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Variational approach to shape derivatives

Kazufumi Ito, Karl Kunisch, Gunther H. Peichl (2008)

ESAIM: Control, Optimisation and Calculus of Variations

A general framework for calculating shape derivatives for optimization problems with partial differential equations as constraints is presented. The proposed technique allows to obtain the shape derivative of the cost without the necessity to involve the shape derivative of the state variable. In fact, the state variable is only required to be Lipschitz continuous with respect to the geometry perturbations. Applications to inverse interface problems, and shape optimization for elliptic systems...

Variational approximation for detecting point-like target problems

Gilles Aubert, Daniele Graziani (2011)

ESAIM: Control, Optimisation and Calculus of Variations

The aim of this paper is to provide a rigorous variational formulation for the detection of points in 2-d biological images. To this purpose we introduce a new functional whose minimizers give the points we want to detect. Then we define an approximating sequence of functionals for which we prove the Γ-convergence to the initial one.

Variational approximation for detecting point-like target problems*

Gilles Aubert, Daniele Graziani (2011)

ESAIM: Control, Optimisation and Calculus of Variations

The aim of this paper is to provide a rigorous variational formulation for the detection of points in 2-d biological images. To this purpose we introduce a new functional whose minimizers give the points we want to detect. Then we define an approximating sequence of functionals for which we prove the Γ-convergence to the initial one.

Variational approximation of a functional of Mumford–Shah type in codimension higher than one

Francesco Ghiraldin (2014)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we consider a new kind of Mumford–Shah functional E(u, Ω) for maps u : ℝm → ℝn with m ≥ n. The most important novelty is that the energy features a singular set Su of codimension greater than one, defined through the theory of distributional jacobians. After recalling the basic definitions and some well established results, we prove an approximation property for the energy E(u, Ω) via Γ −convergence, in the same spirit of the work by Ambrosio and Tortorelli [L. Ambrosio and V.M. Tortorelli,...

Variational problems and PDEs in affine differential geometry

H. Z. Li (2005)

Banach Center Publications

This paper is part of the autumn school on "Variational problems and higher order PDEs for affine hypersurfaces". We discuss variational problems in equiaffine differential geometry, centroaffine differential geometry and relative differential geometry, which have been studied by Blaschke [Bla], Chern [Ch], C. P. Wang [W], Li-Li-Simon [LLS], and Calabi [Ca-II]. We first derive the Euler-Lagrange equations in these settings; these equations are complicated, strongly nonlinear fourth order PDEs. We...

Variations of additive functions

Zoltán Buczolich, Washek Frank Pfeffer (1997)

Czechoslovak Mathematical Journal

We study the relationship between derivates and variational measures of additive functions defined on families of figures or bounded sets of finite perimeter. Our results, valid in all dimensions, include a generalization of Ward’s theorem, a necessary and sufficient condition for derivability, and full descriptive definitions of certain conditionally convergent integrals.

Vers un théorème de Skorohod simultané

Henri Heinich (2008)

Annales de la faculté des sciences de Toulouse Mathématiques

Nous étudions un théorème de Skorohod pour des mesures vectorielles à valeurs ℝ d . En notant X ( ℙ ) la mesure image de ℙ par la variable aléatoire X , nous donnons des classes de mesures ℙ et éventuel-lement de variables telles que, si la suite { X n ( ℙ ) } converge étroitement, il existe une suite { φ n } , φ n ( ℙ ) = X n ( ℙ ) qui converge en mesure, éventuel-lement p.s.Le problème de Monge est abordé comme application. Soit | ℙ | la mesure variation de ℙ , pour un couple ( ℙ , ℚ ) et une fonction coût c , le problème de Monge est l’existence d’une fonction...

Vortex rings for the Gross-Pitaevskii equation

Fabrice Bethuel, G. Orlandi, Didier Smets (2004)

Journal of the European Mathematical Society

We provide a mathematical proof of the existence of traveling vortex rings solutions to the Gross–Pitaevskii (GP) equation in dimension N ≥ 3 . We also extend the asymptotic analysis of the free field Ginzburg–Landau equation to a larger class of equations, including the Ginzburg–Landau equation for superconductivity as well as the traveling wave equation for GP. In particular we rigorously derive a curvature equation for the concentration set (i.e. line vortices if N = 3 ).

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