Journal ArticleDOI
Asymptotic development by Γ-convergence
G. Anzellotti,Sisto Baldo +1 more
TLDR
In this paper, a description of the asymptotic development of a family of minimum problems is proposed by a suitable iteration of Γ-limit procedures, and an example of the development of functionals related to phase transformations is also given.Abstract:
A description of the asymptotic development of a family of minimum problems is proposed by a suitable iteration of Γ-limit procedures. An example of asymptotic development for a family of functionals related to phase transformations is also given.read more
Citations
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Book ChapterDOI
Existence and stability results in the L 1 theory of optimal transportation
Luigi Ambrosio,Aldo Pratelli +1 more
TL;DR: In this article, a 1-dimensional theory and transport rays and transport set are presented. And a stability result is discussed. And the authors discuss the disintegration of measures and a counterexample of measures.
Journal ArticleDOI
A Dynamical Approach to Convex Minimization Coupling Approximation with the Steepest Descent Method
Hedy Attouch,Roberto Cominetti +1 more
TL;DR: In this paper, the authors studied the asymptotic behavior of the solutions to evolution equations of the form 0 # u* (t)+f(u(t), =(t)); u(0)=u0, where [f (}, =): =>0] is a family of strictly convex functions whose minimum is attained at a unique point x(=).
Journal ArticleDOI
Small volume fraction limit of the diblock copolymer problem : II. Diffuse-interface functional
Rustum Choksi,Mark A. Peletier +1 more
TL;DR: This work addresses the limit in which e and the volume fraction tend to zero but the number of minority phases (called particles) remains O(1), and derives first- and second-order effective energies, whose energy landscapes are simpler and more transparent.
Journal ArticleDOI
Asymptotic expansions by Γ-convergence
Andrea Braides,Lev Truskinovsky +1 more
TL;DR: In this article, the authors propose a method based on rectifying the singular points in the parameter space by using a blow-up argument and then asymptotically matching the approximations around such points with the regular approximation away from them.
References
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Book
Geometric Measure Theory
TL;DR: In this article, Grassmann algebras of a vectorspace have been studied in the context of the calculus of variations, and a glossary of some standard notations has been provided.
Book
Minimal surfaces and functions of bounded variation
TL;DR: In this article, a priori estimation of the gradient of the Bernstein problem is given. But the gradient is not a priorimate of the radius of the singular set, and it is not known whether the gradient can be estimated by direct methods.
Journal ArticleDOI
The gradient theory of phase transitions and the minimal interface criterion
TL;DR: In this paper, the authors prove some conjectures of GURTIN concerning the Van der Waals-Cahn-Hilliard theory of phase transitions, and prove the existence of a phase transition in a fluid under isothermal conditions and confined to a bounded container.
Journal ArticleDOI
The effect of a singular perturbation on nonconvex variational problems
TL;DR: In this paper, the effect of singular perturbation on certain nonconvex variational problems was studied. But the focus was on the limit of minimizers, rather than the energy of the variational problem.