Bernard Dacorogna

On a partial differential equation involving the Jacobian determinant

Abstract Let Ω ⊂ ℝn a bounded open set and f > 0 in Ω ¯ satisfying ∫ Ω f ( x ) d x = meas Ω . We study existence and regularity of diffeomorphisms u : Ω ¯ → Ω ¯ such that { det ∇ u ( x ) = f ( x ) , x ∈ Ω u ( x ) = x , x ∈ ∂ Ω .

General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases

Keywords: Dirichlet problem ; Hamilton-Jacobi equation ; existence ; of solutions Reference CAA-ARTICLE-1997-002doi:10.1007/BF02392708View record in Web of Science Record created on 2008-11-25, modified on 2017-05-12

Introduction to the calculus of variations

Introduction Preliminaries Classical Methods Direct Methods: Existence Direct Methods: Regularity Minimal Surfaces Isoperimetric Inequality Solutions to the Exercises Bibliography Index

An example of a quasiconvex function that is not polyconvex in two dimensions

We study the different notions of convexity for the function fγ(ξ) = |ξ|2 (|ξ|2 − 2γ det ξ) where ξ ε ℝ2×2, introduced by Dacorogna & Marcellini. We show that fγ is convex, polyconvex, quasiconvex, rank-one convex, if and only if ¦γ¦≦ 2/3 √2, 1, 1+ɛ (for some ɛ>0), 2/√3, respectively.

Existence of minimizers for non-quasiconvex integrals

Keywords: quasiconvexity ; existence of minimizers Reference CAA-ARTICLE-1995-002doi:10.1007/BF00380915 Record created on 2008-11-25, modified on 2017-05-12

A relaxation theorem and its application to the equilibrium of gases

Keywords: relaxed problem ; lower convex envelope ; gas ; equilibrium ; Van der Waals equation Reference CAA-ARTICLE-1981-001doi:10.1007/BF00280643View record in Web of Science Record created on 2008-11-25, modified on 2017-05-12

The Pullback Equation for Differential Forms

Introduction.- Part I Exterior and Differential Forms.- Exterior Forms and the Notion of Divisibility.- Differential Forms.- Dimension Reduction.- Part II Hodge-Morrey Decomposition and Poincare Lemma.- An Identity Involving Exterior Derivatives and Gaffney Inequality.- The Hodge-Morrey Decomposition.- First-Order Elliptic Systems of Cauchy-Riemann Type.- Poincare Lemma.- The Equation div u = f.- Part III The Case k = n.- The Case f x g > 0.- The Case Without Sign Hypothesis on f.- Part IV The Case 0 <= k <= n-1.- General Considerations on the Flow Method.- The Cases k = 0 and k = 1.- The Case k = 2.- The Case 3 <= k <= n-1.- Part V Holder Spaces.- Holder Continuous Functions.- Part VI Appendix.- Necessary Conditions.- An Abstract Fixed Point Theorem.- Degree Theory.- References.- Further Reading.- Notations.- Index.

Calculus of Variations and Nonlinear Partial Differential Equations

Note: With a historical overview by Elvira Mascolo. Reference CAA-BOOK-2008-007 Record created on 2008-11-25, modified on 2017-05-12

Sur une généralisation de l’inégalité de Wirtinger

Resume Soient α I = α I ( p , q ) = min { ‖ u ′ ‖ L p ‖ u ‖ L q | u ∈ W 1 , p ( − 1 , 1 ) \ { 0 } , u ( − 1 ) = u ( 1 ) , ∫ − 1 1 u | u | q − 2 = 0 } α II = α II ( p , q ) = min { ‖ u ′ ‖ L p ‖ u ‖ L q | u ∈ W 1 , p ( − 1 , 1 ) \ { 0 } , u ( − 1 ) = u ( 1 ) , ∫ − 1 1 u = 0 } . On calcule explicitement αI, et on montre que pour q ≦ 2p, αI = αII, mais pour q suffisamment grand αII

Some examples of rank one convex functions in dimension two

Keywords: Sobolev space ; quasiconvexity ; rank one convexity Reference CAA-ARTICLE-1990-004 Record created on 2008-11-25, modified on 2017-05-12