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

A partial regularity theorem for harmonic maps at a free boundary

1989 ◽  
Vol 2 (4) ◽  
pp. 299-343 ◽  
Author(s):  
Frank Duzaar ◽  
Klaus Steffen
Keyword(s):  
Free Boundary ◽  
Partial Regularity ◽  
Harmonic Maps ◽  
Regularity Theorem

scholarly journals Partial Regularity for Harmonic Maps into Spheres at a Singular or Degenerate Free Boundary

2022 ◽  
Vol 32 (2) ◽  
Author(s):  
Roger Moser ◽  
James Roberts
Keyword(s):  
Free Boundary ◽  
Distance Function ◽  
Boundary Data ◽  
Partial Regularity ◽  
Harmonic Maps

AbstractWe prove partial regularity of weakly stationary harmonic maps with (partially) free boundary data on manifolds where the domain metric may degenerate or become singular along the free boundary at the rate $$d^\alpha $$ d α for the distance function d from the boundary.

EXTREMAL AUTOMATA FOR IMAGE SHARPENING

1994 ◽  
Vol 05 (06) ◽  
pp. 923-931 ◽  
Author(s):  
GONZALO HERNANDEZ ◽  
HANS J. HERRMANN ◽  
ERIC GOLES
Keyword(s):  
Image Processing ◽  
Free Boundary ◽  
Finite Size ◽  
Square Lattice ◽  
Finite Size Effect ◽  
Qualitative Behavior ◽  
Von Neumann ◽  
Damage Spreading ◽  
Free Boundary Conditions ◽  
Parallel Iteration

We study numerically the parallel iteration of Extremal Rules. For four Extremal Rules, conceived for sharpening algorithms for image processing, we measured, on the square lattice with Von Neumann neighborhood and free boundary conditions, the typical transient length, the loss of information and the damage spreading response considering random and smoothening random damage. The same qualitative behavior was found for all the rules, with no noticeable finite size effect. They have a fast logarithmic convergence towards the fixed points of the parallel update. The linear damage spreading response has no discontinuity at zero damage, for both kinds of damage. Three of these rules produce similar effects. We propose these rules as sharpening algorithms for image processing.

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 Regularity and quantification for harmonic maps with free boundary

2017 ◽  
Vol 10 (1) ◽  
pp. 69-82 ◽  
Author(s):  
Paul Laurain ◽  
Romain Petrides
Keyword(s):  
Free Boundary ◽  
Unit Ball ◽  
Harmonic Maps ◽  
Finite Energy ◽  
Riemannian Surfaces

AbstractWe prove a quantification result for harmonic maps with free boundary and finite energy from arbitrary Riemannian surfaces into the unit ball of ${{\mathbb{R}}^{n+1}}$. This generalizes results obtained by Da Lio [1] on the disk.

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