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.

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.

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,...

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

We provide a mathematical proof of the existence of traveling vortex rings solutions to the Gross–Pitaevskii (GP) equation in dimension $N\ge 3$. 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 $N=3$).