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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...
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,...
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...
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.
Nous étudions un théorème de Skorohod pour des mesures vectorielles à valeurs . En notant la mesure image de par la variable aléatoire nous donnons des classes de mesures et éventuel-lement de variables telles que, si la suite converge étroitement, il existe une suite qui converge en mesure, éventuel-lement p.s.Le problème de Monge est abordé comme application. Soit la mesure variation de , pour un couple et une fonction coût 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 . 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 ).
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