Γ-convergence for beginners

By Andrea Braides. 2002

1,174 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/785070384102a40ab279645c2b18a89eb3ce52c9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

Unidimensional and Evolution Methods for Optimal Transportation (2013, unknown): “For more information about Γ-convergence, we refer to Andrea Braides’s book on the subject [12].”
Phase Field: A Variational Method for Structural Topology Optimization (2004, methodology): “This technique belongs to the general theory of Γconvergence, and it has been extensively studied as the subject of gradient theory of phase transitions (Braides (2002)).”
$\Gamma$-convergence of graph Ginzburg-Landau functionals (2012, methodology): “A good introduction to the subject is [15], the standard reference work is [28].”
An adaptive finite element approximation of a generalized Ambrosio-Tortorelli functional (2013, unknown): “…(uh, vh)|2 := ∑ i:xi∈T h2i ‖q − q̄i‖2L2(ωi) + 4ε 2 ∑ e⊆∂T he‖ ∇vh e‖2L2(e), |ν2T (uh, vh)|2 := ∑ i:xi∈T [ h N/2 T ‖q‖L2(T ) + ε ∑ e⊆∂T\∂Ch(vh): xi∈e hN−1e ∇vh e ] osch(χ;ωi), q(uh, vh) := F ′(vh)A∇uh : ∇uh + ε−1G′(vh), osch(χ;ωi) := 1 |(1, ζi)| ∣∣∣∣ ∑ j:xj∈ωi ( χ(xi)− χ(xj) ) ∫ ωi ζjζi dx ∣∣∣∣.”
Γ-limits and relaxations for rate-independent evolutionary problems (2008, background): “Hence, we may pass to the limit for k →∞, since it is well-known that E∞ is the -limit of Ek , see [6,10].”
Atomistic to continuum limits for computational materials science (2007, methodology): “A means to explore this relation is to use Γ-convergence [35, 58] rather than the “pointwise” convergence we have used in [30].”
Phase field approximation of cohesive fracture models (2014, background): “These definitions and main results concerning the -convergence technique can be found in [25,18].”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, background): “Here is a useful approximation lemma in the spirit of -convergence developments (see [77], Section 2.”
Asymptotic expansions by Γ-convergence (2008, methodology): “8 In the case of a general W the ‘effective’ potential W is given by the more complex formula W (z) = 1 2 inf{W (z1)+ W (z2) : z1 + z2 = 2z} − 2J z2 (see [14,21,56]) highlighting oscillations at microscopic scale.”
Γ-convergence for nonlocal phase transitions (2012, unknown): “…as possible with the minimizing features of the energy, and whose limit is capable to capture essential features of the problem We refer to [11,7] for a detailed presentation of several basic aspects and applications of Γ -convergence; see also [24] for applications to homogenization theory.”
Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof (2015, methodology): “, [19], and for methods similar to those employed in the present paper [20]).”