Displaying 21 – 40 of 75

Showing per page

Generalized minimizers of convex integral functionals, Bregman distance, Pythagorean identities

Imre Csiszár, František Matúš (2012)

Kybernetika

Integral functionals based on convex normal integrands are minimized subject to finitely many moment constraints. The integrands are finite on the positive and infinite on the negative numbers, strictly convex but not necessarily differentiable. The minimization is viewed as a primal problem and studied together with a dual one in the framework of convex duality. The effective domain of the value function is described by a conic core, a modification of the earlier concept of convex core. Minimizers...

Global calibrations for the non-homogeneous Mumford-Shah functional

Massimiliano Morini (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Using a calibration method we prove that, if Γ Ω is a closed regular hypersurface and if the function g is discontinuous along Γ and regular outside, then the function u β which solves Δ u β = β ( u β - g ) in Ω Γ ν u β = 0 on Ω Γ is in turn discontinuous along Γ and it is the unique absolute minimizer of the non-homogeneous Mumford-Shah functional Ω S u | u | 2 d x + n - 1 ( S u ) + β Ω S u ( u - g ) 2 d x , over S B V ( Ω ) , for β large enough. Applications of the result to the study of the gradient flow by the method of minimizing movements are shown.

Hamilton’s Principle with Variable Order Fractional Derivatives

Atanackovic, Teodor, Pilipovic, Stevan (2011)

Fractional Calculus and Applied Analysis

MSC 2010: 26A33, 70H25, 46F12, 34K37 Dedicated to 80-th birthday of Prof. Rudolf GorenfloWe propose a generalization of Hamilton’s principle in which the minimization is performed with respect to the admissible functions and the order of the derivation. The Euler–Lagrange equations for such minimization are derived. They generalize the classical Euler-Lagrange equation. Also, a new variational problem is formulated in the case when the order of the derivative is defined through a constitutive equation....

New convexity conditions in the calculus of variations and compensated compactness theory

Krzysztof Chełmiński, Agnieszka Kałamajska (2006)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the lower semicontinuous functional of the form I f ( u ) = Ω f ( u ) d x where u satisfies a given conservation law defined by differential operator of degree one with constant coefficients. We show that under certain constraints the well known Murat and Tartar’s Λ -convexity condition for the integrand f extends to the new geometric conditions satisfied on four dimensional symplexes. Similar conditions on three dimensional symplexes were recently obtained by the second author. New conditions apply to quasiconvex...

New convexity conditions in the calculus of variations and compensated compactness theory

Krzysztof Chełmiński, Agnieszka Kałamajska (2005)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the lower semicontinuous functional of the form I f ( u ) = Ω f ( u ) d x where u satisfies a given conservation law defined by differential operator of degree one with constant coefficients. We show that under certain constraints the well known Murat and Tartar's Λ-convexity condition for the integrand f extends to the new geometric conditions satisfied on four dimensional symplexes. Similar conditions on three dimensional symplexes were recently obtained by the second author. New conditions apply...

Nonsmooth Problems of Calculus of Variations via Codifferentiation

Maxim Dolgopolik (2014)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper multidimensional nonsmooth, nonconvex problems of the calculus of variations with codifferentiable integrand are studied. Special classes of codifferentiable functions, that play an important role in the calculus of variations, are introduced and studied. The codifferentiability of the main functional of the calculus of variations is derived. Necessary conditions for the extremum of a codifferentiable function on a closed convex set and its applications to the nonsmooth problems of...

On a volume constrained variational problem in SBV 2 ( Ω ) : part I

Ana Cristina Barroso, José Matias (2002)

ESAIM: Control, Optimisation and Calculus of Variations

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 u S B V 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 E ( · ) is proved and the asymptotic behaviour of the solutions is investigated.

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

Ana Cristina Barroso, José Matias (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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 u ∈ SBV²(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 E(.) is proved and the asymptotic behaviour of the solutions is investigated.

Currently displaying 21 – 40 of 75