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On convex sets that minimize the average distance

Antoine Lemenant, Edoardo Mainini (2012)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we study the compact and convex sets K ⊆ Ω ⊆ ℝ2that minimize ∫ Ω ( , K ) d + λ 1 Vol ( K ) + λ 2 Per ( K ) ∫ Ω dist ( x ,K ) d x + λ 1 Vol ( K ) + λ 2 Per ( K ) for some constantsλ1 and λ2, that could possibly be zero. We compute in particular the second order derivative of the functional and use it to exclude smooth points of positive curvature for the problem with volume constraint. The problem with perimeter constraint behaves differently since polygons are never minimizers. Finally using a purely geometrical argument from...

On fine properties of mixtures with respect to concentration of measure and Sobolev type inequalities

Djalil Chafaï, Florent Malrieu (2010)

Annales de l'I.H.P. Probabilités et statistiques

Mixtures are convex combinations of laws. Despite this simple definition, a mixture can be far more subtle than its mixed components. For instance, mixing gaussian laws may produce a potential with multiple deep wells. We study in the present work fine properties of mixtures with respect to concentration of measure and Sobolev type functional inequalities. We provide sharp Laplace bounds for Lipschitz functions in the case of generic mixtures, involving a transportation cost diameter of the mixed...

On lower semicontinuity of multiple integrals

Agnieszka Kałamajska (1997)

Colloquium Mathematicae

We give a new short proof of the Morrey-Acerbi-Fusco-Marcellini Theorem on lower semicontinuity of the variational functional ∫ Ω F ( x , u , ∇ u ) d x . The proofs are based on arguments from the theory of Young measures.

On optimal matching measures for matching problems related to the Euclidean distance

José Manuel Mazón, Julio Daniel Rossi, Julián Toledo (2014)

Mathematica Bohemica

We deal with an optimal matching problem, that is, we want to transport two measures to a given place (the target set) where they will match, minimizing the total transport cost that in our case is given by the sum of two different multiples of the Euclidean distance that each measure is transported. We show that such a problem has a solution with an optimal matching measure supported in the target set. This result can be proved by an approximation procedure using a p -Laplacian system. We prove...

On Perelman’s functional with curvature corrections

Rami Ahmad El-Nabulsi (2012)

Annales UMCS, Mathematica

In recent ten years, there has been much concentration and increased research activities on Hamilton’s Ricci flow evolving on a Riemannian metric and Perelman’s functional. In this paper, we extend Perelman’s functional approach to include logarithmic curvature corrections induced by quantum effects. Many interesting consequences are revealed.

On shape optimization problems involving the fractional laplacian

Anne-Laure Dalibard, David Gérard-Varet (2013)

ESAIM: Control, Optimisation and Calculus of Variations

Our concern is the computation of optimal shapes in problems involving (−Δ)1/2. We focus on the energy J(Ω) associated to the solution uΩ of the basic Dirichlet problem ( − Δ)1/2uΩ = 1 in Ω, u = 0 in Ωc. We show that regular minimizers Ω of this energy under a volume constraint are disks. Our proof goes through the explicit computation of the shape derivative (that seems to be completely new in the fractional context), and a refined adaptation of the moving plane method.

On some optimal control problems for the heat radiative transfer equation

Sandro Manservisi, Knut Heusermann (2010)

ESAIM: Control, Optimisation and Calculus of Variations

This paper is concerned with some optimal control problems for the Stefan-Boltzmann radiative transfer equation. The objective of the optimisation is to obtain a desired temperature profile on part of the domain by controlling the source or the shape of the domain. We present two problems with the same objective functional: an optimal control problem for the intensity and the position of the heat sources and an optimal shape design problem where the top surface is sought as control. The problems...

On some properties of three-dimensional minimal sets in ℝ 4

Tien Duc Luu (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

We prove in this paper the Hölder regularity of Almgren minimal sets of dimension 3 in ℝ 4 around a 𝕐 -point and the existence of a point of particular type of a Mumford-Shah minimal set in ℝ 4 , which is very close to a 𝕋 . This will give a local description of minimal sets of dimension 3 in ℝ 4 around a singular point and a property of Mumford-Shah minimal sets in ℝ 4 .

Currently displaying 21 – 40 of 119