Class Of Self-similar Measures Research Articles

Schrödinger equations defined by a class of self-similar measures

Schrödinger equations defined by a class of self-similar measures

Weyl’s asymptotic formula for fractal Laplacians defined by a class of self-similar measures with overlaps

Weyl’s asymptotic formula for fractal Laplacians defined by a class of self-similar measures with overlaps

Strong damping wave equations defined by a class of self-similar measures with overlaps

Strong damping wave equations defined by a class of self-similar measures with overlaps

Weak damped wave equations defined by a class of self-similar measures with overlaps

We study weak damped wave equations defined by fractal Laplacians. These Laplacians are defined by self-similar measures with overlaps, such as the well-known infinite Bernoulli convolution associated with the golden ratio, the three-fold convolution of the Cantor measure, and a class of self-similar measures that we call essentially of finite type. First, we prove the weak well-posedness of these wave equations and some regularity results. Second, by using some important information about the structure of the three measures above, we set up a framework for one-dimensional measures to discretize the wave equations. Then we use the finite element and central difference methods to obtain numerical approximations of the weak solutions for such equations. Finally, we show that the numerical solutions converge to the actual weak solution and obtain the rate of convergence.

On Spectral Asymptotics of the Neumann Problem for the Sturm–Liouville Equation with Arithmetically Self-Similar Weight of a Generalized Cantor Type

Spectral asymptotics of the Sturm–Liouville problem with an arithmetically self-similar singular weight is considered. Previous results by A. A. Vladimirov and I. A. Sheipak, and also by the author, rely on the spectral periodicity property, which imposes significant restrictions on the self-similarity parameters of the weight. This work introduces a new method for estimating the eigenvalue counting function. This makes it possible to consider a much wider class of self-similar measures.

Singularity of a class of self-similar measures

Abstract This paper obtains a sufficient condition for certain self-similar measures being singular. One of its consequences is that the projection of the self-similar measure on Sierpinski carpet onto a line with rational slope is singular.