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The commutative neutrix convolution product of the functions $x^{r} e_{-}^{λ x}$ and $x^{s} e_{+}^{μ x}$ is evaluated for $r, s = 0, 1, 2, ...$ and all $λ, μ$ . Further commutative neutrix convolution products are then deduced.

Compactness of Special Functions of Bounded Higher Variation

Luigi Ambrosio, Francesco Ghiraldin (2013)

Analysis and Geometry in Metric Spaces

Given an open set Ω ⊂ Rm and n > 1, we introduce the new spaces GBnV(Ω) of Generalized functions of bounded higher variation and GSBnV(Ω) of Generalized special functions of bounded higher variation that generalize, respectively, the space BnV introduced by Jerrard and Soner in [43] and the corresponding SBnV space studied by De Lellis in [24]. In this class of spaces, which allow as in [43] the description of singularities of codimension n, the distributional jacobian Ju need not have ﬁnite...

Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents

Fanghua Lin, Tristan Rivière (1999)

Journal of the European Mathematical Society

There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a $S^{1}$ -valued function defined on the boundary of a bounded regular domain of $R^{n}$ . When such extensions do not exist, we use the Ginzburg-Landau relaxation procedure. We prove that, up to a subsequence, a sequence of Ginzburg-Landau minimizers, as the coupling parameter tends to infinity, converges to a unimodular harmonic map away from a codimension-2 minimal current minimizing the area within the homology...

Conformally invariant variational integrals and the removability of isolated singularities.

Michael Grüter (1984)

Manuscripta mathematica

Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane

Yuri L. Sachkov (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is studied. Local and global optimality of extremal trajectories is characterized. Lower and upper bounds on the first conjugate time are proved. The cut time is shown to be equal to the first Maxwell time corresponding to the group of discrete symmetries of the exponential mapping. Optimal synthesis on an open dense subset of the state space is described.

Connectivity properties of the range of a weak diffeomorphism

Mariano Giaquinta, Giuseppe Modica, Jiří Souček (1995)

Annales de l'I.H.P. Analyse non linéaire

Continuity, curvature, and the general covariance of optimal transportation

Young-Heon Kim, Robert McCann (2010)

Journal of the European Mathematical Society

Continuous limits of discrete perimeters

Antonin Chambolle, Alessandro Giacomini, Luca Lussardi (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a class of discrete convex functionals which satisfy a (generalized) coarea formula. These functionals, based on submodular interactions, arise in discrete optimization and are known as a large class of problems which can be solved in polynomial time. In particular, some of them can be solved very efficiently by maximal flow algorithms and are quite popular in the image processing community. We study the limit in the continuum of these functionals, show that they always converge...

Convergence of a variational lagrangian scheme for a nonlinear drift diffusion equation

Daniel Matthes, Horst Osberger (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We study a Lagrangian numerical scheme for solution of a nonlinear drift diffusion equation on an interval. The discretization is based on the equation’s gradient flow structure with respect to the Wasserstein distance. The scheme inherits various properties from the continuous flow, like entropy monotonicity, mass preservation, metric contraction and minimum/ maximum principles. As the main result, we give a proof of convergence in the limit of vanishing mesh size under a CFL-type condition. We...

Convergence of minimizers for the p-Dirichlet integral.

Stephan Luckhaus (1993)

Mathematische Zeitschrift

Convessità della goccia appoggiata

Eduardo Gonzalez, Italo Tamanini (1977)

Rendiconti del Seminario Matematico della Università di Padova

Corrections to the Paper: Variational Problems with Non-Convex Obstacles and an Integral-Constraint for Vector-Valued Functions.

Martin Fuchs, Frank Duzaar (1986)

Mathematische Zeitschrift

Critical points of Ambrosio-Tortorelli converge to critical points of Mumford-Shah in the one-dimensional Dirichlet case

Gilles A. Francfort, Nam Q. Le, Sylvia Serfaty (2009)

ESAIM: Control, Optimisation and Calculus of Variations

Critical points of a variant of the Ambrosio-Tortorelli functional, for which non-zero Dirichlet boundary conditions replace the fidelity term, are investigated. They are shown to converge to particular critical points of the corresponding variant of the Mumford-Shah functional; those exhibit many symmetries. That Dirichlet variant is the natural functional when addressing a problem of brittle fracture in an elastic material.

Critical points of Ambrosio-Tortorelli converge to critical points of Mumford-Shah in the one-dimensional Dirichlet case

Gilles A. Francfort, Nam Q. Le, Sylvia Serfaty (2008)

ESAIM: Control, Optimisation and Calculus of Variations

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