Luc Tartar

H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations

New mathematical objects, called H-measures, are introduced for studying oscillations and concentration effects in partial differential equations. Applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach.

The gradient theory of phase transitions for systems with two potential wells

On generalise la theorie du gradient des transitions de phase au cas a valeur vectoriel. On considere la famille des perturbations E e (u)=∫ Ω W(u)dx+e 2 ∫ Ω |⊇u| 2 dx de la fonctionnelle non convexe E 0 (u):=∫ Ω W(u)dx, ou W:R N →R supporte deux phases et N≥1

Some Remarks on Separately Convex Functions

I want to discuss here some results concerning separately convex functions. Most of these results were obtained some time ago but only mentioned to a few specialists, and I had not taken the time to publish them before, for obvious reasons. The motivation of these studies was nonlinear elasticity, but once I had solved an academic example where quasiconvexity had been replaced by separate convexity, it was not clear to me how to get further on. I find useful to choose this subject now in order to describe the evolution of some ideas during the last fifteen years.

Memory Effects and Homogenization

In the Fall of 1987 I was putting in order some of my results on memory effects induced by homogenization which I used for a contribution to a book in honor of E. De Giorgi on the occasion of his sixtieth birthday, for I had discussed some of these questions with him a long time earlier, during one of my visits to Pisa [7].

Calculus of Variations and Homogenization

In this paper we study a class of problems which includes the following example. Let Ω be an open subset of IRN and let ω be a measurable subset of Ω with given measure γ’. Let a(x)= α on ω and β on Ω\ω, and define u by −div (agradu) = 1 in Ω and u = 0 on the boundary. We want to find an ω which maximizes \( \int {_{\Omega }} u(x) \) u (x) dx among all the measurable sets ω with given measure γ’.

Beyond Young Measures

Young measures and their limitations are discussed. Some relations between Young measures and H-measures are described and used to analyze an example from micromagnetics. The need to improve H-measures and semi-classical measures is stressed.SommarioSi discutono le misure di Young e le loro limitazioni. Alcune relazioni che intercorrono tra le misure di Young e le H-misure sono descritte ed utilizzate nello studio di un esempio tratto dalla meccanica di mezzi micromagnetici. Si evidenzia in particolare la necessità di migliorare la teoria delle H-misure e di altre misure semi-classiche.

Nonlocal Effects Induced by Homogenization

It is indeed surprising that a weak limit of inverses of second order elliptic operators is also the inverse of a second order elliptic operator. Of course in order to obtain this result one has to consider a class larger than the class of operators of the form

On Mathematical Tools for Studying Partial Differential Equations of Continuum Physics: H-Measures and Young Measures

- \sum\limits_{i = 1}^N {{\partial \over {\partial {x_i}}}\left[ {a(x){\partial \over {\partial {x_i}}}} \right]}