal.
We study the eikonal equation on the Sierpinski gasket in the spirit of the construction of the Laplacian in Kigami [8]: we consider graph eikonal equations on the prefractals and we show that the solutions of these problems converge to a function defined on the fractal set. We characterize this limit function as the unique metric viscosity solution to the eikonal equation on the Sierpinski gasket according to the definition introduced in [3].
We apply a recently obtained three critical points theorem of B. Ricceri to prove the existence of at least three solutions of certain two-parameters Dirichlet problems defined on the Sierpinski gasket. We also show the existence of at least three nonzero solutions of certain perturbed two-parameters Dirichlet problems on the Sierpinski gasket, using both the mountain pass theorem of Ambrosetti-Rabinowitz and that of Pucci-Serrin.
We extend and survey results in the theory of analysis on fractal sets from the standard Laplacian on the Sierpiński gasket to the energy Laplacian, which is defined weakly by using the Kusuoka energy measure. We also extend results from the Sierpiński gasket to level-$k$ Sierpiński gaskets, for all $k\geq 2$. We observe that the pointwise formula for the energy Laplacian is valid for all level-$k$ Sierpiński gaskets, $SG_k$, and we provide a proof of a known formula for the renormalization constants of the Dirichlet form for post-critically finite self-similar sets along with a probabilistic …
Aim of this note is to study the infinity Laplace operator and the corresponding Absolutely Minimizing Lipschitz Extension problem on the Sierpinski gasket in the spirit of the classical construction of Kigami for the Laplacian. We introduce a notion of infinity harmonic functions on pre-fractal sets and we show that these functions solve a Lipschitz extension problem in the discrete setting. Then we prove that the limit of the infinity harmonic functions on the pre-fractal sets solves the Absolutely Minimizing Lipschitz Extension problem on the Sierpinski gasket.
In this paper we study a system of boundary value problems involving weak p-Laplacian on the Sierpiński gasket in $\mathbb{R}^2$. Parameters $λ, γ, α, β$ are real and $1<q<p<α+β.$ Functions $a,b,h : \mathcal{S} \rightarrow \mathbb{R}$ are suitably chosen. For $p>1$ we show the existence of at least two nontrivial weak solutions to the system of equations for some $(λ,γ) \in \mathbb{R}^2.$
We study energy measures on SG based on harmonic functions. We characterize the positive energy measures through studying the bounds of Radon-Nikodym derivatives with respect to the Kusuoka measure. We prove a limited continuity of the derivative on the graph $V_*$ and express the average value of the derivative on a whole cell as a weighted average of the values on the boundary vertices. We also prove some characterizations and properties of the weights.
In this paper we study the following boundary value problem involving the weak p-Laplacian. \begin{equation*} \quad -M(\|u\|_{\mathcal{E}_p}^p)Δ_p u = h(x,u) \; \text{in}\; \mathcal{S}\setminus\mathcal{S}_0; \quad u = 0 \; \mbox{on}\; \mathcal{S}_0, \end{equation*} where $\mathcal{S}$ is the Sierpiński gasket in $\mathbb{R}^2$, $\mathcal{S}_0$ is its boundary. $M : \mathbb{R} \to \mathbb{R}$ defined by $M(t) = at^k +b$ and $a,b,k >0$ and $h : \mathcal{S} \times \mathbb{R} \to \mathbb{R}.$ We will show the existence of two nontrivial weak solutions to the above problem.
We derive the macroscopic laws that govern the evolution of the density of particles in the exclusion process on the Sierpinski gasket in the presence of a variable speed boundary. We obtain, at the hydrodynamics level, the heat equation evolving on the Sierpinski gasket with either Dirichlet or Neumann boundary conditions, depending on whether the reservoirs are fast or slow. For a particular strength of the boundary dynamics we obtain linear Robin boundary conditions. As for the fluctuations, we prove that, when starting from the stationary measure, namely the product Bernoulli measure in th…
In the case of some fractals, sampling with average values on cells is more natural than sampling on points. In this paper we investigate this method of sampling on $SG$ and $SG_{3}$. In the former, we show that the cell graph approximations have the spectral decimation property and prove an analog of the Shannon sampling theorem.. We also investigate the numerical properties of these sampling functions and make conjectures which allow us to look at sampling on infinite blowups of $SG$. In the case of $SG_{3}$, we show that the cell graphs have the spectral decimation property, but show that i…
We study boundary value problems for the Laplacian on a domain $Ω$ consisting of the left half of the Sierpinski Gasket ($SG$), whose boundary is essentially a countable set of points $X$. For harmonic functions we give an explicit Poisson integral formula to recover the function from its boundary values, and characterize those that correspond to functions of finite energy. We give an explicit Dirichlet to Neumann map and show that it is invertible. We give an explicit description of the Dirichlet to Neumann spectra of the Laplacian with an exact count of the dimensions of eigenspaces. We comp…
In recent years, relatively sharp quantitative results in the spirit of the Besicovitch projection theorem have been obtained for self-similar sets by studying the $L^p$ norms of the "projection multiplicity" functions, $f_θ$, where $f_θ(x)$ is the number of connected components of the partial fractal set that orthogonally project in the $θ$ direction to cover $x$. In \cite{NPV}, it was shown that $n$-th partial 4-corner Cantor set with self-similar scaling factor 1/4 decays in Favard length at least as fast as $\frac{C}{n^p}$, for $p<1/6$. In \cite{BV}, this same estimate was proved for the 1…
This paper concerns with a class of elliptic equations on fractal domains depending on a real parameter. Our approach is based on variational methods. More precisely, the existence of at least two non-trivial weak (strong) solutions for the treated problem is obtained exploiting a local minimum theorem for differentiable functionals defined on reflexive Banach spaces. A special case of the main result improves a classical application of the Mountain Pass Theorem in the fractal setting, given by Falconer and Hu (1999).
