The qualitative behavior for approximate Dirac-harmonic maps into stationary Lorentzian manifolds

2021 ◽  
Author(s):  
Wanjun Ai ◽  
Miaomiao Zhu
Keyword(s):  
Harmonic Maps ◽  
Qualitative Behavior ◽  
Lorentzian Manifolds

scholarly journals On Finite Energy Solutions of 4-harmonic and ES-4-harmonic Maps

2021 ◽  
Author(s):  
Volker Branding
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Euclidean Space ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Elliptic Partial Differential Equation ◽  
Variational Problems ◽  
Finite Energy ◽  
Qualitative Behavior ◽  
Nonlinear Elliptic

Abstract4-harmonic and ES-4-harmonic maps are two generalizations of the well-studied harmonic map equation which are both given by a nonlinear elliptic partial differential equation of order eight. Due to the large number of derivatives it is very difficult to find any difference in the qualitative behavior of these two variational problems. In this article we prove that finite energy solutions of both 4-harmonic and ES-4-harmonic maps from Euclidean space must be trivial. However, the energy that we require to be finite is different for 4-harmonic and ES-4-harmonic maps pointing out a first difference between these two variational problems.

scholarly journals The qualitative behavior at the free boundary for approximate harmonic maps from surfaces

2018 ◽  
Vol 374 (1-2) ◽  
pp. 133-177 ◽  
Author(s):  
Jürgen Jost ◽  
Lei Liu ◽  
Miaomiao Zhu
Keyword(s):  
Free Boundary ◽  
Harmonic Maps ◽  
Qualitative Behavior

scholarly journals Nonlinear Dirac Equations, Monotonicity Formulas and Liouville Theorems

2019 ◽  
Vol 372 (3) ◽  
pp. 733-767
Author(s):  
Volker Branding
Keyword(s):  
Riemannian Manifolds ◽  
Harmonic Maps ◽  
Qualitative Behavior ◽  
Quantum Field ◽  
Liouville Theorems ◽  
Dirac Equations ◽  
Monotonicity Formulas ◽  
Curvature Term ◽  
Nonlinear Dirac Equations ◽  
Complete Riemannian Manifolds

Abstract We study the qualitative behavior of nonlinear Dirac equations arising in quantum field theory on complete Riemannian manifolds. In particular, we derive monotonicity formulas and Liouville theorems for solutions of these equations. Finally, we extend our analysis to Dirac-harmonic maps with curvature term.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

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