scholarly journals Stationary Boltzmann equation: an approach via Morse theory

2021 ◽  
Vol 40 (6) ◽  
pp. 1473-1487
Author(s):  
Rafael Galeano Andrades ◽  
Joel Torres del Valle
Keyword(s):  
Boltzmann Equation ◽  
Critical Points ◽  
Morse Theory ◽  
Morse Index ◽  
Critical Groups

In this paper we study the unidimensional Stationary Boltzmann Equation by an approach via Morse theory. We define a functional J whose critical points coincide with the solutions of the Stationary Boltzmann Equation. By the calculation of Morse index of J’’0(0)h and the critical groups C2(J, 0) and C2(J, ∞) we prove that J has two different critical points u1 and u2 different from 0, that is, solutions of Boltzmann Equation.

Morse Theory and Critical Groups

2019 ◽  
pp. 457-555
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu ◽  
Dušan D. Repovš
Keyword(s):  
Morse Theory ◽  
Critical Groups

Critical groups and multiple solutions for Kirchhoff type equations with critical exponents

2020 ◽  
pp. 2050031
Author(s):  
Mingzheng Sun ◽  
Jiabao Su ◽  
Binlin Zhang
Keyword(s):  
Critical Exponents ◽  
Morse Theory ◽  
Multiple Solutions ◽  
Nonlinear Term ◽  
Type Equation ◽  
Critical Groups ◽  
First Eigenvalue ◽  
Nontrivial Solutions ◽  
Kirchhoff Type ◽  
The First Eigenvalue

In this paper, by Morse theory we will study the Kirchhoff type equation with an additional critical nonlinear term, and the main results are to compute the critical groups including the cases where zero is a mountain pass solution and the nonlinearity is resonant at zero. As an application, the multiplicity of nontrivial solutions for this equation with the parameter across the first eigenvalue is investigated under appropriate assumptions. To our best knowledge, estimates of our critical groups are new even for the Kirchhoff type equations with subcritical nonlinearities.

ELLIPTIC EQUATIONS IN ℝ2 WITH ONE-SIDED EXPONENTIAL GROWTH

2004 ◽  
Vol 06 (06) ◽  
pp. 947-971 ◽  
Author(s):  
ZHITAO ZHANG ◽  
MARTA CALANCHI ◽  
BERNHARD RUF
Keyword(s):  
Exponential Growth ◽  
Elliptic Equations ◽  
Morse Index ◽  
Linear Growth ◽  
Critical Growth ◽  
The Other ◽  
Critical Groups ◽  
Bounded Domains ◽  
Eigen Value

We consider elliptic equations in bounded domains Ω⊂ℝ2 with nonlinearities which have exponential growth at +∞ (subcritical and critical growth, respectively) and linear growth λ at -∞, with λ>λ1, the first eigen value of the Laplacian. We prove that such equations have at least two solutions for certain forcing terms; one solution is negative, the other one is sign-changing. Some critical groups and Morse index of these solutions are given. Also the case λ<λ1 is considered.

scholarly journals TOPOLOGICAL AND SEMANTIC IMAGES ANALYSIS

2017 ◽  
Vol 13 (3) ◽  
pp. 74-77
Author(s):  
Ol'ga Vladimirovna Samarina
Keyword(s):  
Critical Points ◽  
Morse Theory ◽  
Semantic Analysis ◽  
Single Channel ◽  
Digital Images ◽  
Program Module ◽  
Channel Image

In this paper we investigate method of the digital images topological and semantic analysis based on the Morse theory. The description of the MathLab program module for calculation of statistically significant critical points for single-channel image is also submitted.

Morse Index of Approximating Periodic Solutions for the Billiard Problem. Application to Existence Results

1998 ◽  
Vol 50 (3) ◽  
pp. 497-524
Author(s):  
Philippe Bolle
Keyword(s):  
Periodic Solutions ◽  
Critical Points ◽  
Morse Index ◽  
Approximate Solutions ◽  
Existence Results ◽  
Lagrangian Systems ◽  
Potential Well ◽  
Regular Solutions ◽  
Open Set

AbstractThis paper deals with periodic solutions for the billiard problem in a bounded open set of ℝN which are limits of regular solutions of Lagrangian systems with a potential well. We give a precise link between the Morse index of approximate solutions (regarded as critical points of Lagrangian functionals) and the properties of the bounce trajectory to which they converge.

scholarly journals An application of the Morse theory to foliated manifolds

1974 ◽  
Vol 54 ◽  
pp. 165-178 ◽  
Author(s):  
Kazuhiko Fukui
Keyword(s):  
Critical Points ◽  
Morse Theory ◽  
Topological Properties ◽  
Foliated Manifold ◽  
Variation Equation ◽  
Codimension One ◽  
Foliated Manifolds ◽  
The Given

In [5], R. Thorn has started the study of the foliated structures by using the Morse theory. Recently K. Yamato [7] has studied the topological properties of leaves of a codimension one foliated manifold by investigating the “critical points” of variation equation of the given one-form.

scholarly journals Existence results for double-phase problems via Morse theory

2017 ◽  
Vol 20 (02) ◽  
pp. 1750023 ◽  
Author(s):  
Kanishka Perera ◽  
Marco Squassina
Keyword(s):  
Morse Theory ◽  
Existence Results ◽  
Critical Groups ◽  
Nontrivial Solutions ◽  
Direct Sum ◽  
Double Phase ◽  
Direct Sum Decomposition

We obtain nontrivial solutions for a class of double-phase problems using Morse theory. In the absence of a direct sum decomposition, we use a cohomological local splitting to get an estimate of the critical groups at zero.

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