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Bernard Dacorogna

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Related publications (39)

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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

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

The role of perspective functions in convexity, polyconvexity, rank-one convexity and separate convexity

Bernard Dacorogna

Any finite, separately convex, positively homogeneous function on

\mathbb{R}^2

is convex. This was first established by the first author ["Direct methods in calculus of variations", Springer-Verlag (1989)]. Here we give a new and concise proof of this re ...

2008

A Global Version Of The Darboux Theorem With Optimal Regularity And Dirichlet Condition

Bernard Dacorogna, Olivier Kneuss

Let n > 2 be even; r >= 1 be an integer; 0 < alpha < 1; Omega be a bounded, connected, smooth, open set in R-n; and nu be its exterior unit normal. Let f, g is an element of C-r,C-alpha((Omega) over bar; Lambda(2)) be two symplectic forms (i.e., closed and ...

2011

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

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

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

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

Regularity And Selecting Principles For Implicit Ordinary Differential Equations

Bernard Dacorogna

Implicit Ordinary or Partial Differential Equations have been widely studied in recent times, essentially from the existence of solutions point of view. One of the main issues is to select a meaningful solution among the infinitely many ones. The most cele ...

2009

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

The Pullback Equation For Degenerate Forms

Bernard Dacorogna, Olivier Kneuss, Saugata Bandyopadhyay

We discuss the existence of a diffeomorphism phi : R-n -> R-n such that ...

2010

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

On the equation det del u = f with no sign hypothesis

Bernard Dacorogna, Olivier Kneuss

We prove existence of u is an element of C-k ((Omega) over bar ;R-n) satisfying ...

Springer-Verlag2009

The symmetrized covariant derivative

Bernard Dacorogna

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

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

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

The Two-Well Problem For Piecewise-Affine Maps

Bernard Dacorogna

In the two-well problem we look for a map u which satisfies Dirichlet boundary conditions and whose gradient Du assumes values in SO (2) A boolean OR SO (2) B = S-A boolean OR S-B, for two given invertible matrices A, B (an element of SO (2) A is of the fo ...

2012

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

Qualitative properties of Lipschitz solutions of eikonal type systems

Bernard Dacorogna

2006

A general existence theorem for differential inclusions in the vector valued case

Bernard Dacorogna

2005