Publications de Bernard Dacorogna

The symmetrized covariant derivative

Given matrices Γ1,⋯,Γn and g (g being symmetric), we give necessary and sufficient conditions for the existence of a vector field ψ satisfying (Formula presented.) where ∇Γ denotes the covariant derivative. We then give several applications of our result. ...

2025

Bernhard Riemann 1861 revisited: existence of flat coordinates for an arbitrary bilinear form

Bernard Dacorogna, Marc Troyanov

We generalize the celebrated results of Bernhard Riemann and Gaston Darboux: we give necessary and sufficient conditions for a bilinear form to be flat. More precisely, we give explicit necessary and sufficient conditions for a tensor field of type (0, 2) ...

Springer Heidelberg2023

On the equation (Du)(t) H Du = G

Bernard Dacorogna, Marc Troyanov

In this article, we want to find a map u : (Omega) over bar -> R-n solving, in Omega, the equation ...

PERGAMON-ELSEVIER SCIENCE LTD2022

Integral Convexity And Parabolic Systems

Bernard Dacorogna

In this work we give optimal, i.e., necessary and sufficient, conditions for integrals of the calculus of variations to guarantee the existence of solutions-both weak and variational solutions-to the associated L-2-gradient flow. The initial values are mer ...

SIAM PUBLICATIONS2020

On the equation A del u plus (del u)(t) A = G

Bernard Dacorogna, David Valentin Strütt

Let Omega subset of R-n be an open set, A is an element of R-nxn and G : Omega -> R-nxn be given. We look for a solution u : Omega -> R-n of the equation ...

PERGAMON-ELSEVIER SCIENCE LTD2020

Quasiconvexity and Relaxation in Optimal Transportation of Closed Differential Forms

Bernard Dacorogna

This manuscript extends the relaxation theory from nonlinear elasticity to electromagnetism and to actions defined on paths of differential forms. The introduction of a gauge allows for a reformulation of the notion of quasiconvexity in Bandyopadhyay et al ...

SPRINGER2019

On the best constant in Gaffney inequality

Bernard Dacorogna, Gyula Csató, Swarnendu Sil

2018

Symplectic factorization, Darboux theorem and ellipticity

Bernard Dacorogna, Olivier Kneuss

2018

Transportation of closed differential forms with non-homogeneous convex costs

Bernard Dacorogna

2018

The various stages of Nicola Fusco

Bernard Dacorogna

Pergamon-Elsevier Science Ltd2017

Some Remarks on the Lie Derivative and the Pullback Equation for Contact Forms

Bernard Dacorogna, Olivier Kneuss

Given the contact forms f and g, and the 1-form h, we discuss the existence of a vector field u verifying L-u(f)= d(u (right perpendicular) f)+ u (right perpendicular) df = h. This is closely related to the pullback equation, where we seek for a diffeomorp ...

Walter De Gruyter Gmbh2017

The various stages of Nicola Fusco (Part 2)

Bernard Dacorogna

Elsevier2017

A Dirichlet problem involving the divergence operatord

Bernard Dacorogna, Gyula Csató

We consider the problem {div u + (a; u) = f in Omega u = u(0) on partial derivative Omega. We show that if curl a (x(0)) not equal for some x(0) epsilon Omega, then the problem is solvable without restriction on f. We also discuss the regularity of the sol ...

Elsevier Science Bv2016

Optimal transport of closed differential forms for convex costs

Bernard Dacorogna, Olivier Kneuss

Let c : A(k-1) -> R+ be convex and Omega subset of R-n be a bounded domain. Let f(0) and f(1) be two closed k-forms on Omega satisfying appropriate boundary conditions. We discuss the minimization of integral(Omega) c (A) dx over a subset of (k - 1)-forms ...

Elsevier France-Editions Scientifiques Medicales Elsevier2015

Some New Results On Differential Inclusions For Differential Forms

Bernard Dacorogna, Olivier Kneuss, Saugata Bandyopadhyay

In this article we study some necessary and sufficient conditions for the existence of solutions in W-0(1,infinity) (Omega; Lambda(k)) of the differential inclusion d omega is an element of E a.e. in Omega where E subset of Lambda(k+1) is a prescribed set. ...

Amer Mathematical Soc2015

Calculus of variations with differential forms

Bernard Dacorogna, Saugata Bandyopadhyay, Swarnendu Sil

We study integrals of the form integral(Omega) f (d omega), where 1

European Mathematical Soc2015

Multiple Jacobian Equations

Bernard Dacorogna, Olivier Kneuss

The existence, regularity and uniqueness of a local diffeomorphism phi satisfying g(i)(phi) det del phi = f(i) for 1

Amer Inst Mathematical Sciences2014

The second order pullback equation

Bernard Dacorogna, Olivier Kneuss, Gyula Csató

Let f, g be two closed k-forms over R-n. The pullback equation studies the existence of a diffeomorphism phi : R-n -> R-n such that phi*(g) = f. We prove two types of results. The first one sharpens some of the existing regularity results. The second one d ...

Springer2014

Existence Of Solutions For Jacobian And Hessian Equations Under Smallness Assumptions

Bernard Dacorogna, Sébastien Basterrechea

We discuss three types of problems. The first one involves Jacobian equations and the two others involve Hessian equations. We proceed by fixed point, obtaining the results under a smallness assumption. ...

Taylor & Francis Inc2014

The degenerate two well problem for piecewise affine maps

Bernard Dacorogna

The two well problem consists in finding maps u which satisfy some boundary conditions and whose gradient Du assumes values in the two wells . Here (similarly ) is the well generated by a 2 x 2 matrix A, i.e., is the set of matrices of the form RA, where R ...

Springer Basel Ag2013