# Brouwer Degree

The Experts below are selected from a list of 297 Experts worldwide ranked by ideXlab platform

### Jean Mawhin – 1st expert on this subject based on the ideXlab platform

• ##### Existence and multiplicity results for nonlinear second order difference equations with Dirichlet boundary conditions
, 2020
Co-Authors: Cristian Bereanu, Jean Mawhin

Abstract:

We use Brouwer Degree to prove existence and multiplicity results for the solutions of some nonlinear second order difference equations with Dirichlet boundary conditions. We obtain in particular upper and lower solutions theorems, Ambrosetti-Prodi type results, and sharp existence conditions for nonlinearities which are bounded from below or from above.

• ##### Bolzano’s theorems for holomorphic mappings
Chinese Annals of Mathematics Series B, 2017
Co-Authors: Jean Mawhin

Abstract:

The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign conditions on the boundary generalizing Bolzano’s ones for real functions on an interval is deduced in a very simple way from Cauchy’s theorem for holomorphic functions. A more complicated proof, using Cauchy’s argument principle, provides uniqueness of the zero, when the sign conditions on the boundary are strict. Applications are given to corresponding Brouwer fixed point theorems for holomorphic functions. Extensions to holomorphic mappings from ℂ^ n to ℂ^ n are obtained using Brouwer Degree.

• ##### reduction and continuation theorems for BrouwerDegree and applications to nonlinear difference equations
Opuscula Mathematica, 2008
Co-Authors: Jean Mawhin

Abstract:

The aim of this note is to describe the continuation theorem of [J. Mawhin, Equivalence theorems for nonlinear operator equations and coincidence Degree theory for some mappings in locally convex topological vector spaces, J. Differential Equations 12 (1972), 610–636, J. Mawhin,
Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Reg. Conf. in Math., No 40, American Math. Soc., Providence, RI, 1979] directly in the context of Brouwer Degree, providing in this way a simple frame for multiple applications to nonlinear difference equations, and to show how the corresponding reduction property can be seen as an extension of the well-known reduction formula of Leray and Schauder [J. Leray, J. Schauder,
Topologie et equations fonctionnelles, Ann. Scient. Ecole Normale Sup. (3) 51 (1934), 45–78], which is fundamental for their construction of Leray-Schauder’s Degree in normed vector spaces.

### Yi-shi Duan – 2nd expert on this subject based on the ideXlab platform

• ##### TOPOLOGICAL STRUCTURE OF VORTEX LINES OF QUANTUM ELECTRON PLASMAS IN THREE-DIMENSIONAL SPACE
International Journal of Modern Physics B, 2009
Co-Authors: Yi-shi Duan

Abstract:

The topological properties of quantum electron plasmas in three-dimensional space are presented. Starting from phi-mapping topological current theory, the vortex lines are just at the core of wave function obtained. It is shown that the vorticity of the vortex can be expressed by the Hopg index and the Brouwer Degree. We find the vortex lines are unstable in some conditions and the evolution of vortex lines at the bifurcation points is given.

• ##### TOPOLOGICAL SOLUTION OF BOHMIAN QUANTUM MECHANICS
Modern Physics Letters A, 2009
Co-Authors: Ming Yu, Yi-shi Duan

Abstract:

The topological solutions of the De Broglie–Bohm quantum mechanics are presented. Starting from the Schrodinger equation for one particle system and ϕ-mapping topological current theory, the trajectory of the particle is derived explicitly, and can be used as the world line of the particle. The world line is just at the zero point of the wave function and it is shown that the vorticity of the world line can be expressed by Hopf index and Brouwer Degree. The evolution of the world line at the bifurcation point is given.

• ##### Knotlike χ disclinations in the cholesteric liquid crystals
Physica A-statistical Mechanics and Its Applications, 2006
Co-Authors: Peng-ming Zhang, Yi-shi Duan, Hong Zhang

Abstract:

We investigate the topological properties of N(N⩾1) disclination lines in cholesteric liquid crystals. The topological structure of N disclination lines is obtained with the Hopf index and Brouwer Degree. Furthermore, the knotted χ disclination loops is proposed with the Hopf invariant. And we consider the stability of such configuration based on the higher order interaction. At last, the evolution of the disclinations is discussed.

### Anna Siffert – 3rd expert on this subject based on the ideXlab platform

• ##### Harmonic Self-Maps of $$\mathrm {SU}(3)$$ SU ( 3 )
Journal of Geometric Analysis, 2017
Co-Authors: Anna Siffert

Abstract:

By constructing solutions of a singular boundary value problem, we prove the existence of a countably infinite family of harmonic self-maps of $$\mathrm {SU}(3)$$
with non-trivial, i.e., $$\ne 0,\pm 1$$
, Brouwer Degree.

• ##### Harmonic Self-Maps of \mathrm {SU}(3)
Journal of Geometric Analysis, 2017
Co-Authors: Anna Siffert

Abstract:

By constructing solutions of a singular boundary value problem, we prove the existence of a countably infinite family of harmonic self-maps of $$\mathrm {SU}(3)$$ with non-trivial, i.e., $$\ne 0,\pm 1$$, Brouwer Degree.

• ##### Infinite families of harmonic self-maps of spheres
Journal of Differential Equations, 2016
Co-Authors: Anna Siffert

Abstract:

Abstract For each of the spheres S n , n ≥ 5 , we construct a new infinite family of harmonic self-maps, and prove that their members have Brouwer Degree ±1 or ±3. These self-maps are obtained by solving a singular boundary value problem. As an application we show that for each of the special orthogonal groups SO ( 4 ) , SO ( 5 ) , SO ( 6 ) and SO ( 7 ) there exist two infinite families of harmonic self-maps.