Displaying similar documents to “An example in the gradient theory of phase transitions”

Expansion for the superheating field in a semi-infinite film in the weak- limit

Pierre Del Castillo (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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Dorsey, Di Bartolo and Dolgert (Di Bartolo , 1996; 1997) have constructed asymptotic matched solutions at order two for the half-space Ginzburg-Landau model, in the weak- limit. These authors deduced a formal expansion for the superheating field in powers of κ 1 2 up to order four, extending the formula by De Gennes (De Gennes, 1966) and the two terms in Parr's formula (Parr, 1976). In this paper, we construct asymptotic matched solutions at all orders leading to a complete expansion...

On a Volume Constrained Variational Problem in SBV²(Ω): Part I

Ana Cristina Barroso, José Matias (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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We consider the problem of minimizing the energy E ( u ) : = Ω | u ( x ) | 2 d x + S u Ω 1 + | [ u ] ( x ) | d H N - 1 ( x ) among all functions ∈ ²(Ω) for which two level sets { u = l i } have prescribed Lebesgue measure α i . Subject to this volume constraint the existence of minimizers for (.) is proved and the asymptotic behaviour of the solutions is investigated.

Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces

Stefano Lisini (2008)

ESAIM: Control, Optimisation and Calculus of Variations

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We study existence and approximation of non-negative solutions of partial differential equations of the type 
 t u - div ( A ( ( f ( u ) ) + u V ) ) = 0 in ( 0 , + ) × n , ( 0 . 1 ) where is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition, f : [ 0 , + ) [ 0 , + ) is a suitable non decreasing function, V : n is a convex function. Introducing the energy functional φ ( u ) = n F ( u ( x ) ) d x + n V ( x ) u ( x ) d x , where is a convex function linked to by f ( u ) = u F ' ( u ) - F ( u ) , we show that is the “gradient flow” of with respect to the 2-Wasserstein distance between probability measures on the...

The steepest descent dynamical system with control. Applications to constrained minimization

Alexandre Cabot (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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Let be a real Hilbert space, Φ 1 : H a convex function of class 𝒞 1 that we wish to minimize under the convex constraint . A classical approach consists in following the trajectories of the generalized steepest descent system (  Brézis [CITE]) applied to the non-smooth function  Φ 1 + δ S . Following Antipin [1], it is also possible to use a continuous gradient-projection system. We propose here an alternative method as follows: given a smooth convex function  Φ 0 : H whose critical points coincide with  and...

Controlled functional differential equations: approximate and exact asymptotic tracking with prescribed transient performance

Eugene P. Ryan, Chris J. Sangwin, Philip Townsend (2008)

ESAIM: Control, Optimisation and Calculus of Variations

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A tracking problem is considered in the context of a class 𝒮 of multi-input, multi-output, nonlinear systems modelled by controlled functional differential equations. The class contains, as a prototype, all finite-dimensional, linear, -input, -output, minimum-phase systems with sign-definite “high-frequency gain". The first control objective is tracking of reference signals by the output of any system in 𝒮 : given λ 0 , construct a feedback strategy which ensures that, for every (assumed...