Related papers: Eikonal equations on the Sierpinski gasket
This paper investigates fractional torsional rigidity on compact, connected metric graphs, a novel extension of the classical concept to nonlocal operators. The fractional torsional rigidity is defined as the $L^1$-norm of the fractional…
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…
Dynamic programming equations for mean field control problems with a separable structure are Eikonal equations on the Wasserstein space. Standard differentiation using linear derivatives yield a direct extension of the classical viscosity…
In this article we study a system of eikonal equations. Our aim is to isolate the solutions which minimise the discontinuity set of the gradient.
Viscosity solutions to the eikonal equation |Du|g = 1, known to be exactly distance-like functions, on a non-compact complete Riemannian manifold (M,g) are crucial for understanding the underlying geometric and topological properties. In…
Some problems related to the transition density u(t,x) of the diffusion on the Sierpinski gasket are considerd, based on recent rigorous results and detailed numerical calculations. The main contents are an extension of Flory's formula for…
In this article we consider a system of eikonal equations with a Dirichlet boundary condition. We propose a variational method to select the class of solutions which minimize the discontinuity set of the gradient.
Based upon elements of the modern Pseudoanalytic Function Theory, we analyse a new method for numerically approaching the solution of the Dirichlet boundary value problem, corresponding to the two-dimensional Electrical Impedance Equation.…
We generalize the results in [16] to higher dimensional ramified spaces. For this purpose we introduce ramified manifolds and, as special cases, locally elementary polygonal ramified spaces (LEP spaces). On LEP spaces we develop a theory of…
We consider viscosity solutions of a class of nonlinear degenerate elliptic equations on bounded domains. We prove comparison principles and a priori supremum bounds for the solutions. We also address the eigenvalue problem and, in many…
We provide a method to compute the entropy-satisfying weak solution to the eikonal equation in an arbitrary-order polynomial space. The method uses an artificial viscosity approach and is demonstrated for the signed distance function, where…
In this paper, we propose a Cauchy type problem to the timelike Lorentzian eikonal equation on a globally hyperbolic space-time. For this equation, as the value of the solution on a Cauchy surface is known, we prove the existence of…
Kusuoka and Zhou have defined the Laplacian on the Sierpinski carpet using average values of functions on small cells and the graph structure of cell adjacency. We have implemented an algorithm that uses their method to approximate…
We characterize functions of finite energy in the plane in terms of their traces on the lines that make up "graph paper" with squares of side length $mn$ for all $n$, and certain $\12-$order Sobolev norms on the graph paper lines. We also…
We give a mathematically rigorous construction of a magnetic Schr\"odinger operator corresponding to a field with flux through finitely many holes of the Sierpinski Gasket. The operator is shown to have discrete spectrum accumulating at…
We provide the convergence analysis for a sinc-Galerkin method to solve the fractional Dirichlet problem. This can be understood as a follow-up of an earlier article by the same authors, where the authors presented a sinc-function based…
Numerous work revolve around the Sierpinski gasket. Its three-dimensional analogue, the Sierpinski tetrahedron, obtained by means of an iterative process which consists in repeatedly contracting a regular 3-simplex to one half of its…
We present a new method to approximate the Neumann spectrum of a Laplacian on a fractal K in the plane as a renormalized limit of the Neumann spectra of the standard Laplacian on a sequence of domains that approximate K from the outside.…
Noncommutative geometry provides a framework, via the construction of spectral triples, for the study of the geometry of certain classes of fractals. Many fractals are constructed as natural limits of certain sets with a simpler structure:…
For a family of domains in the Sierpinski gasket, we study harmonic functions of finite energy, characterizing them in terms of their boundary values, and study their normal derivatives on the boundary. We characterize those domains for…