90 research outputs found
On the existence of three solutions for the Dirichlet problem on the Sierpinski gasket
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
Boundary Value Problems on a Half Sierpinski Gasket
We study boundary value problems for the Laplacian on a domain
consisting of the left half of the Sierpinski Gasket (), whose boundary is
essentially a countable set of points . 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 compute the
exact trace spaces on of the and domains of the Laplacian
on . In terms of the these trace spaces, we characterize the functions in
the and domains of the Laplacian on that extend to
the corresponding domains on , and give an explicit linear extension
operator in terms of piecewise biharmonic functions
On the viscous Burgers equation on metric graphs and fractals
We study a formulation of Burgers equation on the Sierpinski gasket, which is
the prototype of a p.c.f. self-similar fractal. One possibility is to implement
Burgers equation as a semilinear heat equation associated with the Laplacian
for scalar functions, just as on the unit interval. Here we propose a second,
different formulation which follows from the Cole-Hopf transform and is
associated with the Laplacian for vector fields. The difference between these
two equations can be understood in terms of different vertex conditions for
Laplacians on metric graphs. For the second formulation we show existence and
uniqueness of solutions and verify the continuous dependence on the initial
condition. We also prove that solutions on the Sierpinski gasket can be
approximated in a weak sense by solutions to corresponding equations on
approximating metric graphs. These results are part of a larger program
discussing nonlinear partial differential equations on fractal spaces
Boundary Value Problems for a Family of Domains in the Sierpinski Gasket
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 which there is an extension operator for functions of finite energy. We
give an explicit construction of the Green's function for these domains.Comment: 19pages, 10 figure
Existence of multiple solutions of a p-Laplacian equation on the Sierpinski Gasket
In this paper we study the following boundary value problem involving the
weak p-Laplacian. \begin{equation*} \quad -M(\|u\|_{\mathcal{E}_p}^p)\Delta_p u
= h(x,u) \; \text{in}\; \mathcal{S}\setminus\mathcal{S}_0; \quad u = 0 \;
\mbox{on}\; \mathcal{S}_0, \end{equation*} where is the
Sierpi\'nski gasket in , is its boundary. defined by and and We will show the existence of
two nontrivial weak solutions to the above problem.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1807.0688
Asymptotic behavior of density in the boundary-driven exclusion process on the Sierpinski gasket
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 the equilibrium
setting, they are governed by Ornstein-Uhlenbeck processes with the respective
boundary conditions.Comment: 45 pages, 2 figures. Final version, to appear in Math. Phys. Anal.
Geo
A System of p-Laplacian Equations on the Sierpinski Gasket
In this paper we study a system of boundary value problems involving weak
p-Laplacian on the Sierpi\'nski gasket in . Parameters are real and Functions are suitably chosen. For we show the
existence of at least two nontrivial weak solutions to the system of equations
for some Comment: 25 pages, 2 figure
Magnetic Laplacians of locally exact forms on the Sierpinski Gasket
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 , and it is shown that the asymptotic distribution of eigenvalues is
the same as that for the Laplacian. Most eigenfunctions may be computed using
gauge transformations corresponding to the magnetic field and the remainder of
the spectrum may be approximated to arbitrary precision by using a sequence of
approximations by magnetic operators on finite graphs.Comment: 20 pages, 5 figure
Extensions and their Minimizations on the Sierpinski Gasket
We study the extension problem on the Sierpinski Gasket (). In the first
part we consider minimizing the functional with prescribed values at a finite
set of points where denotes the energy (the analog of in Euclidean space) and denotes the standard self-similiar
measure on . We explicitly construct the minimizer for some constants , where is the
resolvent for . We minimize the energy over sets in by
calculating the explicit quadratic form of the minimizer .
We consider properties of this quadratic form for arbitrary sets and then
analyze some specific sets. One such set we consider is the bottom row of a
graph approximation of . We describe both the quadratic form and a
discretized form in terms of Haar functions which corresponds to the continuous
result established in a previous paper. In the second part, we study a similar
problem this time minimizing for general
measures. In both cases, by using standard methods we show the existence and
uniqueness to the minimization problem. We then study properties of the unique
minimizers.Comment: 28 pages, 5 figure
Dirichlet forms on self-similar sets with overlaps
We study Dirichlet forms and Laplacians on self-similar sets with overlaps. A
notion of "finitely ramified of finite type() nested structure" for
self-similar sets is introduced. It allows us to reconstruct a class of
self-similar sets in a graph-directed manner by a modified setup of Mauldin and
Williams, which satisfies the property of finite ramification. This makes it
possible to extend the technique developed by Kigami for analysis on
self-similar sets to this more general framework. Some basic properties related
to nested structures are investigated. Several non-trivial examples
and their Dirichlet forms are provided.Comment: 38 pages, 29 figure
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