Energies ofS2-valued harmonic maps on polyhedra with tangent boundary conditions

A. Majumdar; J.M. Robbins; M. Zyskin

doi:10.1016/j.anihpc.2006.11.003

Energies ofS2-valued harmonic maps on polyhedra with tangent boundary conditions

Annales de l Institut Henri Poincaré C Analyse non linéaire ◽

10.1016/j.anihpc.2006.11.003 ◽

2008 ◽

Vol 25 (1) ◽

pp. 77-103 ◽

Cited By ~ 1

Author(s):

A. Majumdar ◽

J.M. Robbins ◽

M. Zyskin

Keyword(s):

Boundary Conditions ◽

Harmonic Maps

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Evolution of harmonic maps with Dirichlet boundary conditions

Communications in Analysis and Geometry ◽

10.4310/cag.1993.v1.n3.a1 ◽

1993 ◽

Vol 1 (3) ◽

pp. 327-346 ◽

Cited By ~ 40

Author(s):

Yunmei Chen ◽

Fang Hua Lin

Keyword(s):

Boundary Conditions ◽

Dirichlet Boundary ◽

Harmonic Maps ◽

Dirichlet Boundary Conditions

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Boundary regularity ofp-harmonic maps with free and partially constrained boundary conditions

Journal of Geometric Analysis ◽

10.1007/bf02921706 ◽

1997 ◽

Vol 7 (1) ◽

pp. 83-107

Author(s):

C. E. Garza-Hume

Keyword(s):

Boundary Conditions ◽

Harmonic Maps ◽

Boundary Regularity

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Sphere-valued harmonic maps with surface energy and the K13 problem

Advances in Calculus of Variations ◽

10.1515/acv-2016-0033 ◽

2019 ◽

Vol 12 (4) ◽

pp. 363-392

Author(s):

Stuart Day ◽

Arghir Dani Zarnescu

Keyword(s):

Liquid Crystals ◽

Surface Energy ◽

Boundary Conditions ◽

Critical Point ◽

Partial Regularity ◽

Harmonic Maps ◽

Nematic Liquid Crystals ◽

Energy Functional ◽

Dirichlet Energy ◽

Surface Term

AbstractWe consider an energy functional motivated by the celebrated {K_{13}} problem in the Oseen–Frank theory of nematic liquid crystals. It is defined for sphere-valued functions and appears as the usual Dirichlet energy with an additional surface term. It is known that this energy is unbounded from below and our aim has been to study the local minimisers. We show that even having a critical point in a suitable energy space imposes severe restrictions on the boundary conditions. Having suitable boundary conditions makes the energy functional bounded and in this case we study the partial regularity of the global minimisers.

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Variationally harmonic maps with general boundary conditions: Boundary regularity

Calculus of Variations and Partial Differential Equations ◽

10.1007/s00526-005-0329-6 ◽

2006 ◽

Vol 25 (4) ◽

pp. 409-429 ◽

Cited By ~ 6

Author(s):

Christoph Scheven

Keyword(s):

Boundary Conditions ◽

Harmonic Maps ◽

Boundary Regularity ◽

General Boundary ◽

General Boundary Conditions

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Renormalized energies for unit-valued harmonic maps in multiply connected domains

Asymptotic Analysis ◽

10.3233/asy-211712 ◽

2021 ◽

pp. 1-32

Author(s):

Rémy Rodiac ◽

Paúl Ubillús

Keyword(s):

Boundary Conditions ◽

Harmonic Maps ◽

Neumann Boundary ◽

Dirichlet Boundary Conditions ◽

Connected Components ◽

Multiply Connected Domains ◽

Smooth Bounded Domain ◽

Ginzburg Landau ◽

Simply Connected ◽

Simply Connected Domains

In this article we derive the expression of renormalized energies for unit-valued harmonic maps defined on a smooth bounded domain in R 2 whose boundary has several connected components. The notion of renormalized energies was introduced by Bethuel–Brezis–Hélein in order to describe the position of limiting Ginzburg–Landau vortices in simply connected domains. We show here, how a non-trivial topology of the domain modifies the expression of the renormalized energies. We treat the case of Dirichlet boundary conditions and Neumann boundary conditions as well.

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Use of the Magnetostatic Analogue In Electromagnetic Lens Engineering

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100068795 ◽

1970 ◽

Vol 28 ◽

pp. 360-361

Author(s):

John W. Coleman

Keyword(s):

Boundary Conditions ◽

High Performance ◽

Force Fields ◽

Optical Design ◽

Direct Conversion ◽

Design Engineering ◽

Design Data ◽

Scalar Potentials ◽

Geometric Elements ◽

At Will

In the design engineering of high performance electromagnetic lenses, the direct conversion of electron optical design data into drawings for reliable hardware is oftentimes difficult, especially in terms of how to mount parts to each other, how to tolerance dimensions, and how to specify finishes. An answer to this is in the use of magnetostatic analytics, corresponding to boundary conditions for the optical design. With such models, the magnetostatic force on a test pole along the axis may be examined, and in this way one may obtain priority listings for holding dimensions, relieving stresses, etc..The development of magnetostatic models most easily proceeds from the derivation of scalar potentials of separate geometric elements. These potentials can then be conbined at will because of the superposition characteristic of conservative force fields.

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