Publications by Bernard Dacorogna

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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 ...

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 ...

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 ...

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 ...

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 ...

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. ...

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) ...

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 ...

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

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

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. ...

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

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. ...

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

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 ...

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 ...

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 ...

Given a k-form f and an l-form g with 0

Bernard Dacorogna

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Quasiconvexity and Relaxation in Optimal Transportation of Closed Differential Forms

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

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

Optimal transport of closed differential forms for convex costs

The degenerate two well problem for piecewise affine maps

Some New Results On Differential Inclusions For Differential Forms

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

Regularity And Selecting Principles For Implicit Ordinary Differential Equations

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

The Pullback Equation For Degenerate Forms

Existence Of Solutions For Jacobian And Hessian Equations Under Smallness Assumptions

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

The symmetrized covariant derivative

Calculus of variations with differential forms

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

The Two-Well Problem For Piecewise-Affine Maps

The second order pullback equation

Qualitative properties of Lipschitz solutions of eikonal type systems

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

Divisibility in Grassmann algebra