Graph convergence for the 
                
                  
                
                
                  
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              -mixed mappingwith an application for solving the system of generalized variationalinclusions

AbstractWe use the theory of selfdual Lagrangians to give a variational approach to the homogenization of equations in divergence form, that are driven by a periodic family of maximal monotone vector fields. The approach has the advantage of using Γ-convergence methods for corresponding functionals just as in the classical case of convex potentials, as opposed to the graph convergence methods used in the absence of potentials. A new variational formulation for the homogenized equation is also given.

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NOTES ON GRAPH-CONVERGENCE FOR MAXIMAL MONOTONE OPERATORS

Bulletin of the Australian Mathematical Society ◽

10.1017/s000497271000184x ◽

2010 ◽

Vol 83 (1) ◽

pp. 22-29 ◽

Cited By ~ 1

Author(s):

FILOMENA CIANCIARUSO ◽

GIUSEPPE MARINO ◽

LUIGI MUGLIA ◽

HONG-KUN XU

Keyword(s):

Monotone Operator ◽

Maximal Monotone Operator ◽

Maximal Monotone ◽

Maximal Monotone Operators ◽

Monotone Operators ◽

Null Sequence ◽

Graph Convergence ◽

Common Domain

AbstractWe construct a sequence {An} of maximal monotone operators with a common domain and converging, uniformly on bounded subsets, to another maximal monotone operator A; however, the sequence {t−1nAn} fails to graph-converge for some null sequence {tn}.

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General Class of Implicit Variational Inclusions and Graph Convergence on A-Maximal Relaxed Monotonicity

Journal of Optimization Theory and Applications ◽

10.1007/s10957-012-0030-9 ◽

2012 ◽

Vol 155 (1) ◽

pp. 196-214 ◽

Cited By ~ 2

Author(s):

Ram U. Verma

Keyword(s):

General Class ◽

Variational Inclusions ◽

Graph Convergence

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A finer singular limit of a single-well Modica–Mortola functional and its applications to the Kobayashi–Warren–Carter energy

Advances in Calculus of Variations ◽

10.1515/acv-2020-0113 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Yoshikazu Giga ◽

Jun Okamoto ◽

Masaaki Uesaka

Keyword(s):

Materials Science ◽

Dimensional Space ◽

Singular Limit ◽

Explicit Representation ◽

Arc Length ◽

Convergence In Measure ◽

One Dimensional ◽

Graph Convergence ◽

Single Well

Abstract An explicit representation of the Gamma limit of a single-well Modica–Mortola functional is given for one-dimensional space under the graph convergence which is finer than conventional L 1 L^{1} -convergence or convergence in measure. As an application, an explicit representation of a singular limit of the Kobayashi–Warren–Carter energy, which is popular in materials science, is given. Some compactness under the graph convergence is also established. Such formulas as well as compactness are useful to characterize the limit of minimizers of the Kobayashi–Warren–Carter energy. To characterize the Gamma limit under the graph convergence, a new idea which is especially useful for one-dimensional problems is introduced. It is a change of parameter of the variable by arc-length parameter of its graph, which is called unfolding by the arc-length parameter in this paper.

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Pointwise topological convergence and topological graph convergence of set-valued maps

Filomat ◽

10.2298/fil1709779h ◽

2017 ◽

Vol 31 (9) ◽

pp. 2779-2785

Author(s):

L. Holá ◽

G. Kwiecińka

Keyword(s):

Topological Spaces ◽

Topological Graph ◽

Graph Convergence ◽

Topological Convergence ◽

Topological Limit ◽

Graph Limit

Let X,Y be topological spaces and {Fn : n ? ?} be a sequence of set-valued maps from X to Y with the pointwise topological limit G and with the topological graph limit F. We give an answer to the question from ([19]): which conditions on X,Y and/or {F,G,Fn : n ? ?} are needed to F = G.

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