Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2024
Volume
Vol. 44
Number
No. 2
Description
Journal Volume
Opuscula Mathematica
Vol. 44 (2024)
Projects
Pages
Articles
On the structure of the diffusion distance induced by the fractional dyadic Laplacian
(Wydawnictwa AGH, 2024) Acosta, María Florencia; Aimar, Hugo; Gómez, Ivana; Morana, Federico
In this note we explore the structure of the diffusion metric of Coifman-Lafon determined by fractional dyadic Laplacians. The main result is that, for each $t \gt 0$, the diffusion metric is a function of the dyadic distance, given in $\mathbb{R}^{+}$ by $\delta(x,y) = \inf\{|I|\colon I \text{ is a dyadic interval containing } x \text{ and } y\}$. Even if these functions of $\delta$ are not equivalent to $\delta$, the families of balls are the same, to wit, the dyadic intervals.
Green's functions and existence of solutions of nonlinear fractional implicit difference equations with Dirichlet boundary conditions
(Wydawnictwa AGH, 2024) Cabada, Alberto; Dimitrov, Nikolay D.; Jonnalagadda, Jagan Mohan
This article is devoted to deduce the expression of the Green's function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In this case, due to the points where some of the fractional operators are applied, we are in presence of an implicit fractional difference equation. So, due to such a property, it is more complicated to calculate and manage the expression of the Green's function than in the explicit case studied in a previous work of the authors. Contrary to the explicit case, where it is shown that the Green's function is constructed as finite sums, the Green's function constructed here is an infinite series. This fact makes necessary to impose more restrictive assumptions on the parameters that appear in the equation. The expression of the Green's function will be deduced from the Laplace transform on the time scales of the integers. We point out that, despite the implicit character of the considered equation, we can have an explicit expression of the solution by means of the expression of the Green's function. These two facts are not incompatible. Even more, this method allows us to have an explicit expression of the solution of an implicit problem. Finally, we prove two existence results for nonlinear problems, via suitable fixed point theorems.
Parabolic turbulence k-epsilon model with applications in fluid flows through permeable media
(Wydawnictwa AGH, 2024) Oliveira, Hermenegildo Borges de
In this work, we study a one-equation turbulence $k$-epsilon model that governs fluid flows through permeable media. The model problem under consideration here is derived from the incompressible Navier-Stokes equations by the application of a time-averaging operator used in the $k$-epsilon modeling and a volume-averaging operator that is characteristic of modeling unsteady porous media flows. For the associated initial- and boundary-value problem, we prove the existence of suitable weak solutions (average velocity field and turbulent kinetic energy) in the space dimensions of physics interest.
An inequality for imaginary parts of eigenvalues of non-compact operators with Hilbert-Schmidt Hermitian components
(Wydawnictwa AGH, 2024) Gil', Michael
Let $A$ be a bounded linear operator in a complex separable Hilbert space, $A^{*}$ be its adjoint one and $A_I:=(A-A^*)/(2i)$. Assuming that $A_I$ is a Hilbert-Schmidt operator, we investigate perturbations of the imaginary parts of the eigenvalues of $A$. Our results are formulated in terms of the »extended« eigenvalue sets in the sense introduced by T. Kato. Besides, we refine the classical Weyl inequality $\sum_{k=1}^\infty (\operatorname{Im} \lambda_k(A))^2 \leq N_2^2(A_I)$, where $\lambda_{k}(A)$ $(k=1,2, \ldots )$ are the eigenvalues of $A$ and $N_2(\cdot)$ is the Hilbert-Schmidt norm. In addition, we discuss applications of our results to the Jacobi operators.
Anisotropic p-Laplace Equations on long cylindrical domain
(Wydawnictwa AGH, 2024) Jana, Purbita
The main aim of this article is to study the Poisson type problem for anisotropic $p$-Laplace type equation on long cylindrical domains. The rate of convergence is shown to be exponential, thereby improving earlier known results for similar type of operators. The Poincaré inequality for a pseudo $p$-Laplace operator on an infinite strip-like domain is also studied and the best constant, like in many other situations in literature for other operators, is shown to be the same with the best Poincaré constant of an analogous problem set on a lower dimension.