Benchmark Problems for the Numerical Discretization of the Cahn–Hilliard Equation with a Source Term

In this paper, we present benchmark problems for the numerical discretization of the Cahn–Hilliard equation with a source term. If the source term includes an isotropic growth term, then initially circular and spherical shapes should grow with their original shapes. However, there is numerical anisotropic error and this error results in anisotropic evolutions. Therefore, it is essential to use isotropic space discretization in the simulation of growth phenomenon such as tumor growth. To test numerical discretization, we present two benchmark problems: one is the growth of a disk or a sphere and the other is the growth of a rotated ellipse or a rotated ellipsoid. The computational results show that the standard discrete Laplace operator has severe grid orientation dependence. However, the isotropic discrete Laplace operator generates good results.

ON THE SPECTRUM OF THE TWO-PARTICLE SCHRÖDINGER OPERATOR WITH POINT POTENTIAL: ONE DIMENTIONAL CASE

In the paper a one-dimensional two-particle quantum system interacted by two identical point interactions is considered. The corresponding Schr\"{o}\-dinger operator (energy operator) $h_\varepsilon$ depending on $\varepsilon,$ is constructed as a self-adjoint extension of the symmetric Laplace operator. The main results of the work are based to the study of the operator $h_\varepsilon.$ First the essential spectrum is described. The existence of unique negative eigenvalue of the Schr\"{o}dinger operator is proved. Further, this eigenvalue and corresponding eigenfunction are found.

A theorem of Roe and Strichartz on homogeneous trees

Let 𝔛 {\mathfrak{X}} be a homogeneous tree and let ℒ {\mathcal{L}} be the Laplace operator on 𝔛 {\mathfrak{X}} . In this paper, we address problems of the following form: Suppose that { f k } k ∈ ℤ {\{f_{k}\}_{k\in\mathbb{Z}}} is a doubly infinite sequence of functions in 𝔛 {\mathfrak{X}} such that for all k ∈ ℤ {k\in\mathbb{Z}} one has ℒ ⁢ f k = A ⁢ f k + 1 {\mathcal{L}f_{k}=Af_{k+1}} and ∥ f k ∥ ≤ M {\lVert f_{k}\rVert\leq M} for some constants A ∈ ℂ {A\in\mathbb{C}} , M > 0 {M>0} and a suitable norm ∥ ⋅ ∥ {\lVert\,\cdot\,\rVert} . From this hypothesis, we try to infer that f 0 {f_{0}} , and hence every f k {f_{k}} , is an eigenfunction of ℒ {\mathcal{L}} . Moreover, we express f 0 {f_{0}} as the Poisson transform of functions defined on the boundary of 𝔛 {\mathfrak{X}} .

High perturbations of a new Kirchhoff problem involving the p-Laplace operator

In the present work we are concerned with the existence and multiplicity of solutions for the following new Kirchhoff problem involving the p-Laplace operator: $$ \textstyle\begin{cases} - (a-b\int _{\Omega } \vert \nabla u \vert ^{p}\,dx ) \Delta _{p}u = \lambda \vert u \vert ^{q-2}u + g(x, u), & x \in \Omega , \\ u = 0, & x \in \partial \Omega , \end{cases} $$ { − ( a − b ∫ Ω | ∇ u | p d x ) Δ p u = λ | u | q − 2 u + g ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where $a, b > 0$ a , b > 0 , $\Delta _{p} u := \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ Δ p u : = div ( | ∇ u | p − 2 ∇ u ) is the p-Laplace operator, $1 < p < N$ 1 < p < N , $p < q < p^{\ast }:=(Np)/(N-p)$ p < q < p ∗ : = ( N p ) / ( N − p ) , $\Omega \subset \mathbb{R}^{N}$ Ω ⊂ R N ($N \geq 3$ N ≥ 3 ) is a bounded smooth domain. Under suitable conditions on g, we show the existence and multiplicity of solutions in the case of high perturbations (λ large enough). The novelty of our work is the appearance of new nonlocal terms which present interesting difficulties.

Optimal control for cooperative systems involving fractional Laplace operators

In this work, the elliptic $2\times 2$ 2 × 2 cooperative systems involving fractional Laplace operators are studied. Due to the nonlocality of the fractional Laplace operator, we reformulate the problem into a local problem by an extension problem. Then, the existence and uniqueness of the weak solution for these systems are proved. Hence, the existence and optimality conditions are deduced.

Infinite Schrödinger networks

Finite-difference models of partial differential equations such as Laplace or Poisson equations lead to a finite network. A discretized equation on an unbounded plane or space results in an infinite network. In an infinite network, Schrödinger operator (perturbed Laplace operator, $q$-Laplace) is defined to develop a discrete potential theory which has a model in the Schrödinger equation in the Euclidean spaces. The relation between Laplace operator $\Delta$-theory and the $\Delta_q$-theory is investigated. In the $\Delta_q$-theory the Poisson equation is solved if the network is a tree and a canonical representation for non-negative $q$-superharmonic functions is obtained in general case.

Small Object Signal Formation Model for Optical System with Matrix Photodetector When Processed by the Laplace Operator

The authors review the formation of a small object signal by an optical system with a matrix photodetector. They develop a mathematical model of signal formation and test it. The authors calculate the signal/noise proportion for various small object signal shapes and amplitudes processed by the Laplace operator.