Annual Report of Antoine LEMENANT

During the year 2009/2010 I was principally working on 3 different problems in collaboration with 4 different people, 3 of them working in Pisa and one in Parma. Those works led to 3 scientific papers and 1 review paper submitted during the year 2009/2010. The first work is a collaboration with Pablo Àlvarez-Caudevilla, postdoc at the center De Giorgi. We studied a problem in PDEs, about the asymptotic behavior of some linear operators, answering a question of Norman Dancer and where the Γ-convergence theory played a main role. The second work is a collaboration with Luigi Ambrosio, Professor at the Scuola Normale, and Gianni Royer-Carfagni, Professor of Structural Mechanics at the University of Parma. We studied a model in elasticity with both mathematical and physical aspects. Finally, the last work is about the average distance problem. This problem was suggested to me last year by Pr. Ambrosio yielding a first paper containing a partial regularity result (early 2009). This year I wrote a review paper on this very nice subject and I continued some investigations together with Edoardo Mainini, Phd student at the Scuola Normale. This was my last year as a postdoc at the center De Giorgi. I would like to say that the center offered me a perfect environment of work during the last 2 years, either from material and human point of view. The ambiance was really encouraging and my experience in Pisa influenced my research carrier in a very deep way. I hope I could always keep a link with Pisa in the future.