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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 .

On some recent developments of the theory of sets of finite perimeter

Luigi Ambrosio (2003)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this paper we describe some recent progress on the theory of sets of finite perimeter, currents, and rectifiability in metric spaces. We discuss the relation between intrinsic and extrinsic theories for rectifiability

On the Curvature and Heat Flow on Hamiltonian Systems

Shin-ichi Ohta (2014)

Analysis and Geometry in Metric Spaces

We develop the differential geometric and geometric analytic studies of Hamiltonian systems. Key ingredients are the curvature operator, the weighted Laplacian, and the associated Riccati equation.We prove appropriate generalizations of the Bochner-Weitzenböck formula and Laplacian comparison theorem, and study the heat flow.

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